FAQ

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General

How can I subscribe to the PETSc mailing lists?

See mailing list documentation

Any useful books on numerical computing?

Bueler, PETSc for Partial Differential Equations: Numerical Solutions in C and Python

Oliveira and Stewart, Writing Scientific Software: A Guide to Good Style

What kind of parallel computers or clusters are needed to use PETSc? Or why do I get little speedup?

Important

PETSc can be used with any kind of parallel system that supports MPI BUT for any decent performance one needs:

• Fast, low-latency interconnect; any ethernet (even 10 GigE) simply cannot provide the needed performance.

• High per-core memory performance. Each core needs to have its own memory bandwidth of at least 2 or more gigabytes/second. Most modern computers are not bottlenecked by how fast they can perform calculations; rather, they are usually restricted by how quickly they can get their data.

To obtain good performance it is important that you know your machine! I.e. how many compute nodes (generally, how many motherboards), how many memory sockets per node and how many cores per memory socket and how much memory bandwidth for each.

If you do not know this and can run MPI programs with mpiexec (that is, you don’t have batch system), run the following from $PETSC_DIR:

$ make streams [NPMAX=maximum_number_of_mpi_processes_you_plan_to_use]

This will provide a summary of the bandwidth with different number of MPI processes and potential speedups. If you have a batch system:

$ cd $PETSC_DIR/src/benchmarks/streams
$ make MPIVersion
submit MPIVersion to the batch system a number of times with 1, 2, 3, etc MPI processes
collecting all of the output from the runs into the single file scaling.log. Copy
scaling.log into the src/benchmarks/streams directory.
$ ./process.py createfile ; process.py

Even if you have enough memory bandwidth if the OS switches processes between cores performance can degrade. Smart process to core/socket binding (this just means locking a process to a particular core or memory socket) may help you. For example, consider using fewer processes than cores and binding processes to separate sockets so that each process uses a different memory bus:

• MPICH2 binding with the Hydra process manager

  $ mpiexec.hydra -n 4 --binding cpu:sockets
  

• Open MPI binding

  $ mpiexec -n 4 --map-by socket --bind-to socket --report-bindings
  

• taskset, part of the util-linux package

  Check man taskset for details. Make sure to set affinity for your program, not for the mpiexec program.

• numactl

  In addition to task affinity, this tool also allows changing the default memory affinity policy. On Linux, the default policy is to attempt to find memory on the same memory bus that serves the core that a thread is running on when the memory is faulted (not when malloc() is called). If local memory is not available, it is found elsewhere, possibly leading to serious memory imbalances.

  The option --localalloc allocates memory on the local NUMA node, similar to the numa_alloc_local() function in the libnuma library. The option --cpunodebind=nodes binds the process to a given NUMA node (note that this can be larger or smaller than a CPU (socket); a NUMA node usually has multiple cores).

  The option --physcpubind=cpus binds the process to a given processor core (numbered according to /proc/cpuinfo, therefore including logical cores if Hyper-threading is enabled).

  With Open MPI, you can use knowledge of the NUMA hierarchy and core numbering on your machine to calculate the correct NUMA node or processor number given the environment variable OMPI_COMM_WORLD_LOCAL_RANK. In most cases, it is easier to make mpiexec or a resource manager set affinities.

The software Open-MX provides faster speed for ethernet systems, we have not tried it but it claims it can dramatically reduce latency and increase bandwidth on Linux system. You must first install this software and then install MPICH or Open MPI to use it.

What kind of license is PETSc released under?

See licensing documentation

Why is PETSc written in C, instead of Fortran or C++?

When this decision was made, in the early 1990s, C enabled us to build data structures for storing sparse matrices, solver information, etc. in ways that Fortran simply did not allow. ANSI C was a complete standard that all modern C compilers supported. The language was identical on all machines. C++ was still evolving and compilers on different machines were not identical. Using C function pointers to provide data encapsulation and polymorphism allowed us to get many of the advantages of C++ without using such a large and more complicated language. It would have been natural and reasonable to have coded PETSc in C++; we opted to use C instead.

Does all the PETSc error checking and logging reduce PETSc’s efficiency?

No

How do such a small group of people manage to write and maintain such a large and marvelous package as PETSc?

1. We work very efficiently.

   • We use powerful editors and programming environments.

   • Our manual pages are generated automatically from formatted comments in the code, thus alleviating the need for creating and maintaining manual pages.

   • We employ continuous integration testing of the entire PETSc library on many different machine architectures. This process significantly protects (no bug-catching process is perfect) against inadvertently introducing bugs with new additions. Every new feature must pass our suite of thousands of tests as well as formal code review before it may be included.

2. We are very careful in our design (and are constantly revising our design)

   • PETSc as a package should be easy to use, write, and maintain. Our mantra is to write code like everyone is using it.

3. We are willing to do the grunt work

   • PETSc is regularly checked to make sure that all code conforms to our interface design. We will never keep in a bad design decision simply because changing it will require a lot of editing; we do a lot of editing.

4. We constantly seek out and experiment with new design ideas

   • We retain the useful ones and discard the rest. All of these decisions are based not just on performance, but also on practicality.

5. Function and variable names must adhere to strict guidelines

   • Even the rules about capitalization are designed to make it easy to figure out the name of a particular object or routine. Our memories are terrible, so careful consistent naming puts less stress on our limited human RAM.

6. The PETSc directory tree is designed to make it easy to move throughout the entire package

7. We have a rich, robust, and fast bug reporting system

   • petsc-maint@mcs.anl.gov is always checked, and we pride ourselves on responding quickly and accurately. Email is very lightweight, and so bug reports system retains an archive of all reported problems and fixes, so it is easy to re-find fixes to previously discovered problems.

8. We contain the complexity of PETSc by using powerful object-oriented programming techniques

   • Data encapsulation serves to abstract complex data formats or movement to human-readable format. This is why your program cannot, for example, look directly at what is inside the object Mat.

   • Polymorphism makes changing program behavior as easy as possible, and further abstracts the intent of your program from what is written in code. You call MatMult() regardless of whether your matrix is dense, sparse, parallel or sequential; you don’t call a different routine for each format.

9. We try to provide the functionality requested by our users

For complex numbers will I get better performance with C++?

To use PETSc with complex numbers you may use the following configure options; --with-scalar-type=complex and either --with-clanguage=c++ or (the default) --with-clanguage=c. In our experience they will deliver very similar performance (speed), but if one is concerned they should just try both and see if one is faster.

How come when I run the same program on the same number of processes I get a “different” answer?

Inner products and norms in PETSc are computed using the MPI_Allreduce() command. For different runs the order at which values arrive at a given process (via MPI) can be in a different order, thus the order in which some floating point arithmetic operations are performed will be different. Since floating point arithmetic is not associative, the computed quantity may be slightly different.

Over a run the many slight differences in the inner products and norms will effect all the computed results. It is important to realize that none of the computed answers are any less right or wrong (in fact the sequential computation is no more right then the parallel ones). All answers are equal, but some are more equal than others.

The discussion above assumes that the exact same algorithm is being used on the different number of processes. When the algorithm is different for the different number of processes (almost all preconditioner algorithms except Jacobi are different for different number of processes) then one expects to see (and does) a greater difference in results for different numbers of processes. In some cases (for example block Jacobi preconditioner) it may be that the algorithm works for some number of processes and does not work for others.

How come when I run the same linear solver on a different number of processes it takes a different number of iterations?

The convergence of many of the preconditioners in PETSc including the default parallel preconditioner block Jacobi depends on the number of processes. The more processes the (slightly) slower convergence it has. This is the nature of iterative solvers, the more parallelism means the more “older” information is used in the solution process hence slower convergence.

Can PETSc use GPUs to speed up computations?

See also

See GPU development roadmap for the latest information regarding the state of PETSc GPU integration.

See GPU install documentation for up-to-date information on installing PETSc to use GPU’s.

Quick summary of usage with CUDA:

• The VecType VECSEQCUDA, VECMPICUDA, or VECCUDA may be used with VecSetType() or -vec_type seqcuda, mpicuda, or cuda when VecSetFromOptions() is used.

• The MatType MATSEQAIJCUSPARSE, MATMPIAIJCUSPARSE, or MATAIJCUSPARSE maybe used with MatSetType() or -mat_type seqaijcusparse, mpiaijcusparse, or aijcusparse when MatSetFromOptions() is used.

• If you are creating the vectors and matrices with a DM, you can use -dm_vec_type cuda and -dm_mat_type aijcusparse.

Quick summary of usage with OpenCL (provided by the ViennaCL library):

• The VecType VECSEQVIENNACL, VECMPIVIENNACL, or VECVIENNACL may be used with VecSetType() or -vec_type seqviennacl, mpiviennacl, or viennacl when VecSetFromOptions() is used.

• The MatType MATSEQAIJVIENNACL, MATMPIAIJVIENNACL, or MATAIJVIENNACL maybe used with MatSetType() or -mat_type seqaijviennacl, mpiaijviennacl, or aijviennacl when MatSetFromOptions() is used.

• If you are creating the vectors and matrices with a DM, you can use -dm_vec_type viennacl and -dm_mat_type aijviennacl.

General hints:

• It is useful to develop your code with the default vectors and then run production runs with the command line options to use the GPU since debugging on GPUs is difficult.

• All of the Krylov methods except KSPIBCGS run on the GPU.

• Parts of most preconditioners run directly on the GPU. After setup, PCGAMG runs fully on GPUs, without any memory copies between the CPU and GPU.

Some GPU systems (for example many laptops) only run with single precision; thus, PETSc must be built with the configure option --with-precision=single.

Can I run PETSc with extended precision?

Yes, with gcc and gfortran. configure PETSc using the options --with-precision=__float128 and `` –download-f2cblaslapack``.

Warning

External packages are not guaranteed to work in this mode!

Why doesn’t PETSc use Qd to implement support for extended precision?

We tried really hard but could not. The problem is that the QD c++ classes, though they try to, implement the built-in data types of double are not native types and cannot “just be used” in a general piece of numerical source code. Ratherm the code has to rewritten to live within the limitations of QD classes. However PETSc can be built to use quad precision, as detailed here.

How do I cite PETSc?

Use these citations.

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Installation

How do I begin using PETSc if the software has already been completely built and installed by someone else?

Assuming that the PETSc libraries have been successfully built for a particular architecture and level of optimization, a new user must merely:

1. Set PETSC_DIR to the full path of the PETSc home directory. This will be the location of the configure script, and usually called “petsc” or some variation of that (for example, /home/username/petsc).

2. Set PETSC_ARCH, which indicates the configuration on which PETSc will be used. Note that $PETSC_ARCH is simply a name the installer used when installing the libraries. There will exist a directory within $PETSC_DIR that is named after its corresponding $PETSC_ARCH. There many be several on a single system, for example “linux-c-debug” for the debug versions compiled by a C compiler or “linux-c-opt” for the optimized version.

Still Stuck?

See the quick-start tutorial for a step-by-step guide on installing PETSc, in case you have missed a step.

See the users manual section on getting started.

The PETSc distribution is SO Large. How can I reduce my disk space usage?

1. The PETSc users manual is provided in PDF format at $PETSC_DIR/manual.pdf. You can delete this.

2. The PETSc test suite contains sample output for many of the examples. These are contained in the PETSc directories $PETSC_DIR/src/*/tutorials/output and $PETSC_DIR/src/*/tests/output. Once you have run the test examples, you may remove all of these directories to save some disk space. You can locate the largest with e.g. find . -name output -type d | xargs du -sh | sort -hr on a Unix-based system.

3. The debugging versions of the libraries are larger than the optimized versions. In a pinch you can work with the optimized version, although we bid you good luck in finnding bugs as it is much easier with the debug version.

I want to use PETSc only for uniprocessor programs. Must I still install and use a version of MPI?

No, run configure with the option --with-mpi=0

Can I install PETSc to not use X windows (either under Unix or Microsoft Windows with GCC)?

Yes. Run configure with the additional flag --with-x=0

Why do you use MPI?

MPI is the message-passing standard. Because it is a standard, it will not frequently change over time; thus, we do not have to change PETSc every time the provider of the message-passing system decides to make an interface change. MPI was carefully designed by experts from industry, academia, and government labs to provide the highest quality performance and capability.

For example, the careful design of communicators in MPI allows the easy nesting of different libraries; no other message-passing system provides this support. All of the major parallel computer vendors were involved in the design of MPI and have committed to providing quality implementations.

In addition, since MPI is a standard, several different groups have already provided complete free implementations. Thus, one does not have to rely on the technical skills of one particular group to provide the message-passing libraries. Today, MPI is the only practical, portable approach to writing efficient parallel numerical software.

What do I do if my MPI compiler wrappers are invalid?

Most MPI implementations provide compiler wrappers (such as mpicc) which give the include and link options necessary to use that verson of MPI to the underlying compilers . Configuration will fail if these wrappers are either absent or broken in the MPI pointed to by --with-mpi-dir. You can rerun the configure with the additional option --with-mpi-compilers=0, which will try to auto-detect working compilers; however, these compilers may be incompatible with the particular MPI build. If this fix does not work, run with --with-cc=[your_c_compiler] where you know your_c_compiler works with this particular MPI, and likewise for C++ (--with-cxx=[your_cxx_compiler]) and Fortran (--with-fc=[your_fortran_compiler]).

When should/can I use the configure option --with-64-bit-indices?

By default the type that PETSc uses to index into arrays and keep sizes of arrays is a PetscInt defined to be a 32-bit int. If your problem:

• Involves more than 2^31 - 1 unknowns (around 2 billion).

• Your matrix might contain more than 2^31 - 1 nonzeros on a single process.

Then you need to use this option. Otherwise you will get strange crashes.

This option can be used when you are using either 32 or 64-bit pointers. You do not need to use this option if you are using 64-bit pointers unless the two conditions above hold.

What if I get an internal compiler error?

You can rebuild the offending file individually with a lower optimization level. Then make sure to complain to the compiler vendor and file a bug report. For example, if the compiler chokes on src/mat/impls/baij/seq/baijsolvtrannat.c you can run the following from $PETSC_DIR:

$ make -f gmakefile PCC_FLAGS="-O1" $PETSC_ARCH/obj/src/mat/impls/baij/seq/baijsolvtrannat.o
$ make all

How do I enable Python bindings (petsc4py) with PETSc?

1. Install Cython.

2. configure PETSc with the --with-petsc4py=1 option.

3. set PYTHONPATH=$PETSC_DIR/$PETSC_ARCH/lib

What Fortran compiler do you recommend on macOS?

We recommend using homebrew to install gfortran, see Installing On macOS

How can I find the URL locations of the packages you install using --download-PACKAGE?

$ grep "self.download " $PETSC_DIR/config/BuildSystem/config/packages/*.py

How to fix the problem: PETSc was configured with one MPICH (or Open MPI) mpi.h version but now appears to be compiling using a different MPICH (or Open MPI) mpi.h version

This happens for generally one of two reasons:

• You previously ran configure with the option --download-mpich (or --download-openmpi) but later ran configure to use a version of MPI already installed on the machine. Solution:

  $ rm -rf $PETSC_DIR/$PETSC_ARCH
  $ ./configure --your-args
  

What does it mean when make check hangs or errors on PetscOptionsInsertFile()?

For example:

Possible error running C/C++ src/snes/tutorials/ex19 with 2 MPI processes
See https://petsc.org/release/faq/
[0]PETSC ERROR: #1 PetscOptionsInsertFile() line 563 in /Users/barrysmith/Src/PETSc/src/sys/objects/options.c
[0]PETSC ERROR: #2 PetscOptionsInsert() line 720 in /Users/barrysmith/Src/PETSc/src/sys/objects/options.c
[0]PETSC ERROR: #3 PetscInitialize() line 828 in /Users/barrysmith/Src/PETSc/src/sys/objects/pinit.c

or

$ make check
Running check examples to verify correct installation
Using PETSC_DIR=/Users/barrysmith/Src/petsc and PETSC_ARCH=arch-fix-mpiexec-hang-2-ranks
C/C++ example src/snes/tutorials/ex19 run successfully with 1 MPI process
PROGRAM SEEMS TO BE HANGING HERE

This usually occurs when network settings are misconfigured (perhaps due to VPN) resulting in a failure or hang in system call gethostbyname().

• Verify you are using the correct mpiexec for the MPI you have linked PETSc with.

• If you have a VPN enabled on your machine, try turning it off and then running make check to verify that it is not the VPN playing poorly with MPI.

• If ping `hostname` `` (/sbin/ping`` on macOS) fails or hangs do:

  echo 127.0.0.1 `hostname` | sudo tee -a /etc/hosts
  

  and try make check again.

• Try completely disconnecting your machine from the network and see if make check then works

• Try the PETSc configure option --download-mpich-device=ch3:nemesis with --download-mpich.

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Usage

How can I redirect PETSc’s stdout and stderr when programming with a GUI interface in Windows Developer Studio or to C++ streams?

To overload just the error messages write your own MyPrintError() function that does whatever you want (including pop up windows etc) and use it like below.

extern "C" {
  int PASCAL WinMain(HINSTANCE,HINSTANCE,LPSTR,int);
};

#include <petscsys.h>
#include <mpi.h>

const char help[] = "Set up from main";

int MyPrintError(const char error[], ...)
{
  printf("%s", error);
  return 0;
}

int main(int ac, char *av[])
{
  char           buf[256];
  HINSTANCE      inst;
  PetscErrorCode ierr;

  inst = (HINSTANCE)GetModuleHandle(NULL);
  PetscErrorPrintf = MyPrintError;

  buf[0] = 0;
  for (int i = 1; i < ac; ++i) {
    strcat(buf, av[i]);
    strcat(buf, " ");
  }

  PetscCall(PetscInitialize(&ac, &av, NULL, help));

  return WinMain(inst, NULL, buf, SW_SHOWNORMAL);
}

Place this file in the project and compile with this preprocessor definitions:

WIN32
_DEBUG
_CONSOLE
_MBCS
USE_PETSC_LOG
USE_PETSC_BOPT_g
USE_PETSC_STACK
_AFXDLL

And these link options:

/nologo
/subsystem:console
/incremental:yes
/debug
/machine:I386
/nodefaultlib:"libcmtd.lib"
/nodefaultlib:"libcd.lib"
/nodefaultlib:"mvcrt.lib"
/pdbtype:sept

Note

The above is compiled and linked as if it was a console program. The linker will search for a main, and then from it the WinMain will start. This works with MFC templates and derived classes too.

When writing a Window’s console application you do not need to do anything, the stdout and stderr is automatically output to the console window.

To change where all PETSc stdout and stderr go, (you can also reassign PetscVFPrintf() to handle stdout and stderr any way you like) write the following function:

PetscErrorCode mypetscvfprintf(FILE *fd, const char format[], va_list Argp)
{
  PetscFunctionBegin;
  if (fd != stdout && fd != stderr) { /* handle regular files */
    PetscCall(PetscVFPrintfDefault(fd, format, Argp));
  } else {
    char buff[1024]; /* Make sure to assign a large enough buffer */
    int  length;

    PetscCall(PetscVSNPrintf(buff, 1024, format, &length, Argp));

    /* now send buff to whatever stream or whatever you want */
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

Then assign PetscVFPrintf = mypetscprintf before PetscInitialize() in your main program.

I want to use Hypre boomerAMG without GMRES but when I run -pc_type hypre -pc_hypre_type boomeramg -ksp_type preonly I don’t get a very accurate answer!

You should run with -ksp_type richardson to have PETSc run several V or W cycles. -ksp_type preonly causes boomerAMG to use only one V/W cycle. You can control how many cycles are used in a single application of the boomerAMG preconditioner with -pc_hypre_boomeramg_max_iter <it> (the default is 1). You can also control the tolerance boomerAMG uses to decide if to stop before max_iter with -pc_hypre_boomeramg_tol <tol> (the default is 1.e-7). Run with -ksp_view to see all the hypre options used and -help | grep boomeramg to see all the command line options.

How do I use PETSc for Domain Decomposition?

PETSc includes Additive Schwarz methods in the suite of preconditioners under the umbrella of PCASM. These may be activated with the runtime option -pc_type asm. Various other options may be set, including the degree of overlap -pc_asm_overlap <number> the type of restriction/extension -pc_asm_type [basic,restrict,interpolate,none] sets ASM type and several others. You may see the available ASM options by using -pc_type asm -help. See the procedural interfaces in the manual pages, for example PCASMType() and check the index of the users manual for PCASMCreateSubdomains().

PETSc also contains a domain decomposition inspired wirebasket or face based two level method where the coarse mesh to fine mesh interpolation is defined by solving specific local subdomain problems. It currently only works for 3D scalar problems on structured grids created with PETSc DMDA. See the manual page for PCEXOTIC and src/ksp/ksp/tutorials/ex45.c for an example.

PETSc also contains a balancing Neumann-Neumann type preconditioner, see the manual page for PCBDDC. This requires matrices be constructed with MatCreateIS() via the finite element method. See src/ksp/ksp/tests/ex59.c for an example.

You have MATAIJ and MATBAIJ matrix formats, and MATSBAIJ for symmetric storage, how come no MATSAIJ?

Just for historical reasons; the MATSBAIJ format with blocksize one is just as efficient as a MATSAIJ would be.

Can I Create BAIJ matrices with different size blocks for different block rows?

No. The MATBAIJ format only supports a single fixed block size on the entire matrix. But the MATAIJ format automatically searches for matching rows and thus still takes advantage of the natural blocks in your matrix to obtain good performance.

Note

If you use MATAIJ you cannot use the MatSetValuesBlocked().

How do I access the values of a remote parallel PETSc Vec?

1. On each process create a local Vec large enough to hold all the values it wishes to access.

2. Create a VecScatter that scatters from the parallel Vec into the local Vec.

3. Use VecGetArray() to access the values in the local Vec.

For example, assuming we have distributed a vector vecGlobal of size \(N\) to \(R\) ranks and each remote rank holds \(N/R = m\) values (similarly assume that \(N\) is cleanly divisible by \(R\)). We want each rank \(r\) to gather the first \(n\) (also assume \(n \leq m\)) values from its immediately superior neighbor \(r+1\) (final rank will retrieve from rank 0).

Vec            vecLocal;
IS             isLocal, isGlobal;
VecScatter     ctx;
PetscScalar    *arr;
PetscInt       N, firstGlobalIndex;
MPI_Comm       comm;
PetscMPIInt    r, R;

/* Create sequential local vector, big enough to hold local portion */
PetscCall(VecCreateSeq(PETSC_COMM_SELF, n, &vecLocal));

/* Create IS to describe where we want to scatter to */
PetscCall(ISCreateStride(PETSC_COMM_SELF, n, 0, 1, &isLocal));

/* Compute the global indices */
PetscCall(VecGetSize(vecGlobal, &N));
PetscCall(PetscObjectGetComm((PetscObject) vecGlobal, &comm));
PetscCallMPI(MPI_Comm_rank(comm, &r));
PetscCallMPI(MPI_Comm_size(comm, &R));
firstGlobalIndex = r == R-1 ? 0 : (N/R)*(r+1);

/* Create IS that describes where we want to scatter from */
PetscCall(ISCreateStride(comm, n, firstGlobalIndex, 1, &isGlobal));

/* Create the VecScatter context */
PetscCall(VecScatterCreate(vecGlobal, isGlobal, vecLocal, isLocal, &ctx));

/* Gather the values */
PetscCall(VecScatterBegin(ctx, vecGlobal, vecLocal, INSERT_VALUES, SCATTER_FORWARD));
PetscCall(VecScatterEnd(ctx, vecGlobal, vecLocal, INSERT_VALUES, SCATTER_FORWARD));

/* Retrieve and do work */
PetscCall(VecGetArray(vecLocal, &arr));
/* Work */
PetscCall(VecRestoreArray(vecLocal, &arr));

/* Don't forget to clean up */
PetscCall(ISDestroy(&isLocal));
PetscCall(ISDestroy(&isGlobal));
PetscCall(VecScatterDestroy(&ctx));
PetscCall(VecDestroy(&vecLocal));

How do I collect to a single processor all the values from a parallel PETSc Vec?

1. Create the VecScatter context that will do the communication:

   Vec        in_par,out_seq;
   VecScatter ctx;
   
   PetscCall(VecScatterCreateToAll(in_par, &ctx, &out_seq));
   

2. Initiate the communication (this may be repeated if you wish):

   PetscCall(VecScatterBegin(ctx, in_par, out_seq, INSERT_VALUES, SCATTER_FORWARD));
   PetscCall(VecScatterEnd(ctx, in_par, out_seq, INSERT_VALUES, SCATTER_FORWARD));
   /* May destroy context now if no additional scatters are needed, otherwise reuse it */
   PetscCall(VecScatterDestroy(&ctx));
   

Note that this simply concatenates in the parallel ordering of the vector (computed by the MPI_Comm_rank of the owning process). If you are using a Vec from DMCreateGlobalVector() you likely want to first call DMDAGlobalToNaturalBegin() followed by DMDAGlobalToNaturalEnd() to scatter the original Vec into the natural ordering in a new global Vec before calling VecScatterBegin()/VecScatterEnd() to scatter the natural Vec onto all processes.

How do I collect all the values from a parallel PETSc Vec on the 0th rank?

See FAQ entry on collecting to an arbitrary processor, but replace

PetscCall(VecScatterCreateToAll(in_par, &ctx, &out_seq));

with

PetscCall(VecScatterCreateToZero(in_par, &ctx, &out_seq));

Note

The same ordering considerations as discussed in the aforementioned entry also apply here.

How can I read in or write out a sparse matrix in Matrix Market, Harwell-Boeing, Slapc or other ASCII format?

If you can read or write your matrix using Python or MATLAB/Octave, PetscBinaryIO modules are provided at $PETSC_DIR/lib/petsc/bin for each language that can assist with reading and writing. If you just want to convert MatrixMarket, you can use:

$ python -m $PETSC_DIR/lib/petsc/bin/PetscBinaryIO convert matrix.mtx

To produce matrix.petsc.

You can also call the script directly or import it from your Python code. There is also a PETScBinaryIO.jl Julia package.

For other formats, either adapt one of the above libraries or see the examples in $PETSC_DIR/src/mat/tests, specifically ex72.c or ex78.c. You will likely need to modify the code slightly to match your required ASCII format.

Note

Never read or write in parallel an ASCII matrix file.

Instead read in sequentially with a standalone code based on ex72.c or ex78.c then save the matrix with the binary viewer PetscViewerBinaryOpen() and load the matrix in parallel in your “real” PETSc program with MatLoad().

For writing save with the binary viewer and then load with the sequential code to store it as ASCII.

Does TSSetFromOptions(), SNESSetFromOptions(), or KSPSetFromOptions() reset all the parameters I previously set or how come do they not seem to work?

If XXSetFromOptions() is used (with -xxx_type aaaa) to change the type of the object then all parameters associated with the previous type are removed. Otherwise it does not reset parameters.

TS/SNES/KSPSetXXX() commands that set properties for a particular type of object (such as KSPGMRESSetRestart()) ONLY work if the object is ALREADY of that type. For example, with

KSP ksp;

PetscCall(KSPCreate(PETSC_COMM_WORLD, &ksp));
PetscCall(KSPGMRESSetRestart(ksp, 10));

the restart will be ignored since the type has not yet been set to KSPGMRES. To have those values take effect you should do one of the following:

• Allow setting the type from the command line, if it is not on the command line then the default type is automatically set.

/* Create generic object */
XXXCreate(..,&obj);
/* Must set all settings here, or default */
XXXSetFromOptions(obj);

• Hardwire the type in the code, but allow the user to override it via a subsequent XXXSetFromOptions() call. This essentially allows the user to customize what the “default” type to of the object.

/* Create generic object */
XXXCreate(..,&obj);
/* Set type directly */
XXXSetYYYYY(obj,...);
/* Can always change to different type */
XXXSetFromOptions(obj);

How do I compile and link my own PETSc application codes and can I use my own makefile or rules for compiling code, rather than PETSc’s?

See the section of the users manual on writing application codes with PETSc.

Can I use CMake to build my own project that depends on PETSc?

See the section of the users manual on writing application codes with PETSc.

How can I put carriage returns in PetscPrintf() statements from Fortran?

You can use the same notation as in C, just put a \n in the string. Note that no other C format instruction is supported.

Or you can use the Fortran concatination // and char(10); for example 'some string'//char(10)//'another string on the next line.

How can I implement callbacks using C++ class methods?

Declare the class method static. Static methods do not have a this pointer, but the void* context parameter will usually be cast to a pointer to the class where it can serve the same function.

Remember

All PETSc callbacks return PetscErrorCode.

Everyone knows that when you code Newton’s Method you should compute the function and its Jacobian at the same time. How can one do this in PETSc?

The update in Newton’s method is computed as

\[u^{n+1} = u^n - \lambda * \left[J(u^n)] * F(u^n) \right]^{\dagger}\]

The reason PETSc doesn’t default to computing both the function and Jacobian at the same time is:

• In order to do the line search \(F \left(u^n - \lambda * \text{step} \right)\) may need to be computed for several \(\lambda\). The Jacobian is not needed for each of those and one does not know in advance which will be the final \(\lambda\) until after the function value is computed, so many extra Jacobians may be computed.

• In the final step if \(|| F(u^p)||\) satisfies the convergence criteria then a Jacobian need not be computed.

You are free to have your FormFunction() compute as much of the Jacobian at that point as you like, keep the information in the user context (the final argument to FormFunction() and FormJacobian()) and then retrieve the information in your FormJacobian() function.

Computing the Jacobian or preconditioner is time consuming. Is there any way to compute it less often?

PETSc has a variety of ways of lagging the computation of the Jacobian or the preconditioner. They are documented in the manual page for SNESComputeJacobian() and in the users manual:

-snes_lag_jacobian

(SNESSetLagJacobian()) How often Jacobian is rebuilt (use -1 to never rebuild, use -2 to rebuild the next time requested and then never again).

-snes_lag_jacobian_persists

(SNESSetLagJacobianPersists()) Forces lagging of Jacobian through multiple SNES solves , same as passing -2 to -snes_lag_jacobian. By default, each new SNES solve normally triggers a recomputation of the Jacobian.

-snes_lag_preconditioner

(SNESSetLagPreconditioner()) how often the preconditioner is rebuilt. Note: if you are lagging the Jacobian the system will know that the matrix has not changed and will not recompute the (same) preconditioner.

-snes_lag_preconditioner_persists

(SNESSetLagPreconditionerPersists()) Preconditioner lags through multiple SNES solves

Note

These are often (but does not need to be) used in combination with -snes_mf_operator which applies the fresh Jacobian matrix-free for every matrix-vector product. Otherwise the out-of-date matrix vector product, computed with the lagged Jacobian will be used.

By using KSPMonitorSet() and/or SNESMonitorSet() one can provide code that monitors the convergence rate and automatically triggers an update of the Jacobian or preconditioner based on decreasing convergence of the iterative method. For example if the number of SNES iterations doubles one might trigger a new computation of the Jacobian. Experimentation is the only general purpose way to determine which approach is best for your problem.

Important

It is also vital to experiment on your true problem at the scale you will be solving the problem since the performance benefits depend on the exact problem and the problem size!

How can I use Newton’s Method Jacobian free? Can I difference a different function than provided with SNESSetFunction()?

The simplest way is with the option -snes_mf, this will use finite differencing of the function provided to SNESComputeFunction() to approximate the action of Jacobian.

Important

Since no matrix-representation of the Jacobian is provided the -pc_type used with this option must be -pc_type none. You may provide a custom preconditioner with SNESGetKSP(), KSPGetPC(), and PCSetType() and use PCSHELL.

The option -snes_mf_operator will use Jacobian free to apply the Jacobian (in the Krylov solvers) but will use whatever matrix you provided with SNESSetJacobian() (assuming you set one) to compute the preconditioner.

To write the code (rather than use the options above) use MatCreateSNESMF() and pass the resulting matrix object to SNESSetJacobian().

For purely matrix-free (like -snes_mf) pass the matrix object for both matrix arguments and pass the function MatMFFDComputeJacobian().

To provide your own approximate Jacobian matrix to compute the preconditioner (like -snes_mf_operator), pass this other matrix as the second matrix argument to SNESSetJacobian(). Make sure your provided computejacobian() function calls MatAssemblyBegin() and MatAssemblyEnd() separately on BOTH matrix arguments to this function. See src/snes/tests/ex7.c.

To difference a different function than that passed to SNESSetJacobian() to compute the matrix-free Jacobian multiply call MatMFFDSetFunction() to set that other function. See src/snes/tests/ex7.c.h.

How can I determine the condition number of a matrix?

For small matrices, the condition number can be reliably computed using

-pc_type svd -pc_svd_monitor

For larger matrices, you can run with

-pc_type none -ksp_type gmres -ksp_monitor_singular_value -ksp_gmres_restart 1000

to get approximations to the condition number of the operator. This will generally be accurate for the largest singular values, but may overestimate the smallest singular value unless the method has converged. Make sure to avoid restarts. To estimate the condition number of the preconditioned operator, use -pc_type somepc in the last command.

You can use SLEPc for highly scalable, efficient, and quality eigenvalue computations.

How can I compute the inverse of a matrix in PETSc?

Are you sure?

It is very expensive to compute the inverse of a matrix and very rarely needed in practice. We highly recommend avoiding algorithms that need it.

The inverse of a matrix (dense or sparse) is essentially always dense, so begin by creating a dense matrix B and fill it with the identity matrix (ones along the diagonal), also create a dense matrix X of the same size that will hold the solution. Then factor the matrix you wish to invert with MatLUFactor() or MatCholeskyFactor(), call the result A. Then call MatMatSolve(A,B,X) to compute the inverse into X. See also section on Schur's complement.

How can I compute the Schur complement in PETSc?

Are you sure?

It is very expensive to compute the Schur complement of a matrix and very rarely needed in practice. We highly recommend avoiding algorithms that need it.

The Schur complement of the matrix \(M \in \mathbb{R}^{\left(p+q \right) \times \left(p + q \right)}\)

\[M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\]

where

\[A \in \mathbb{R}^{p \times p}, \quad B \in \mathbb{R}^{p \times q}, \quad C \in \mathbb{R}^{q \times p}, \quad D \in \mathbb{R}^{q \times q}\]

is given by

\[S_D := A - BD^{-1}C \\ S_A := D - CA^{-1}B\]

Like the inverse, the Schur complement of a matrix (dense or sparse) is essentially always dense, so assuming you wish to calculate \(S_A = D - C \underbrace{ \overbrace{(A^{-1})}^{U} B}_{V}\) begin by:

1. Forming a dense matrix \(B\)

2. Also create another dense matrix \(V\) of the same size.

3. Then either factor the matrix \(A\) directly with MatLUFactor() or MatCholeskyFactor(), or use MatGetFactor() followed by MatLUFactorSymbolic() followed by MatLUFactorNumeric() if you wish to use and external solver package like SuperLU_Dist. Call the result \(U\).

4. Then call MatMatSolve(U,B,V).

5. Then call MatMatMult(C,V,MAT_INITIAL_MATRIX,1.0,&S).

6. Now call MatAXPY(S,-1.0,D,MAT_SUBSET_NONZERO).

7. Followed by MatScale(S,-1.0).

For computing Schur complements like this it does not make sense to use the KSP iterative solvers since for solving many moderate size problems using a direct factorization is much faster than iterative solvers. As you can see, this requires a great deal of work space and computation so is best avoided.

However, it is not necessary to assemble the Schur complement \(S\) in order to solve systems with it. Use MatCreateSchurComplement(A,A_pre,B,C,D,&S) to create a matrix that applies the action of \(S\) (using A_pre to solve with A), but does not assemble.

Alternatively, if you already have a block matrix M = [A, B; C, D] (in some ordering), then you can create index sets (IS) isa and isb to address each block, then use MatGetSchurComplement() to create the Schur complement and/or an approximation suitable for preconditioning.

Since \(S\) is generally dense, standard preconditioning methods cannot typically be applied directly to Schur complements. There are many approaches to preconditioning Schur complements including using the SIMPLE approximation

\[D - C \text{diag}(A)^{-1} B\]

to create a sparse matrix that approximates the Schur complement (this is returned by default for the optional “preconditioning” matrix in MatGetSchurComplement()).

An alternative is to interpret the matrices as differential operators and apply approximate commutator arguments to find a spectrally equivalent operation that can be applied efficiently (see the “PCD” preconditioners [ESW14]). A variant of this is the least squares commutator, which is closely related to the Moore-Penrose pseudoinverse, and is available in PCLSC which operates on matrices of type MATSCHURCOMPLEMENT.

Do you have examples of doing unstructured grid Finite Element Method (FEM) with PETSc?

There are at least three ways to write finite element codes using PETSc:

1. Use DMPLEX, which is a high level approach to manage your mesh and discretization. See the tutorials sections for further information, or see src/snes/tutorial/ex62.c.

2. Use packages such as deal.ii, libMesh, or Firedrake, which use PETSc for the solvers.

3. Manage the grid data structure yourself and use PETSc PetscSF, IS, and VecScatter to communicate the required ghost point communication. See src/snes/tutorials/ex10d/ex10.c.

DMDA decomposes the domain differently than the MPI_Cart_create() command. How can one use them together?

The MPI_Cart_create() first divides the mesh along the z direction, then the y, then the x. DMDA divides along the x, then y, then z. Thus, for example, rank 1 of the processes will be in a different part of the mesh for the two schemes. To resolve this you can create a new MPI communicator that you pass to DMDACreate() that renumbers the process ranks so that each physical process shares the same part of the mesh with both the DMDA and the MPI_Cart_create(). The code to determine the new numbering was provided by Rolf Kuiper:

// the numbers of processors per direction are (int) x_procs, y_procs, z_procs respectively
// (no parallelization in direction 'dir' means dir_procs = 1)

MPI_Comm    NewComm;
int         x, y, z;
PetscMPIInt MPI_Rank, NewRank;

// get rank from MPI ordering:
PetscCallMPI(MPI_Comm_rank(MPI_COMM_WORLD, &MPI_Rank));

// calculate coordinates of cpus in MPI ordering:
x = MPI_rank / (z_procs*y_procs);
y = (MPI_rank % (z_procs*y_procs)) / z_procs;
z = (MPI_rank % (z_procs*y_procs)) % z_procs;

// set new rank according to PETSc ordering:
NewRank = z*y_procs*x_procs + y*x_procs + x;

// create communicator with new ranks according to PETSc ordering
PetscCallMPI(MPI_Comm_split(PETSC_COMM_WORLD, 1, NewRank, &NewComm));

// override the default communicator (was MPI_COMM_WORLD as default)
PETSC_COMM_WORLD = NewComm;

When solving a system with Dirichlet boundary conditions I can use MatZeroRows() to eliminate the Dirichlet rows but this results in a non-Symmetric system. How can I apply Dirichlet boundary conditions but keep the matrix symmetric?

• For nonsymmetric systems put the appropriate boundary solutions in the x vector and use MatZeroRows() followed by KSPSetOperators().

• For symmetric problems use MatZeroRowsColumns().

• If you have many Dirichlet locations you can use MatZeroRows() (not MatZeroRowsColumns()) and -ksp_type preonly -pc_type redistribute (see PCREDISTRIBUTE) and PETSc will repartition the parallel matrix for load balancing. In this case the new matrix solved remains symmetric even though MatZeroRows() is used.

An alternative approach is, when assembling the matrix (generating values and passing them to the matrix), never to include locations for the Dirichlet grid points in the vector and matrix, instead taking them into account as you put the other values into the load.

How can I get PETSc vectors and matrices to MATLAB or vice versa?

There are numerous ways to work with PETSc and MATLAB. All but the first approach require PETSc to be configured with –with-matlab.

1. To save PETSc Mat and Vec to files that can be read from MATLAB use PetscViewerBinaryOpen() viewer and VecView() or MatView() to save objects for MATLAB and VecLoad() and MatLoad() to get the objects that MATLAB has saved. See share/petsc/matlab/PetscBinaryRead.m and share/petsc/matlab/PetscBinaryWrite.m for loading and saving the objects in MATLAB.

2. Using the MATLAB Engine, allows PETSc to automatically call MATLAB to perform some specific computations. This does not allow MATLAB to be used interactively by the user. See the PetscMatlabEngine.

3. You can open a socket connection between MATLAB and PETSc to allow sending objects back and forth between an interactive MATLAB session and a running PETSc program. See PetscViewerSocketOpen() for access from the PETSc side and share/petsc/matlab/PetscReadBinary.m for access from the MATLAB side.

4. You can save PETSc Vec (not Mat) with the PetscViewerMatlabOpen() viewer that saves .mat files can then be loaded into MATLAB using the load() command

How do I get started with Cython so that I can extend petsc4py?

1. Learn how to build a Cython module.

2. Go through the simple example. Note also the next comment that shows how to create numpy arrays in the Cython and pass them back.

3. Check out this page which tells you how to get fast indexing.

4. Have a look at the petsc4py array source.

How do I compute a custom norm for KSP to use as a convergence test or for monitoring?

You need to call KSPBuildResidual() on your KSP object and then compute the appropriate norm on the resulting residual. Note that depending on the KSPSetNormType() of the method you may not return the same norm as provided by the method. See also KSPSetPCSide().

If I have a sequential program can I use a PETSc parallel solver?

Important

Do not expect to get great speedups! Much of the speedup gained by using parallel solvers comes from building the underlying matrices and vectors in parallel to begin with. You should see some reduction in the time for the linear solvers.

Yes, you must set up PETSc with MPI (even though you will not use MPI) with at least the following options:

$ ./configure --download-superlu_dist --download-parmetis --download-metis --with-openmp

Your compiler must support OpenMP. To have the linear solver run in parallel run your program with

$ OMP_NUM_THREADS=n ./myprog -pc_type lu -pc_factor_mat_solver superlu_dist

where n is the number of threads and should be less than or equal to the number of cores available.

Note

If your code is MPI parallel you can also use these same options to have SuperLU_dist utilize multiple threads per MPI process for the direct solver. Make sure that the $OMP_NUM_THREADS you use per MPI process is less than or equal to the number of cores available for each MPI process. For example if your compute nodes have 6 cores and you use 2 MPI processes per node then set $OMP_NUM_THREADS to 2 or 3.

Another approach that allows using a PETSc parallel solver is to use PCMPI.

TS or SNES produces infeasible (out of domain) solutions or states. How can I prevent this?

For TS call TSSetFunctionDomainError(). For both TS and SNES call SNESSetFunctionDomainError() when the solver passes an infeasible (out of domain) solution or state to your routines.

If it occurs for DAEs, it is important to insure the algebraic constraints are well satisfied, which can prevent “breakdown” later. Thus, one can try using a tight tolerance for SNES, using a direct linear solver (PCType of PCLU) when possible, and reducing the timestep (or tightening TS tolerances for adaptive time stepping).

Can PETSc work with Hermitian matrices?

PETSc’s support of Hermitian matrices is limited. Many operations and solvers work for symmetric (MATSBAIJ) matrices and operations on transpose matrices but there is little direct support for Hermitian matrices and Hermitian transpose (complex conjugate transpose) operations. There is KSPSolveTranspose() for solving the transpose of a linear system but no KSPSolveHermitian().

For creating known Hermitian matrices:

• MatCreateNormalHermitian()

• MatCreateHermitianTranspose()

For determining or setting Hermitian status on existing matrices:

• MatIsHermitian()

• MatIsHermitianKnown()

• MatIsStructurallySymmetric()

• MatIsSymmetricKnown()

• MatIsSymmetric()

• MatSetOption() (use with MAT_SYMMETRIC or MAT_HERMITIAN to assert to PETSc that either is the case).

For performing matrix operations on known Hermitian matrices (note that regular Mat functions such as MatMult() will of course also work on Hermitian matrices):

• MatMultHermitianTranspose()

• MatMultHermitianTransposeAdd() (very limited support)

How can I assemble a bunch of similar matrices?

You can first add the values common to all the matrices, then use MatStoreValues() to stash the common values. Each iteration you call MatRetrieveValues(), then set the unique values in a matrix and assemble.

Can one resize or change the size of PETSc matrices or vectors?

No, once the vector or matrices sizes have been set and the matrices or vectors are fully usuable one cannot change the size of the matrices or vectors or number of processors they live on. One may create new vectors and copy, for example using VecScatterCreate(), the old values from the previous vector.

How can one compute the nullspace of a sparse matrix with MUMPS?

Assuming you have an existing matrix \(A\) whose nullspace \(V\) you want to find:

Mat      F, work, V;
PetscInt N, rows;

/* Determine factorability */
PetscCall(MatGetFactor(A, MATSOLVERMUMPS, MAT_FACTOR_LU, &F));
PetscCall(MatGetLocalSize(A, &rows, NULL));

/* Set MUMPS options, see MUMPS documentation for more information */
PetscCall(MatMumpsSetIcntl(F, 24, 1));
PetscCall(MatMumpsSetIcntl(F, 25, 1));

/* Perform factorization */
PetscCall(MatLUFactorSymbolic(F, A, NULL, NULL, NULL));
PetscCall(MatLUFactorNumeric(F, A, NULL));

/* This is the dimension of the null space */
PetscCall(MatMumpsGetInfog(F, 28, &N));

/* This will contain the null space in the columns */
PetscCall(MatCreateDense(comm, rows, N, PETSC_DETERMINE, PETSC_DETERMINE, NULL, &V));

PetscCall(MatDuplicate(V, MAT_DO_NOT_COPY_VALUES, &work));
PetscCall(MatMatSolve(F, work, V));

---

Execution

PETSc executables are SO big and take SO long to link

Note

See shared libraries section for more information.

We find this annoying as well. On most machines PETSc can use shared libraries, so executables should be much smaller, run configure with the additional option --with-shared-libraries (this is the default). Also, if you have room, compiling and linking PETSc on your machine’s /tmp disk or similar local disk, rather than over the network will be much faster.

How does PETSc’s -help option work? Why is it different for different programs?

There are 2 ways in which one interacts with the options database:

• PetscOptionsGetXXX() where XXX is some type or data structure (for example PetscOptionsGetBool() or PetscOptionsGetScalarArray()). This is a classic “getter” function, which queries the command line options for a matching option name, and returns the specificied value.

• PetscOptionsXXX() where XXX is some type or data structure (for example PetscOptionsBool() or PetscOptionsScalarArray()). This is a so-called “provider” function. It first records the option name in an internal list of previously encountered options before calling PetscOptionsGetXXX() to query the status of said option.

While users generally use the first option, developers will always use the second (provider) variant of functions. Thus, as the program runs, it will build up a list of encountered option names which are then printed in the order of their appearance on the root rank. Different programs may take different paths through PETSc source code, so they will encounter different providers, and therefore have different -help output.

PETSc has so many options for my program that it is hard to keep them straight

Running the PETSc program with the option -help will print out many of the options. To print the options that have been specified within a program, employ -options_left to print any options that the user specified but were not actually used by the program and all options used; this is helpful for detecting typo errors. The PETSc website has a search option, in the upper right hand corner, that quickly finds answers to most PETSc questions.

PETSc automatically handles many of the details in parallel PDE solvers. How can I understand what is really happening within my program?

You can use the option -info to get more details about the solution process. The option -log_view provides details about the distribution of time spent in the various phases of the solution process. You can run with -ts_view or -snes_view or -ksp_view to see what solver options are being used. Run with -ts_monitor, -snes_monitor, or -ksp_monitor to watch convergence of the methods. -snes_converged_reason and -ksp_converged_reason will indicate why and if the solvers have converged.

Assembling large sparse matrices takes a long time. What can I do to make this process faster? Or MatSetValues() is so slow; what can I do to speed it up?

You probably need to do preallocation, as explained in Sparse Matrices. See also the performance chapter of the users manual.

For GPUs (and even CPUs) you can use MatSetPreallocationCOO() and MatSetValuesCOO() for more rapid assembly.

How can I generate performance summaries with PETSc?

Use this option at runtime:

-log_view

Outputs a comprehensive timing, memory consumption, and communications digest for your program. See the profiling chapter of the users manual for information on interpreting the summary data.

How do I know the amount of time spent on each level of the multigrid solver/preconditioner?

Run with -log_view and -pc_mg_log

Where do I get the input matrices for the examples?

Some examples use $DATAFILESPATH/matrices/medium and other files. These test matrices in PETSc binary format can be found in the datafiles repository.

When I dump some matrices and vectors to binary, I seem to be generating some empty files with .info extensions. What’s the deal with these?

PETSc binary viewers put some additional information into .info files like matrix block size. It is harmless but if you really don’t like it you can use -viewer_binary_skip_info or PetscViewerBinarySkipInfo().

Note

You need to call PetscViewerBinarySkipInfo() before PetscViewerFileSetName(). In other words you cannot use PetscViewerBinaryOpen() directly.

Why is my parallel solver slower than my sequential solver, or I have poor speed-up?

This can happen for many reasons:

1. Make sure it is truly the time in KSPSolve() that is slower (by running the code with -log_view). Often the slower time is in generating the matrix or some other operation.

2. There must be enough work for each process to outweigh the communication time. We recommend an absolute minimum of about 10,000 unknowns per process, better is 20,000 or more. This is even more true when using multiple GPUs, where you need to have millions of unknowns per GPU.

3. Make sure the communication speed of the parallel computer is good enough for parallel solvers.

4. Check the number of solver iterates with the parallel solver against the sequential solver. Most preconditioners require more iterations when used on more processes, this is particularly true for block Jacobi (the default parallel preconditioner). You can try -pc_type asm (PCASM) its iterations scale a bit better for more processes. You may also consider multigrid preconditioners like PCMG or BoomerAMG in PCHYPRE.

What steps are necessary to make the pipelined solvers execute efficiently?

Pipelined solvers like KSPPGMRES, KSPPIPECG, KSPPIPECR, and KSPGROPPCG may require special MPI configuration to effectively overlap reductions with computation. In general, this requires an MPI-3 implementation, an implementation that supports multiple threads, and use of a “progress thread”. Consult the documentation from your vendor or computing facility for more details.

Cray MPI

Cray MPT-5.6 supports MPI-3, but setting $MPICH_MAX_THREAD_SAFETY to “multiple” for threads, plus either $MPICH_ASYNC_PROGRESS or $MPICH_NEMESIS_ASYNC_PROGRESS. E.g.

$ export MPICH_MAX_THREAD_SAFETY=multiple
$ export MPICH_ASYNC_PROGRESS=1
$ export MPICH_NEMESIS_ASYNC_PROGRESS=1

MPICH

MPICH version 3.0 and later implements the MPI-3 standard and the default configuration supports use of threads. Use of a progress thread is configured by setting the environment variable $MPICH_ASYNC_PROGRESS. E.g.

$ export MPICH_ASYNC_PROGRESS=1

When using PETSc in single precision mode (--with-precision=single when running configure) are the operations done in single or double precision?

PETSc does NOT do any explicit conversion of single precision to double before performing computations; it depends on the hardware and compiler for what happens. For example, the compiler could choose to put the single precision numbers into the usual double precision registers and then use the usual double precision floating point unit. Or it could use SSE2 instructions that work directly on the single precision numbers. It is a bit of a mystery what decisions get made sometimes. There may be compiler flags in some circumstances that can affect this.

Why is Newton’s method (SNES) not converging, or converges slowly?

Newton’s method may not converge for many reasons, here are some of the most common:

• The Jacobian is wrong (or correct in sequential but not in parallel).

• The linear system is not solved or is not solved accurately enough.

• The Jacobian system has a singularity that the linear solver is not handling.

• There is a bug in the function evaluation routine.

• The function is not continuous or does not have continuous first derivatives (e.g. phase change or TVD limiters).

• The equations may not have a solution (e.g. limit cycle instead of a steady state) or there may be a “hill” between the initial guess and the steady state (e.g. reactants must ignite and burn before reaching a steady state, but the steady-state residual will be larger during combustion).

Here are some of the ways to help debug lack of convergence of Newton:

• Run on one processor to see if the problem is only in parallel.

• Run with -info to get more detailed information on the solution process.

• Run with the options

  -snes_monitor -ksp_monitor_true_residual -snes_converged_reason -ksp_converged_reason
  

  • If the linear solve does not converge, check if the Jacobian is correct, then see this question.

  • If the preconditioned residual converges, but the true residual does not, the preconditioner may be singular.

  • If the linear solve converges well, but the line search fails, the Jacobian may be incorrect.

• Run with -pc_type lu or -pc_type svd to see if the problem is a poor linear solver.

• Run with -mat_view or -mat_view draw to see if the Jacobian looks reasonable.

• Run with -snes_test_jacobian -snes_test_jacobian_view to see if the Jacobian you are using is wrong. Compare the output when you add -mat_fd_type ds to see if the result is sensitive to the choice of differencing parameter.

• Run with -snes_mf_operator -pc_type lu to see if the Jacobian you are using is wrong. If the problem is too large for a direct solve, try

  -snes_mf_operator -pc_type ksp -ksp_ksp_rtol 1e-12.
  

  Compare the output when you add -mat_mffd_type ds to see if the result is sensitive to choice of differencing parameter.

• Run with -snes_linesearch_monitor to see if the line search is failing (this is usually a sign of a bad Jacobian). Use -info in PETSc 3.1 and older versions, -snes_ls_monitor in PETSc 3.2 and -snes_linesearch_monitor in PETSc 3.3 and later.

Here are some ways to help the Newton process if everything above checks out:

• Run with grid sequencing (-snes_grid_sequence if working with a DM is all you need) to generate better initial guess on your finer mesh.

• Run with quad precision, i.e.

  $ ./configure --with-precision=__float128 --download-f2cblaslapack
  

  Note

  quad precision requires PETSc 3.2 and later and recent versions of the GNU compilers.

• Change the units (nondimensionalization), boundary condition scaling, or formulation so that the Jacobian is better conditioned. See Buckingham pi theorem and Dimensional and Scaling Analysis.

• Mollify features in the function that do not have continuous first derivatives (often occurs when there are “if” statements in the residual evaluation, e.g. phase change or TVD limiters). Use a variational inequality solver (SNESVINEWTONRSLS) if the discontinuities are of fundamental importance.

• Try a trust region method (-ts_type tr, may have to adjust parameters).

• Run with some continuation parameter from a point where you know the solution, see TSPSEUDO for steady-states.

• There are homotopy solver packages like PHCpack that can get you all possible solutions (and tell you that it has found them all) but those are not scalable and cannot solve anything but small problems.

Why is the linear solver (KSP) not converging, or converges slowly?

Tip

Always run with -ksp_converged_reason -ksp_monitor_true_residual when trying to learn why a method is not converging!

Common reasons for KSP not converging are:

• A symmetric method is being used for a non-symmetric problem.

• The equations are singular by accident (e.g. forgot to impose boundary conditions). Check this for a small problem using -pc_type svd -pc_svd_monitor.

• The equations are intentionally singular (e.g. constant null space), but the Krylov method was not informed, see MatSetNullSpace(). Always inform your local Krylov subspace solver of any change of singularity. Failure to do so will result in the immediate revocation of your computing and keyboard operator licenses, as well as a stern talking-to by the nearest Krylov Subspace Method representative.

• The equations are intentionally singular and MatSetNullSpace() was used, but the right-hand side is not consistent. You may have to call MatNullSpaceRemove() on the right-hand side before calling KSPSolve(). See MatSetTransposeNullSpace().

• The equations are indefinite so that standard preconditioners don’t work. Usually you will know this from the physics, but you can check with

  -ksp_compute_eigenvalues -ksp_gmres_restart 1000 -pc_type none
  

  For simple saddle point problems, try

  -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_detect_saddle_point
  

  For more difficult problems, read the literature to find robust methods and ask petsc-users@mcs.anl.gov or petsc-maint@mcs.anl.gov if you want advice about how to implement them.

• If the method converges in preconditioned residual, but not in true residual, the preconditioner is likely singular or nearly so. This is common for saddle point problems (e.g. incompressible flow) or strongly nonsymmetric operators (e.g. low-Mach hyperbolic problems with large time steps).

• The preconditioner is too weak or is unstable. See if -pc_type asm -sub_pc_type lu improves the convergence rate. If GMRES is losing too much progress in the restart, see if longer restarts help -ksp_gmres_restart 300. If a transpose is available, try -ksp_type bcgs or other methods that do not require a restart.

  Note

  Unfortunately convergence with these methods is frequently erratic.

• The preconditioner is nonlinear (e.g. a nested iterative solve), try -ksp_type fgmres or -ksp_type gcr.

• You are using geometric multigrid, but some equations (often boundary conditions) are not scaled compatibly between levels. Try -pc_mg_galerkin both to algebraically construct a correctly scaled coarse operator or make sure that all the equations are scaled in the same way if you want to use rediscretized coarse levels.

• The matrix is very ill-conditioned. Check the condition number.

  • Try to improve it by choosing the relative scaling of components/boundary conditions.

  • Try -ksp_diagonal_scale -ksp_diagonal_scale_fix.

  • Perhaps change the formulation of the problem to produce more friendly algebraic equations.

• Change the units (nondimensionalization), boundary condition scaling, or formulation so that the Jacobian is better conditioned. See Buckingham pi theorem and Dimensional and Scaling Analysis.

• Classical Gram-Schmidt is becoming unstable, try -ksp_gmres_modifiedgramschmidt or use a method that orthogonalizes differently, e.g. -ksp_type gcr.

I get the error message: Actual argument at (1) to assumed-type dummy is of derived type with type-bound or FINAL procedures

Use the following code-snippet:

module context_module
#include petsc/finclude/petsc.h
use petsc
implicit none
private
type, public ::  context_type
  private
  PetscInt :: foo
contains
  procedure, public :: init => context_init
end type context_type
contains
subroutine context_init(self, foo)
  class(context_type), intent(in out) :: self
  PetscInt, intent(in) :: foo
  self%foo = foo
end subroutine context_init
end module context_module

!------------------------------------------------------------------------

program test_snes
use,intrinsic :: iso_c_binding
use petsc
use context_module
implicit none

SNES :: snes
type(context_type),target :: context
type(c_ptr) :: contextOut
PetscErrorCode :: ierr

call PetscInitialize(PETSC_NULL_CHARACTER, ierr)
call SNESCreate(PETSC_COMM_WORLD, snes, ierr)
call context%init(1)

contextOut = c_loc(context) ! contextOut is a C pointer on context

call SNESSetConvergenceTest(snes, convergence, contextOut, PETSC_NULL_FUNCTION, ierr)

call SNESDestroy(snes, ierr)
call PetscFinalize(ierr)

contains

subroutine convergence(snes, num_iterations, xnorm, pnorm,fnorm, reason, contextIn, ierr)
SNES, intent(in) :: snes

PetscInt, intent(in) :: num_iterations
PetscReal, intent(in) :: xnorm, pnorm, fnorm
SNESConvergedReason, intent(out) :: reason
type(c_ptr), intent(in out) :: contextIn
type(context_type), pointer :: context
PetscErrorCode, intent(out) :: ierr

call c_f_pointer(contextIn,context)  ! convert the C pointer to a Fortran pointer to use context as in the main program
reason = 0
ierr = 0
end subroutine convergence
end program test_snes

In C++ I get a crash on VecDestroy() (or some other PETSc object) at the end of the program

This can happen when the destructor for a C++ class is automatically called at the end of the program after PetscFinalize(). Use the following code-snippet:

main()
{
  PetscCall(PetscInitialize());
  {
    your variables
    your code

    ...   /* all your destructors are called here automatically by C++ so they work correctly */
  }
  PetscCall(PetscFinalize());
  return 0
}

---

Debugging

What does the message hwloc/linux: Ignoring PCI device with non-16bit domain mean?

This is printed by the hwloc library that is used by some MPI implementations. It can be ignored. To prevent the message from always being printed set the environmental variable HWLOC_HIDE_ERRORS to 2. For example

export HWLOC_HIDE_ERRORS=2

which can be added to your login profile file such as ~/.bashrc.

How do I turn off PETSc signal handling so I can use the -C Option On xlf?

Immediately after calling PetscInitialize() call PetscPopSignalHandler().

Some Fortran compilers including the IBM xlf, xlF etc compilers have a compile option (-C for IBM’s) that causes all array access in Fortran to be checked that they are in-bounds. This is a great feature but does require that the array dimensions be set explicitly, not with a *.

How do I debug if -start_in_debugger does not work on my machine?

The script Azrael3000/tmpi can be used to run multiple MPI ranks in the debugger using tmux.

On newer macOS machines - one has to be in admin group to be able to use debugger.

On newer Ubuntu linux machines - one has to disable ptrace_scope with

$ sudo echo 0 > /proc/sys/kernel/yama/ptrace_scope

to get start in debugger working.

If -start_in_debugger does not work on your OS, for a uniprocessor job, just try the debugger directly, for example: gdb ex1. You can also use TotalView which is a good graphical parallel debugger.

How do I see where my code is hanging?

You can use the -start_in_debugger option to start all processes in the debugger (each will come up in its own xterm) or run in TotalView. Then use cont (for continue) in each xterm. Once you are sure that the program is hanging, hit control-c in each xterm and then use ‘where’ to print a stack trace for each process.

How can I inspect PETSc vector and matrix values when in the debugger?

I will illustrate this with gdb, but it should be similar on other debuggers. You can look at local Vec values directly by obtaining the array. For a Vec v, we can print all local values using:

(gdb) p ((Vec_Seq*) v->data)->array[0]@v->map.n

However, this becomes much more complicated for a matrix. Therefore, it is advisable to use the default viewer to look at the object. For a Vec v and a Mat m, this would be:

(gdb) call VecView(v, 0)
(gdb) call MatView(m, 0)

or with a communicator other than MPI_COMM_WORLD:

(gdb) call MatView(m, PETSC_VIEWER_STDOUT_(m->comm))

Totalview 8.8.0+ has a new feature that allows libraries to provide their own code to display objects in the debugger. Thus in theory each PETSc object, Vec, Mat etc could have custom code to print values in the object. We have only done this for the most elementary display of Vec and Mat. See the routine TV_display_type() in src/vec/vec/interface/vector.c for an example of how these may be written. Contact us if you would like to add more.

How can I find the cause of floating point exceptions like not-a-number (NaN) or infinity?

The best way to locate floating point exceptions is to use a debugger. On supported architectures (including Linux and glibc-based systems), just run in a debugger and pass -fp_trap to the PETSc application. This will activate signaling exceptions and the debugger will break on the line that first divides by zero or otherwise generates an exceptions.

Without a debugger, running with -fp_trap in debug mode will only identify the function in which the error occurred, but not the line or the type of exception. If -fp_trap is not supported on your architecture, consult the documentation for your debugger since there is likely a way to have it catch exceptions.

Error while loading shared libraries: libimf.so: cannot open shared object file: No such file or directory

The Intel compilers use shared libraries (like libimf) that cannot be found, by default, at run time. When using the Intel compilers (and running the resulting code) you must make sure that the proper Intel initialization scripts are run. This is usually done by adding some code into your .cshrc, .bashrc, .profile etc file. Sometimes on batch file systems that do now access your initialization files (like .cshrc) you must include the initialization calls in your batch file submission.

For example, on my Mac using csh I have the following in my .cshrc file:

source /opt/intel/cc/10.1.012/bin/iccvars.csh
source /opt/intel/fc/10.1.012/bin/ifortvars.csh
source /opt/intel/idb/10.1.012/bin/idbvars.csh

And in my .profile I have

source /opt/intel/cc/10.1.012/bin/iccvars.sh
source /opt/intel/fc/10.1.012/bin/ifortvars.sh
source /opt/intel/idb/10.1.012/bin/idbvars.sh

What does “Object Type Not Set: Argument # N” Mean?

Many operations on PETSc objects require that the specific type of the object be set before the operations is performed. You must call XXXSetType() or XXXSetFromOptions() before you make the offending call. For example

Mat A;

PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
PetscCall(MatSetValues(A,....));

will not work. You must add MatSetType() or MatSetFromOptions() before the call to MatSetValues(). I.e.

Mat A;

PetscCall(MatCreate(PETSC_COMM_WORLD, &A));

PetscCall(MatSetType(A, MATAIJ));
/* Will override MatSetType() */
PetscCall(MatSetFromOptions());

PetscCall(MatSetValues(A,....));

What does error detected in PetscSplitOwnership() about “sum of local lengths …”: mean?

In a previous call to VecSetSizes(), MatSetSizes(), VecCreateXXX() or MatCreateXXX() you passed in local and global sizes that do not make sense for the correct number of processors. For example if you pass in a local size of 2 and a global size of 100 and run on two processors, this cannot work since the sum of the local sizes is 4, not 100.

What does “corrupt argument” or “caught signal” Or “SEGV” Or “segmentation violation” Or “bus error” mean? Can I use Valgrind or CUDA-Memcheck to debug memory corruption issues?

Sometimes it can mean an argument to a function is invalid. In Fortran this may be caused by forgetting to list an argument in the call, especially the final PetscErrorCode.

Otherwise it is usually caused by memory corruption; that is somewhere the code is writing out of array bounds. To track this down rerun the debug version of the code with the option -malloc_debug. Occasionally the code may crash only with the optimized version, in that case run the optimized version with -malloc_debug. If you determine the problem is from memory corruption you can put the macro CHKMEMQ in the code near the crash to determine exactly what line is causing the problem.

If -malloc_debug does not help: on NVIDIA CUDA systems you can use https://docs.nvidia.com/cuda/cuda-memcheck/index.html

If -malloc_debug does not help: on GNU/Linux (not macOS machines) - you can use valgrind. Follow the below instructions:

1. configure PETSc with --download-mpich --with-debugging (You can use other MPI implementations but most produce spurious Valgrind messages)

2. Compile your application code with this build of PETSc.

3. Run with valgrind.

   $ $PETSC_DIR/lib/petsc/bin/petscmpiexec -valgrind -n NPROC PETSCPROGRAMNAME PROGRAMOPTIONS
   

   or

   $ mpiexec -n NPROC valgrind --tool=memcheck -q --num-callers=20 \
   --suppressions=$PETSC_DIR/share/petsc/suppressions/valgrind \
   --log-file=valgrind.log.%p PETSCPROGRAMNAME -malloc off PROGRAMOPTIONS
   

Note

• option --with-debugging enables valgrind to give stack trace with additional source-file:line-number info.

• option --download-mpich is valgrind clean, other MPI builds are not valgrind clean.

• when --download-mpich is used - mpiexec will be in $PETSC_ARCH/bin

• --log-file=valgrind.log.%p option tells valgrind to store the output from each process in a different file [as %p i.e PID, is different for each MPI process.

• memcheck will not find certain array access that violate static array declarations so if memcheck runs clean you can try the --tool=exp-ptrcheck instead.

You might also consider using http://drmemory.org which has support for GNU/Linux, Apple Mac OS and Microsoft Windows machines. (Note we haven’t tried this ourselves).

What does “detected zero pivot in LU factorization” mean?

A zero pivot in LU, ILU, Cholesky, or ICC sparse factorization does not always mean that the matrix is singular. You can use

-pc_factor_shift_type nonzero -pc_factor_shift_amount [amount]

or

-pc_factor_shift_type positive_definite -[level]_pc_factor_shift_type nonzero
 -pc_factor_shift_amount [amount]

or

-[level]_pc_factor_shift_type positive_definite

to prevent the zero pivot. [level] is “sub” when lu, ilu, cholesky, or icc are employed in each individual block of the bjacobi or ASM preconditioner. [level] is “mg_levels” or “mg_coarse” when lu, ilu, cholesky, or icc are used inside multigrid smoothers or to the coarse grid solver. See PCFactorSetShiftType(), PCFactorSetShiftAmount().

This error can also happen if your matrix is singular, see MatSetNullSpace() for how to handle this. If this error occurs in the zeroth row of the matrix, it is likely you have an error in the code that generates the matrix.

You create draw windows or PETSCVIEWERDRAW windows or use options -ksp_monitor draw::draw_lg or -snes_monitor draw::draw_lg and the program seems to run OK but windows never open

The libraries were compiled without support for X windows. Make sure that configure was run with the option --with-x.

The program seems to use more and more memory as it runs, even though you don’t think you are allocating more memory

Some of the following may be occurring:

• You are creating new PETSc objects but never freeing them.

• There is a memory leak in PETSc or your code.

• Something much more subtle: (if you are using Fortran). When you declare a large array in Fortran, the operating system does not allocate all the memory pages for that array until you start using the different locations in the array. Thus, in a code, if at each step you start using later values in the array your virtual memory usage will “continue” to increase as measured by ps or top.

• You are running with the -log, -log_mpe, or -log_all option. With these options, a great deal of logging information is stored in memory until the conclusion of the run.

• You are linking with the MPI profiling libraries; these cause logging of all MPI activities. Another symptom is at the conclusion of the run it may print some message about writing log files.

The following may help:

• Run with the -malloc_debug option and -malloc_view. Or use PetscMallocDump() and PetscMallocView() sprinkled about your code to track memory that is allocated and not later freed. Use the commands PetscMallocGetCurrentUsage() and PetscMemoryGetCurrentUsage() to monitor memory allocated and PetscMallocGetMaximumUsage() and PetscMemoryGetMaximumUsage() for total memory used as the code progresses.

• This is just the way Unix works and is harmless.

• Do not use the -log, -log_mpe, or -log_all option, or use PetscLogEventDeactivate() or PetscLogEventDeactivateClass() to turn off logging of specific events.

• Make sure you do not link with the MPI profiling libraries.

When calling MatPartitioningApply() you get a message “Error! Key 16615 Not Found”

The graph of the matrix you are using is not symmetric. You must use symmetric matrices for partitioning.

With GMRES at restart the second residual norm printed does not match the first

I.e.

26 KSP Residual norm 3.421544615851e-04
27 KSP Residual norm 2.973675659493e-04
28 KSP Residual norm 2.588642948270e-04
29 KSP Residual norm 2.268190747349e-04
30 KSP Residual norm 1.977245964368e-04
30 KSP Residual norm 1.994426291979e-04 <----- At restart the residual norm is printed a second time

This is actually not surprising! GMRES computes the norm of the residual at each iteration via a recurrence relation between the norms of the residuals at the previous iterations and quantities computed at the current iteration. It does not compute it via directly \(|| b - A x^{n} ||\).

Sometimes, especially with an ill-conditioned matrix, or computation of the matrix-vector product via differencing, the residual norms computed by GMRES start to “drift” from the correct values. At the restart, we compute the residual norm directly, hence the “strange stuff,” the difference printed. The drifting, if it remains small, is harmless (doesn’t affect the accuracy of the solution that GMRES computes).

If you use a more powerful preconditioner the drift will often be smaller and less noticeable. Of if you are running matrix-free you may need to tune the matrix-free parameters.

Why do some Krylov methods seem to print two residual norms per iteration?

I.e.

1198 KSP Residual norm 1.366052062216e-04
1198 KSP Residual norm 1.931875025549e-04
1199 KSP Residual norm 1.366026406067e-04
1199 KSP Residual norm 1.931819426344e-04

Some Krylov methods, for example KSPTFQMR, actually have a “sub-iteration” of size 2 inside the loop. Each of the two substeps has its own matrix vector product and application of the preconditioner and updates the residual approximations. This is why you get this “funny” output where it looks like there are two residual norms per iteration. You can also think of it as twice as many iterations.

Unable to locate PETSc dynamic library libpetsc

When using DYNAMIC libraries - the libraries cannot be moved after they are installed. This could also happen on clusters - where the paths are different on the (run) nodes - than on the (compile) front-end. Do not use dynamic libraries & shared libraries. Run configure with --with-shared-libraries=0 --with-dynamic-loading=0.

Important

This option has been removed in petsc v3.5

How do I determine what update to PETSc broke my code?

If at some point (in PETSc code history) you had a working code - but the latest PETSc code broke it, its possible to determine the PETSc code change that might have caused this behavior. This is achieved by:

• Using Git to access PETSc sources

• Knowing the Git commit for the known working version of PETSc

• Knowing the Git commit for the known broken version of PETSc

• Using the bisect functionality of Git

Git bisect can be done as follows:

1. Get PETSc development (main branch in git) sources

   $ git clone https://gitlab.com/petsc/petsc.git
   

2. Find the good and bad markers to start the bisection process. This can be done either by checking git log or gitk or petsc/petsc or the web history of petsc-release clones. Lets say the known bad commit is 21af4baa815c and known good commit is 5ae5ab319844.

3. Start the bisection process with these known revisions. Build PETSc, and test your code to confirm known good/bad behavior:

   $ git bisect start 21af4baa815c 5ae5ab319844
   

   build/test, perhaps discover that this new state is bad

   $ git bisect bad
   

   build/test, perhaps discover that this state is good

   $ git bisect good
   

   Now until done - keep bisecting, building PETSc, and testing your code with it and determine if the code is working or not. After something like 5-15 iterations, git bisect will pin-point the exact code change that resulted in the difference in application behavior.

Tip

See git-bisect(1) and the debugging section of the Git Book for more debugging tips.

How to fix the error “PMIX Error: error in file gds_ds12_lock_pthread.c”?

This seems to be an error when using Open MPI and OpenBLAS with threads (or perhaps other packages that use threads).

$ export PMIX_MCA_gds=hash

Should resolve the problem.

---

Shared Libraries

Can I install PETSc libraries as shared libraries?

Yes. Use

$ ./configure --with-shared-libraries

Why should I use shared libraries?

When you link to shared libraries, the function symbols from the shared libraries are not copied in the executable. This way the size of the executable is considerably smaller than when using regular libraries. This helps in a couple of ways:

• Saves disk space when more than one executable is created

• Improves the compile time immensly, because the compiler has to write a much smaller file (executable) to the disk.

How do I link to the PETSc shared libraries?

By default, the compiler should pick up the shared libraries instead of the regular ones. Nothing special should be done for this.

What if I want to link to the regular .a library files?

You must run configure with the option --with-shared-libraries=0 (you can use a different $PETSC_ARCH for this build so you can easily switch between the two).

What do I do if I want to move my executable to a different machine?

You would also need to have access to the shared libraries on this new machine. The other alternative is to build the exeutable without shared libraries by first deleting the shared libraries, and then creating the executable.

[ESW14]

Howard C. Elman, David J. Silvester, and Andrew J. Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, 2014. URL: https://global.oup.com/academic/product/finite-elements-and-fast-iterative-solvers-9780199678792?cc=us&lang=en&.