TSSetIHessianProduct#

Sets the function that computes the vector-Hessian-vector product. The Hessian is the second-order derivative of F (IFunction) w.r.t. the state variable.

Synopsis#

#include "petscts.h"  
PetscErrorCode TSSetIHessianProduct(TS ts, Vec *ihp1, PetscErrorCode (*ihessianproductfunc1)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *ihp2, PetscErrorCode (*ihessianproductfunc2)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *ihp3, PetscErrorCode (*ihessianproductfunc3)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *ihp4, PetscErrorCode (*ihessianproductfunc4)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), void *ctx)

Logically Collective

Input Parameters#

ts - TS context obtained from TSCreate()
ihp1 - an array of vectors storing the result of vector-Hessian-vector product for F_UU
hessianproductfunc1 - vector-Hessian-vector product function for F_UU
ihp2 - an array of vectors storing the result of vector-Hessian-vector product for F_UP
hessianproductfunc2 - vector-Hessian-vector product function for F_UP
ihp3 - an array of vectors storing the result of vector-Hessian-vector product for F_PU
hessianproductfunc3 - vector-Hessian-vector product function for F_PU
ihp4 - an array of vectors storing the result of vector-Hessian-vector product for F_PP
hessianproductfunc4 - vector-Hessian-vector product function for F_PP

Calling sequence of `ihessianproductfunc`#

PetscErrorCode ihessianproductfunc(TS ts, PetscReal t, Vec U, Vec *Vl, Vec Vr, Vec *VHV, void *ctx);

t - current timestep
U - input vector (current ODE solution)
Vl - an array of input vectors to be left-multiplied with the Hessian
Vr - input vector to be right-multiplied with the Hessian
VHV - an array of output vectors for vector-Hessian-vector product
ctx - [optional] user-defined function context

Notes#

The first Hessian function and the working array are required. As an example to implement the callback functions, the second callback function calculates the vector-Hessian-vector product $ Vl_n^T*F_UP*Vr where the vector Vl_n (n-th element in the array Vl) and Vr are of size N and M respectively, and the Hessian F_UP is of size N x N x M. Each entry of F_UP corresponds to the derivative $ F_UP[i][j][k] = \frac{\partial^2 F[i]}{\partial U[j] \partial P[k]}. The result of the vector-Hessian-vector product for Vl_n needs to be stored in vector VHV_n with the j-th entry being $ VHV_n[j] = \sum_i \sum_k {Vl_n[i] * F_UP[i][j][k] * Vr[k]} If the cost function is a scalar, there will be only one vector in Vl and VHV.