TSSetRHSHessianProduct#

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

Synopsis#

#include "petscts.h"  
PetscErrorCode TSSetRHSHessianProduct(TS ts, Vec *rhshp1, PetscErrorCode (*rhshessianproductfunc1)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *rhshp2, PetscErrorCode (*rhshessianproductfunc2)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *rhshp3, PetscErrorCode (*rhshessianproductfunc3)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), Vec *rhshp4, PetscErrorCode (*rhshessianproductfunc4)(TS, PetscReal, Vec, Vec *, Vec, Vec *, void *), void *ctx)

Logically Collective

Input Parameters#

ts - TS context obtained from TSCreate()
rhshp1 - an array of vectors storing the result of vector-Hessian-vector product for G_UU
hessianproductfunc1 - vector-Hessian-vector product function for G_UU
rhshp2 - an array of vectors storing the result of vector-Hessian-vector product for G_UP
hessianproductfunc2 - vector-Hessian-vector product function for G_UP
rhshp3 - an array of vectors storing the result of vector-Hessian-vector product for G_PU
hessianproductfunc3 - vector-Hessian-vector product function for G_PU
rhshp4 - an array of vectors storing the result of vector-Hessian-vector product for G_PP
hessianproductfunc4 - vector-Hessian-vector product function for G_PP

Calling sequence of `ihessianproductfunc`#

PetscErrorCode rhshessianproductfunc(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*G_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 G_UP is of size N x N x M. Each entry of G_UP corresponds to the derivative $ G_UP[i][j][k] = \frac{\partial^2 G[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 j-th entry being $ VHV_n[j] = \sum_i \sum_k {Vl_n[i] * G_UP[i][j][k] * Vr[k]} If the cost function is a scalar, there will be only one vector in Vl and VHV.