PETSc version 3.16.5
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# 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 on TS

### 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

    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.

### See Also

### Level

intermediate

### Location

src/ts/interface/sensitivity/tssen.c

### Examples

src/ts/tutorials/ex20opt_p.c.html
src/ts/tutorials/ex20opt_ic.c.html
Index of all Sensitivity routines
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