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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 ts, PetscReal t, Vec U, Vec *Vl, Vec Vr, Vec *VHV, void *ctx), 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

- ctx - [optional] user-defined function context

## Calling sequence of `rhshessianproductfunc1`

- ts - the `TS` context

- 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

All other functions have the same calling sequence as `rhhessianproductfunc1`, so their descriptions are omitted for brevity.

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

`TS`, `TSAdjoint`

## Level

intermediate

## Location

src/ts/interface/sensitivity/tssen.c

## Examples

src/ts/tutorials/ex20opt_p.c
src/ts/tutorials/ex20opt_ic.c

Index of all Sensitivity routines
Table of Contents for all manual pages
Index of all manual pages