Actual source code: ex16fwd.c

      1: static char help[] = "Performs adjoint sensitivity analysis for the van der Pol equation.\n\
      2: Input parameters include:\n\
      3:       -mu : stiffness parameter\n\n";

      5: /* ------------------------------------------------------------------------

      7:    This program solves the van der Pol equation
      8:        y'' - \mu (1-y^2)*y' + y = 0        (1)
      9:    on the domain 0 <= x <= 1, with the boundary conditions
     10:        y(0) = 2, y'(0) = 0,
     11:    and computes the sensitivities of the final solution w.r.t. initial conditions and parameter \mu with an explicit Runge-Kutta method and its discrete tangent linear model.

     13:    Notes:
     14:    This code demonstrates the TSForward interface to a system of ordinary differential equations (ODEs) in the form of u_t = f(u,t).

     16:    (1) can be turned into a system of first order ODEs
     17:    [ y' ] = [          z          ]
     18:    [ z' ]   [ \mu (1 - y^2) z - y ]

     20:    which then we can write as a vector equation

     22:    [ u_1' ] = [             u_2           ]  (2)
     23:    [ u_2' ]   [ \mu (1 - u_1^2) u_2 - u_1 ]

     25:    which is now in the form of u_t = F(u,t).

     27:    The user provides the right-hand-side function

     29:    [ f(u,t) ] = [ u_2                       ]
     30:                 [ \mu (1 - u_1^2) u_2 - u_1 ]

     32:    the Jacobian function

     34:    df   [       0           ;         1        ]
     35:    -- = [                                      ]
     36:    du   [ -2 \mu u_1*u_2 - 1;  \mu (1 - u_1^2) ]

     38:    and the JacobainP (the Jacobian w.r.t. parameter) function

     40:    df      [  0;   0;     0             ]
     41:    ---   = [                            ]
     42:    d\mu    [  0;   0;  (1 - u_1^2) u_2  ]

     44:   ------------------------------------------------------------------------- */

     46: #include <petscts.h>
     47: #include <petscmat.h>
     48: typedef struct _n_User *User;
     49: struct _n_User {
     50:   PetscReal mu;
     51:   PetscReal next_output;
     52:   PetscReal tprev;
     53: };

     55: /*
     56:    User-defined routines
     57: */
     58: static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, void *ctx)
     59: {
     60:   User               user = (User)ctx;
     61:   PetscScalar       *f;
     62:   const PetscScalar *x;

     64:   PetscFunctionBeginUser;
     65:   PetscCall(VecGetArrayRead(X, &x));
     66:   PetscCall(VecGetArray(F, &f));
     67:   f[0] = x[1];
     68:   f[1] = user->mu * (1. - x[0] * x[0]) * x[1] - x[0];
     69:   PetscCall(VecRestoreArrayRead(X, &x));
     70:   PetscCall(VecRestoreArray(F, &f));
     71:   PetscFunctionReturn(PETSC_SUCCESS);
     72: }

     74: static PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec X, Mat A, Mat B, void *ctx)
     75: {
     76:   User               user     = (User)ctx;
     77:   PetscReal          mu       = user->mu;
     78:   PetscInt           rowcol[] = {0, 1};
     79:   PetscScalar        J[2][2];
     80:   const PetscScalar *x;

     82:   PetscFunctionBeginUser;
     83:   PetscCall(VecGetArrayRead(X, &x));
     84:   J[0][0] = 0;
     85:   J[1][0] = -2. * mu * x[1] * x[0] - 1.;
     86:   J[0][1] = 1.0;
     87:   J[1][1] = mu * (1.0 - x[0] * x[0]);
     88:   PetscCall(MatSetValues(A, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
     89:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
     90:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
     91:   if (A != B) {
     92:     PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
     93:     PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
     94:   }
     95:   PetscCall(VecRestoreArrayRead(X, &x));
     96:   PetscFunctionReturn(PETSC_SUCCESS);
     97: }

     99: static PetscErrorCode RHSJacobianP(TS ts, PetscReal t, Vec X, Mat A, void *ctx)
    100: {
    101:   PetscInt           row[] = {0, 1}, col[] = {2};
    102:   PetscScalar        J[2][1];
    103:   const PetscScalar *x;

    105:   PetscFunctionBeginUser;
    106:   PetscCall(VecGetArrayRead(X, &x));
    107:   J[0][0] = 0;
    108:   J[1][0] = (1. - x[0] * x[0]) * x[1];
    109:   PetscCall(VecRestoreArrayRead(X, &x));
    110:   PetscCall(MatSetValues(A, 2, row, 1, col, &J[0][0], INSERT_VALUES));

    112:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    113:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    114:   PetscFunctionReturn(PETSC_SUCCESS);
    115: }

    117: /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
    118: static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, void *ctx)
    119: {
    120:   const PetscScalar *x;
    121:   PetscReal          tfinal, dt, tprev;
    122:   User               user = (User)ctx;

    124:   PetscFunctionBeginUser;
    125:   PetscCall(TSGetTimeStep(ts, &dt));
    126:   PetscCall(TSGetMaxTime(ts, &tfinal));
    127:   PetscCall(TSGetPrevTime(ts, &tprev));
    128:   PetscCall(VecGetArrayRead(X, &x));
    129:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "[%.1f] %" PetscInt_FMT " TS %.6f (dt = %.6f) X % 12.6e % 12.6e\n", (double)user->next_output, step, (double)t, (double)dt, (double)PetscRealPart(x[0]), (double)PetscRealPart(x[1])));
    130:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "t %.6f (tprev = %.6f) \n", (double)t, (double)tprev));
    131:   PetscCall(VecRestoreArrayRead(X, &x));
    132:   PetscFunctionReturn(PETSC_SUCCESS);
    133: }

    135: int main(int argc, char **argv)
    136: {
    137:   TS             ts;   /* nonlinear solver */
    138:   Vec            x;    /* solution, residual vectors */
    139:   Mat            A;    /* Jacobian matrix */
    140:   Mat            Jacp; /* JacobianP matrix */
    141:   PetscInt       steps;
    142:   PetscReal      ftime   = 0.5;
    143:   PetscBool      monitor = PETSC_FALSE;
    144:   PetscScalar   *x_ptr;
    145:   PetscMPIInt    size;
    146:   struct _n_User user;
    147:   Mat            sp;

    149:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    150:      Initialize program
    151:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    152:   PetscFunctionBeginUser;
    153:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    154:   PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
    155:   PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "This is a uniprocessor example only!");

    157:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    158:     Set runtime options
    159:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    160:   user.mu          = 1;
    161:   user.next_output = 0.0;

    163:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL));
    164:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL));

    166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    167:     Create necessary matrix and vectors, solve same ODE on every process
    168:     - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    169:   PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
    170:   PetscCall(MatSetSizes(A, PETSC_DECIDE, PETSC_DECIDE, 2, 2));
    171:   PetscCall(MatSetFromOptions(A));
    172:   PetscCall(MatSetUp(A));
    173:   PetscCall(MatCreateVecs(A, &x, NULL));

    175:   PetscCall(MatCreate(PETSC_COMM_WORLD, &Jacp));
    176:   PetscCall(MatSetSizes(Jacp, PETSC_DECIDE, PETSC_DECIDE, 2, 3));
    177:   PetscCall(MatSetFromOptions(Jacp));
    178:   PetscCall(MatSetUp(Jacp));

    180:   PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, 2, 3, NULL, &sp));
    181:   PetscCall(MatZeroEntries(sp));
    182:   PetscCall(MatShift(sp, 1.0));

    184:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    185:      Create timestepping solver context
    186:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    187:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
    188:   PetscCall(TSSetType(ts, TSRK));
    189:   PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &user));
    190:   /*   Set RHS Jacobian for the adjoint integration */
    191:   PetscCall(TSSetRHSJacobian(ts, A, A, RHSJacobian, &user));
    192:   PetscCall(TSSetMaxTime(ts, ftime));
    193:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
    194:   if (monitor) PetscCall(TSMonitorSet(ts, Monitor, &user, NULL));
    195:   PetscCall(TSForwardSetSensitivities(ts, 3, sp));
    196:   PetscCall(TSSetRHSJacobianP(ts, Jacp, RHSJacobianP, &user));

    198:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    199:      Set initial conditions
    200:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    201:   PetscCall(VecGetArray(x, &x_ptr));

    203:   x_ptr[0] = 2;
    204:   x_ptr[1] = 0.66666654321;
    205:   PetscCall(VecRestoreArray(x, &x_ptr));
    206:   PetscCall(TSSetTimeStep(ts, .001));

    208:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    209:      Set runtime options
    210:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    211:   PetscCall(TSSetFromOptions(ts));

    213:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    214:      Solve nonlinear system
    215:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    216:   PetscCall(TSSolve(ts, x));
    217:   PetscCall(TSGetSolveTime(ts, &ftime));
    218:   PetscCall(TSGetStepNumber(ts, &steps));
    219:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "mu %g, steps %" PetscInt_FMT ", ftime %g\n", (double)user.mu, steps, (double)ftime));
    220:   PetscCall(VecView(x, PETSC_VIEWER_STDOUT_WORLD));

    222:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "\n forward sensitivity: d[y(tf) z(tf)]/d[y0 z0 mu]\n"));
    223:   PetscCall(MatView(sp, PETSC_VIEWER_STDOUT_WORLD));

    225:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    226:      Free work space.  All PETSc objects should be destroyed when they
    227:      are no longer needed.
    228:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    229:   PetscCall(MatDestroy(&A));
    230:   PetscCall(MatDestroy(&Jacp));
    231:   PetscCall(VecDestroy(&x));
    232:   PetscCall(MatDestroy(&sp));
    233:   PetscCall(TSDestroy(&ts));
    234:   PetscCall(PetscFinalize());
    235:   return 0;
    236: }

    238: /*TEST

    240:     test:
    241:       args: -monitor 0 -ts_adapt_type none

    243: TEST*/