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;

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

 83:   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:   MatSetValues(A, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES);
 89:   MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
 90:   MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
 91:   if (A != B) {
 92:     MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY);
 93:     MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY);
 94:   }
 95:   VecRestoreArrayRead(X, &x);
 96:   return 0;
 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;

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

112:   MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
113:   MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
114:   return 0;
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;

125:   TSGetTimeStep(ts, &dt);
126:   TSGetMaxTime(ts, &tfinal);
127:   TSGetPrevTime(ts, &tprev);
128:   VecGetArrayRead(X, &x);
129:   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:   PetscPrintf(PETSC_COMM_WORLD, "t %.6f (tprev = %.6f) \n", (double)t, (double)tprev);
131:   VecRestoreArrayRead(X, &x);
132:   return 0;
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:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
153:   PetscInitialize(&argc, &argv, NULL, help);
154:   MPI_Comm_size(PETSC_COMM_WORLD, &size);

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

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

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

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

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

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

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

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

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

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

222:   PetscPrintf(PETSC_COMM_WORLD, "\n forward sensitivity: d[y(tf) z(tf)]/d[y0 z0 mu]\n");
223:   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:   MatDestroy(&A);
230:   MatDestroy(&Jacp);
231:   VecDestroy(&x);
232:   MatDestroy(&sp);
233:   TSDestroy(&ts);
234:   PetscFinalize();
235:   return 0;
236: }

238: /*TEST

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

243: TEST*/