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;
66:   PetscCall(VecGetArray(F, &f));
67:   f[0] = x[1];
68:   f[1] = user->mu * (1. - x[0] * x[0]) * x[1] - x[0];
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;
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:   }
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;
107:   J[0][0] = 0;
108:   J[1][0] = (1. - x[0] * x[0]) * x[1];
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));
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));
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*/