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";
4
5 /* ------------------------------------------------------------------------
6
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.
12
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).
15
16 (1) can be turned into a system of first order ODEs
17 [ y' ] = [ z ]
18 [ z' ] [ \mu (1 - y^2) z - y ]
19
20 which then we can write as a vector equation
21
22 [ u_1' ] = [ u_2 ] (2)
23 [ u_2' ] [ \mu (1 - u_1^2) u_2 - u_1 ]
24
25 which is now in the form of u_t = F(u,t).
26
27 The user provides the right-hand-side function
28
29 [ f(u,t) ] = [ u_2 ]
30 [ \mu (1 - u_1^2) u_2 - u_1 ]
31
32 the Jacobian function
33
34 df [ 0 ; 1 ]
35 -- = [ ]
36 du [ -2 \mu u_1*u_2 - 1; \mu (1 - u_1^2) ]
37
38 and the JacobainP (the Jacobian w.r.t. parameter) function
39
40 df [ 0; 0; 0 ]
41 --- = [ ]
42 d\mu [ 0; 0; (1 - u_1^2) u_2 ]
43
44 ------------------------------------------------------------------------- */
45
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 };
54
55 /*
56 User-defined routines
57 */
RHSFunction(TS ts,PetscReal t,Vec X,Vec F,PetscCtx ctx)58 static PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec X, Vec F, PetscCtx ctx)
59 {
60 User user = (User)ctx;
61 PetscScalar *f;
62 const PetscScalar *x;
63
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 }
73
RHSJacobian(TS ts,PetscReal t,Vec X,Mat A,Mat B,PetscCtx ctx)74 static PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec X, Mat A, Mat B, PetscCtx 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;
81
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 }
98
RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,PetscCtx ctx)99 static PetscErrorCode RHSJacobianP(TS ts, PetscReal t, Vec X, Mat A, PetscCtx ctx)
100 {
101 PetscInt row[] = {0, 1}, col[] = {2};
102 PetscScalar J[2][1];
103 const PetscScalar *x;
104
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));
111
112 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
113 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
114 PetscFunctionReturn(PETSC_SUCCESS);
115 }
116
117 /* Monitor timesteps and use interpolation to output at integer multiples of 0.1 */
Monitor(TS ts,PetscInt step,PetscReal t,Vec X,PetscCtx ctx)118 static PetscErrorCode Monitor(TS ts, PetscInt step, PetscReal t, Vec X, PetscCtx ctx)
119 {
120 const PetscScalar *x;
121 PetscReal tfinal, dt, tprev;
122 User user = (User)ctx;
123
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 }
134
main(int argc,char ** argv)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;
148
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!");
156
157 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
158 Set runtime options
159 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
160 user.mu = 1;
161 user.next_output = 0.0;
162
163 PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &user.mu, NULL));
164 PetscCall(PetscOptionsGetBool(NULL, NULL, "-monitor", &monitor, NULL));
165
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));
174
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));
179
180 PetscCall(MatCreateDense(PETSC_COMM_WORLD, PETSC_DECIDE, PETSC_DECIDE, 2, 3, NULL, &sp));
181 PetscCall(MatZeroEntries(sp));
182 PetscCall(MatShift(sp, 1.0));
183
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));
197
198 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
199 Set initial conditions
200 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201 PetscCall(VecGetArray(x, &x_ptr));
202
203 x_ptr[0] = 2;
204 x_ptr[1] = 0.66666654321;
205 PetscCall(VecRestoreArray(x, &x_ptr));
206 PetscCall(TSSetTimeStep(ts, .001));
207
208 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
209 Set runtime options
210 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
211 PetscCall(TSSetFromOptions(ts));
212
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));
221
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));
224
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 }
237
238 /*TEST
239
240 test:
241 args: -monitor 0 -ts_adapt_type none
242
243 TEST*/
244