1 static char help[] = "Adjoint sensitivity of a hybrid system with state-dependent switchings.\n";
2
3 /*
4 The dynamics is described by the ODE
5 u_t = A_i u
6
7 where A_1 = [ 1 -100
8 10 1 ],
9 A_2 = [ 1 10
10 -100 1 ].
11 The index i changes from 1 to 2 when u[1]=2.75u[0] and from 2 to 1 when u[1]=0.36u[0].
12 Initially u=[0 1]^T and i=1.
13
14 References:
15 + * - H. Zhang, S. Abhyankar, E. Constantinescu, M. Mihai, Discrete Adjoint Sensitivity Analysis of Hybrid Dynamical Systems With Switching,
16 IEEE Transactions on Circuits and Systems I: Regular Papers, 64(5), May 2017
17 - * - I. A. Hiskens, M.A. Pai, Trajectory Sensitivity Analysis of Hybrid Systems, IEEE Transactions on Circuits and Systems, Vol 47, No 2, February 2000
18 */
19
20 #include <petscts.h>
21
22 typedef struct {
23 PetscScalar lambda1;
24 PetscScalar lambda2;
25 PetscInt mode; /* mode flag*/
26 } AppCtx;
27
EventFunction(TS ts,PetscReal t,Vec U,PetscReal * fvalue,PetscCtx ctx)28 PetscErrorCode EventFunction(TS ts, PetscReal t, Vec U, PetscReal *fvalue, PetscCtx ctx)
29 {
30 AppCtx *actx = (AppCtx *)ctx;
31 const PetscScalar *u;
32
33 PetscFunctionBegin;
34 PetscCall(VecGetArrayRead(U, &u));
35 if (actx->mode == 1) {
36 fvalue[0] = PetscRealPart(u[1] - actx->lambda1 * u[0]);
37 } else if (actx->mode == 2) {
38 fvalue[0] = PetscRealPart(u[1] - actx->lambda2 * u[0]);
39 }
40 PetscCall(VecRestoreArrayRead(U, &u));
41 PetscFunctionReturn(PETSC_SUCCESS);
42 }
43
ShiftGradients(TS ts,Vec U,AppCtx * actx)44 PetscErrorCode ShiftGradients(TS ts, Vec U, AppCtx *actx)
45 {
46 Vec *lambda, *mu;
47 PetscScalar *x, *y;
48 const PetscScalar *u;
49 PetscScalar tmp[2], A1[2][2], A2[2], denorm;
50 PetscInt numcost;
51
52 PetscFunctionBegin;
53 PetscCall(TSGetCostGradients(ts, &numcost, &lambda, &mu));
54 PetscCall(VecGetArrayRead(U, &u));
55
56 if (actx->mode == 2) {
57 denorm = -actx->lambda1 * (u[0] - 100. * u[1]) + 1. * (10. * u[0] + u[1]);
58 A1[0][0] = 110. * u[1] * (-actx->lambda1) / denorm + 1.;
59 A1[0][1] = -110. * u[0] * (-actx->lambda1) / denorm;
60 A1[1][0] = 110. * u[1] * 1. / denorm;
61 A1[1][1] = -110. * u[0] * 1. / denorm + 1.;
62
63 A2[0] = 110. * u[1] * (-u[0]) / denorm;
64 A2[1] = -110. * u[0] * (-u[0]) / denorm;
65 } else {
66 denorm = -actx->lambda2 * (u[0] + 10. * u[1]) + 1. * (-100. * u[0] + u[1]);
67 A1[0][0] = 110. * u[1] * (actx->lambda2) / denorm + 1;
68 A1[0][1] = -110. * u[0] * (actx->lambda2) / denorm;
69 A1[1][0] = -110. * u[1] * 1. / denorm;
70 A1[1][1] = 110. * u[0] * 1. / denorm + 1.;
71
72 A2[0] = 0;
73 A2[1] = 0;
74 }
75
76 PetscCall(VecRestoreArrayRead(U, &u));
77
78 PetscCall(VecGetArray(lambda[0], &x));
79 PetscCall(VecGetArray(mu[0], &y));
80 tmp[0] = A1[0][0] * x[0] + A1[0][1] * x[1];
81 tmp[1] = A1[1][0] * x[0] + A1[1][1] * x[1];
82 y[0] = y[0] + A2[0] * x[0] + A2[1] * x[1];
83 x[0] = tmp[0];
84 x[1] = tmp[1];
85 PetscCall(VecRestoreArray(mu[0], &y));
86 PetscCall(VecRestoreArray(lambda[0], &x));
87
88 PetscCall(VecGetArray(lambda[1], &x));
89 PetscCall(VecGetArray(mu[1], &y));
90 tmp[0] = A1[0][0] * x[0] + A1[0][1] * x[1];
91 tmp[1] = A1[1][0] * x[0] + A1[1][1] * x[1];
92 y[0] = y[0] + A2[0] * x[0] + A2[1] * x[1];
93 x[0] = tmp[0];
94 x[1] = tmp[1];
95 PetscCall(VecRestoreArray(mu[1], &y));
96 PetscCall(VecRestoreArray(lambda[1], &x));
97 PetscFunctionReturn(PETSC_SUCCESS);
98 }
99
PostEventFunction(TS ts,PetscInt nevents,PetscInt event_list[],PetscReal t,Vec U,PetscBool forwardsolve,PetscCtx ctx)100 PetscErrorCode PostEventFunction(TS ts, PetscInt nevents, PetscInt event_list[], PetscReal t, Vec U, PetscBool forwardsolve, PetscCtx ctx)
101 {
102 AppCtx *actx = (AppCtx *)ctx;
103
104 PetscFunctionBegin;
105 /* PetscCall(VecView(U,PETSC_VIEWER_STDOUT_WORLD)); */
106 if (!forwardsolve) PetscCall(ShiftGradients(ts, U, actx));
107 if (actx->mode == 1) {
108 actx->mode = 2;
109 /* PetscCall(PetscPrintf(PETSC_COMM_SELF,"Change from mode 1 to 2 at t = %f \n",(double)t)); */
110 } else if (actx->mode == 2) {
111 actx->mode = 1;
112 /* PetscCall(PetscPrintf(PETSC_COMM_SELF,"Change from mode 2 to 1 at t = %f \n",(double)t)); */
113 }
114 PetscFunctionReturn(PETSC_SUCCESS);
115 }
116
117 /*
118 Defines the ODE passed to the ODE solver
119 */
IFunction(TS ts,PetscReal t,Vec U,Vec Udot,Vec F,PetscCtx ctx)120 static PetscErrorCode IFunction(TS ts, PetscReal t, Vec U, Vec Udot, Vec F, PetscCtx ctx)
121 {
122 AppCtx *actx = (AppCtx *)ctx;
123 PetscScalar *f;
124 const PetscScalar *u, *udot;
125
126 PetscFunctionBegin;
127 /* The next three lines allow us to access the entries of the vectors directly */
128 PetscCall(VecGetArrayRead(U, &u));
129 PetscCall(VecGetArrayRead(Udot, &udot));
130 PetscCall(VecGetArray(F, &f));
131
132 if (actx->mode == 1) {
133 f[0] = udot[0] - u[0] + 100 * u[1];
134 f[1] = udot[1] - 10 * u[0] - u[1];
135 } else if (actx->mode == 2) {
136 f[0] = udot[0] - u[0] - 10 * u[1];
137 f[1] = udot[1] + 100 * u[0] - u[1];
138 }
139
140 PetscCall(VecRestoreArrayRead(U, &u));
141 PetscCall(VecRestoreArrayRead(Udot, &udot));
142 PetscCall(VecRestoreArray(F, &f));
143 PetscFunctionReturn(PETSC_SUCCESS);
144 }
145
146 /*
147 Defines the Jacobian of the ODE passed to the ODE solver. See TSSetIJacobian() for the meaning of a and the Jacobian.
148 */
IJacobian(TS ts,PetscReal t,Vec U,Vec Udot,PetscReal a,Mat A,Mat B,PetscCtx ctx)149 static PetscErrorCode IJacobian(TS ts, PetscReal t, Vec U, Vec Udot, PetscReal a, Mat A, Mat B, PetscCtx ctx)
150 {
151 AppCtx *actx = (AppCtx *)ctx;
152 PetscInt rowcol[] = {0, 1};
153 PetscScalar J[2][2];
154 const PetscScalar *u, *udot;
155
156 PetscFunctionBegin;
157 PetscCall(VecGetArrayRead(U, &u));
158 PetscCall(VecGetArrayRead(Udot, &udot));
159
160 if (actx->mode == 1) {
161 J[0][0] = a - 1;
162 J[0][1] = 100;
163 J[1][0] = -10;
164 J[1][1] = a - 1;
165 } else if (actx->mode == 2) {
166 J[0][0] = a - 1;
167 J[0][1] = -10;
168 J[1][0] = 100;
169 J[1][1] = a - 1;
170 }
171 PetscCall(MatSetValues(B, 2, rowcol, 2, rowcol, &J[0][0], INSERT_VALUES));
172
173 PetscCall(VecRestoreArrayRead(U, &u));
174 PetscCall(VecRestoreArrayRead(Udot, &udot));
175
176 PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
177 PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
178 if (A != B) {
179 PetscCall(MatAssemblyBegin(B, MAT_FINAL_ASSEMBLY));
180 PetscCall(MatAssemblyEnd(B, MAT_FINAL_ASSEMBLY));
181 }
182 PetscFunctionReturn(PETSC_SUCCESS);
183 }
184
185 /* Matrix JacobianP is constant so that it only needs to be evaluated once */
RHSJacobianP(TS ts,PetscReal t,Vec X,Mat A,PetscCtx ctx)186 static PetscErrorCode RHSJacobianP(TS ts, PetscReal t, Vec X, Mat A, PetscCtx ctx)
187 {
188 PetscFunctionBeginUser;
189 PetscFunctionReturn(PETSC_SUCCESS);
190 }
191
main(int argc,char ** argv)192 int main(int argc, char **argv)
193 {
194 TS ts; /* ODE integrator */
195 Vec U; /* solution will be stored here */
196 Mat A; /* Jacobian matrix */
197 Mat Ap; /* dfdp */
198 PetscMPIInt size;
199 PetscInt n = 2;
200 PetscScalar *u, *v;
201 AppCtx app;
202 PetscInt direction[1];
203 PetscBool terminate[1];
204 Vec lambda[2], mu[2];
205 PetscReal tend;
206
207 FILE *f;
208 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
209 Initialize program
210 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
211 PetscFunctionBeginUser;
212 PetscCall(PetscInitialize(&argc, &argv, NULL, help));
213 PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
214 PetscCheck(size == 1, PETSC_COMM_WORLD, PETSC_ERR_WRONG_MPI_SIZE, "Only for sequential runs");
215 app.mode = 1;
216 app.lambda1 = 2.75;
217 app.lambda2 = 0.36;
218 tend = 0.125;
219 PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "ex1adj options", "");
220 {
221 PetscCall(PetscOptionsReal("-lambda1", "", "", app.lambda1, &app.lambda1, NULL));
222 PetscCall(PetscOptionsReal("-lambda2", "", "", app.lambda2, &app.lambda2, NULL));
223 PetscCall(PetscOptionsReal("-tend", "", "", tend, &tend, NULL));
224 }
225 PetscOptionsEnd();
226
227 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
228 Create necessary matrix and vectors
229 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
230 PetscCall(MatCreate(PETSC_COMM_WORLD, &A));
231 PetscCall(MatSetSizes(A, n, n, PETSC_DETERMINE, PETSC_DETERMINE));
232 PetscCall(MatSetType(A, MATDENSE));
233 PetscCall(MatSetFromOptions(A));
234 PetscCall(MatSetUp(A));
235
236 PetscCall(MatCreateVecs(A, &U, NULL));
237
238 PetscCall(MatCreate(PETSC_COMM_WORLD, &Ap));
239 PetscCall(MatSetSizes(Ap, n, 1, PETSC_DETERMINE, PETSC_DETERMINE));
240 PetscCall(MatSetType(Ap, MATDENSE));
241 PetscCall(MatSetFromOptions(Ap));
242 PetscCall(MatSetUp(Ap));
243 PetscCall(MatZeroEntries(Ap)); /* initialize to zeros */
244
245 PetscCall(VecGetArray(U, &u));
246 u[0] = 0;
247 u[1] = 1;
248 PetscCall(VecRestoreArray(U, &u));
249 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
250 Create timestepping solver context
251 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
252 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
253 PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
254 PetscCall(TSSetType(ts, TSCN));
255 PetscCall(TSSetIFunction(ts, NULL, (TSIFunctionFn *)IFunction, &app));
256 PetscCall(TSSetIJacobian(ts, A, A, (TSIJacobianFn *)IJacobian, &app));
257 PetscCall(TSSetRHSJacobianP(ts, Ap, RHSJacobianP, &app));
258
259 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
260 Set initial conditions
261 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
262 PetscCall(TSSetSolution(ts, U));
263
264 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
265 Save trajectory of solution so that TSAdjointSolve() may be used
266 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
267 PetscCall(TSSetSaveTrajectory(ts));
268
269 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
270 Set solver options
271 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
272 PetscCall(TSSetMaxTime(ts, tend));
273 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
274 PetscCall(TSSetTimeStep(ts, 1. / 256.));
275 PetscCall(TSSetFromOptions(ts));
276
277 /* Set directions and terminate flags for the two events */
278 direction[0] = 0;
279 terminate[0] = PETSC_FALSE;
280 PetscCall(TSSetEventHandler(ts, 1, direction, terminate, EventFunction, PostEventFunction, (void *)&app));
281
282 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
283 Run timestepping solver
284 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
285 PetscCall(TSSolve(ts, U));
286
287 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
288 Adjoint model starts here
289 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
290 PetscCall(MatCreateVecs(A, &lambda[0], NULL));
291 PetscCall(MatCreateVecs(A, &lambda[1], NULL));
292 /* Set initial conditions for the adjoint integration */
293 PetscCall(VecZeroEntries(lambda[0]));
294 PetscCall(VecZeroEntries(lambda[1]));
295 PetscCall(VecGetArray(lambda[0], &u));
296 u[0] = 1.;
297 PetscCall(VecRestoreArray(lambda[0], &u));
298 PetscCall(VecGetArray(lambda[1], &u));
299 u[1] = 1.;
300 PetscCall(VecRestoreArray(lambda[1], &u));
301
302 PetscCall(MatCreateVecs(Ap, &mu[0], NULL));
303 PetscCall(MatCreateVecs(Ap, &mu[1], NULL));
304 PetscCall(VecZeroEntries(mu[0]));
305 PetscCall(VecZeroEntries(mu[1]));
306 PetscCall(TSSetCostGradients(ts, 2, lambda, mu));
307
308 PetscCall(TSAdjointSolve(ts));
309
310 /*
311 PetscCall(VecView(lambda[0],PETSC_VIEWER_STDOUT_WORLD));
312 PetscCall(VecView(lambda[1],PETSC_VIEWER_STDOUT_WORLD));
313 PetscCall(VecView(mu[0],PETSC_VIEWER_STDOUT_WORLD));
314 PetscCall(VecView(mu[1],PETSC_VIEWER_STDOUT_WORLD));
315 */
316 PetscCall(VecGetArray(mu[0], &u));
317 PetscCall(VecGetArray(mu[1], &v));
318 f = fopen("adj_mu.out", "a");
319 PetscCall(PetscFPrintf(PETSC_COMM_WORLD, f, "%20.15lf %20.15lf %20.15lf\n", (double)tend, (double)PetscRealPart(u[0]), (double)PetscRealPart(v[0])));
320 PetscCall(VecRestoreArray(mu[0], &u));
321 PetscCall(VecRestoreArray(mu[1], &v));
322 fclose(f);
323 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
324 Free work space. All PETSc objects should be destroyed when they are no longer needed.
325 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
326 PetscCall(MatDestroy(&A));
327 PetscCall(VecDestroy(&U));
328 PetscCall(TSDestroy(&ts));
329
330 PetscCall(MatDestroy(&Ap));
331 PetscCall(VecDestroy(&lambda[0]));
332 PetscCall(VecDestroy(&lambda[1]));
333 PetscCall(VecDestroy(&mu[0]));
334 PetscCall(VecDestroy(&mu[1]));
335 PetscCall(PetscFinalize());
336 return 0;
337 }
338
339 /*TEST
340
341 build:
342 requires: !complex
343
344 test:
345 args: -ts_monitor -ts_adjoint_monitor
346
347 TEST*/
348