Actual source code: burgers_spectral.c
1: static char help[] = "Solves a simple data assimilation problem with one dimensional Burger's equation using TSAdjoint\n\n";
3: /*
5: Not yet tested in parallel
7: */
9: /* ------------------------------------------------------------------------
11: This program uses the one-dimensional Burger's equation
12: u_t = mu*u_xx - u u_x,
13: on the domain 0 <= x <= 1, with periodic boundary conditions
15: to demonstrate solving a data assimilation problem of finding the initial conditions
16: to produce a given solution at a fixed time.
18: The operators are discretized with the spectral element method
20: See the paper PDE-CONSTRAINED OPTIMIZATION WITH SPECTRAL ELEMENTS USING PETSC AND TAO
21: by OANA MARIN, EMIL CONSTANTINESCU, AND BARRY SMITH for details on the exact solution
22: used
24: ------------------------------------------------------------------------- */
26: #include <petsctao.h>
27: #include <petscts.h>
28: #include <petscdt.h>
29: #include <petscdraw.h>
30: #include <petscdmda.h>
32: /*
33: User-defined application context - contains data needed by the
34: application-provided call-back routines.
35: */
37: typedef struct {
38: PetscInt n; /* number of nodes */
39: PetscReal *nodes; /* GLL nodes */
40: PetscReal *weights; /* GLL weights */
41: } PetscGLL;
43: typedef struct {
44: PetscInt N; /* grid points per elements*/
45: PetscInt E; /* number of elements */
46: PetscReal tol_L2, tol_max; /* error norms */
47: PetscInt steps; /* number of timesteps */
48: PetscReal Tend; /* endtime */
49: PetscReal mu; /* viscosity */
50: PetscReal L; /* total length of domain */
51: PetscReal Le;
52: PetscReal Tadj;
53: } PetscParam;
55: typedef struct {
56: Vec obj; /* desired end state */
57: Vec grid; /* total grid */
58: Vec grad;
59: Vec ic;
60: Vec curr_sol;
61: Vec true_solution; /* actual initial conditions for the final solution */
62: } PetscData;
64: typedef struct {
65: Vec grid; /* total grid */
66: Vec mass; /* mass matrix for total integration */
67: Mat stiff; /* stifness matrix */
68: Mat keptstiff;
69: Mat grad;
70: PetscGLL gll;
71: } PetscSEMOperators;
73: typedef struct {
74: DM da; /* distributed array data structure */
75: PetscSEMOperators SEMop;
76: PetscParam param;
77: PetscData dat;
78: TS ts;
79: PetscReal initial_dt;
80: } AppCtx;
82: /*
83: User-defined routines
84: */
85: extern PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal *, Vec, void *);
86: extern PetscErrorCode RHSMatrixLaplaciangllDM(TS, PetscReal, Vec, Mat, Mat, void *);
87: extern PetscErrorCode RHSMatrixAdvectiongllDM(TS, PetscReal, Vec, Mat, Mat, void *);
88: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
89: extern PetscErrorCode TrueSolution(Vec, AppCtx *);
90: extern PetscErrorCode ComputeObjective(PetscReal, Vec, AppCtx *);
91: extern PetscErrorCode MonitorError(Tao, void *);
92: extern PetscErrorCode RHSFunction(TS, PetscReal, Vec, Vec, void *);
93: extern PetscErrorCode RHSJacobian(TS, PetscReal, Vec, Mat, Mat, void *);
95: int main(int argc, char **argv)
96: {
97: AppCtx appctx; /* user-defined application context */
98: Tao tao;
99: Vec u; /* approximate solution vector */
100: PetscInt i, xs, xm, ind, j, lenglob;
101: PetscReal x, *wrk_ptr1, *wrk_ptr2;
102: MatNullSpace nsp;
103: PetscMPIInt size;
105: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106: Initialize program and set problem parameters
107: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
108: PetscFunctionBeginUser;
109: PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));
111: /*initialize parameters */
112: appctx.param.N = 10; /* order of the spectral element */
113: appctx.param.E = 10; /* number of elements */
114: appctx.param.L = 4.0; /* length of the domain */
115: appctx.param.mu = 0.01; /* diffusion coefficient */
116: appctx.initial_dt = 5e-3;
117: appctx.param.steps = PETSC_MAX_INT;
118: appctx.param.Tend = 4;
120: PetscCall(PetscOptionsGetInt(NULL, NULL, "-N", &appctx.param.N, NULL));
121: PetscCall(PetscOptionsGetInt(NULL, NULL, "-E", &appctx.param.E, NULL));
122: PetscCall(PetscOptionsGetReal(NULL, NULL, "-Tend", &appctx.param.Tend, NULL));
123: PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &appctx.param.mu, NULL));
124: appctx.param.Le = appctx.param.L / appctx.param.E;
126: PetscCallMPI(MPI_Comm_size(PETSC_COMM_WORLD, &size));
127: PetscCheck((appctx.param.E % size) == 0, PETSC_COMM_WORLD, PETSC_ERR_ARG_WRONG, "Number of elements must be divisible by number of processes");
129: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
130: Create GLL data structures
131: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132: PetscCall(PetscMalloc2(appctx.param.N, &appctx.SEMop.gll.nodes, appctx.param.N, &appctx.SEMop.gll.weights));
133: PetscCall(PetscDTGaussLobattoLegendreQuadrature(appctx.param.N, PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA, appctx.SEMop.gll.nodes, appctx.SEMop.gll.weights));
134: appctx.SEMop.gll.n = appctx.param.N;
135: lenglob = appctx.param.E * (appctx.param.N - 1);
137: /*
138: Create distributed array (DMDA) to manage parallel grid and vectors
139: and to set up the ghost point communication pattern. There are E*(Nl-1)+1
140: total grid values spread equally among all the processors, except first and last
141: */
143: PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, lenglob, 1, 1, NULL, &appctx.da));
144: PetscCall(DMSetFromOptions(appctx.da));
145: PetscCall(DMSetUp(appctx.da));
147: /*
148: Extract global and local vectors from DMDA; we use these to store the
149: approximate solution. Then duplicate these for remaining vectors that
150: have the same types.
151: */
153: PetscCall(DMCreateGlobalVector(appctx.da, &u));
154: PetscCall(VecDuplicate(u, &appctx.dat.ic));
155: PetscCall(VecDuplicate(u, &appctx.dat.true_solution));
156: PetscCall(VecDuplicate(u, &appctx.dat.obj));
157: PetscCall(VecDuplicate(u, &appctx.SEMop.grid));
158: PetscCall(VecDuplicate(u, &appctx.SEMop.mass));
159: PetscCall(VecDuplicate(u, &appctx.dat.curr_sol));
161: PetscCall(DMDAGetCorners(appctx.da, &xs, NULL, NULL, &xm, NULL, NULL));
162: PetscCall(DMDAVecGetArray(appctx.da, appctx.SEMop.grid, &wrk_ptr1));
163: PetscCall(DMDAVecGetArray(appctx.da, appctx.SEMop.mass, &wrk_ptr2));
165: /* Compute function over the locally owned part of the grid */
167: xs = xs / (appctx.param.N - 1);
168: xm = xm / (appctx.param.N - 1);
170: /*
171: Build total grid and mass over entire mesh (multi-elemental)
172: */
174: for (i = xs; i < xs + xm; i++) {
175: for (j = 0; j < appctx.param.N - 1; j++) {
176: x = (appctx.param.Le / 2.0) * (appctx.SEMop.gll.nodes[j] + 1.0) + appctx.param.Le * i;
177: ind = i * (appctx.param.N - 1) + j;
178: wrk_ptr1[ind] = x;
179: wrk_ptr2[ind] = .5 * appctx.param.Le * appctx.SEMop.gll.weights[j];
180: if (j == 0) wrk_ptr2[ind] += .5 * appctx.param.Le * appctx.SEMop.gll.weights[j];
181: }
182: }
183: PetscCall(DMDAVecRestoreArray(appctx.da, appctx.SEMop.grid, &wrk_ptr1));
184: PetscCall(DMDAVecRestoreArray(appctx.da, appctx.SEMop.mass, &wrk_ptr2));
186: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
187: Create matrix data structure; set matrix evaluation routine.
188: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
189: PetscCall(DMSetMatrixPreallocateOnly(appctx.da, PETSC_TRUE));
190: PetscCall(DMCreateMatrix(appctx.da, &appctx.SEMop.stiff));
191: PetscCall(DMCreateMatrix(appctx.da, &appctx.SEMop.grad));
192: /*
193: For linear problems with a time-dependent f(u,t) in the equation
194: u_t = f(u,t), the user provides the discretized right-hand side
195: as a time-dependent matrix.
196: */
197: PetscCall(RHSMatrixLaplaciangllDM(appctx.ts, 0.0, u, appctx.SEMop.stiff, appctx.SEMop.stiff, &appctx));
198: PetscCall(RHSMatrixAdvectiongllDM(appctx.ts, 0.0, u, appctx.SEMop.grad, appctx.SEMop.grad, &appctx));
199: /*
200: For linear problems with a time-dependent f(u,t) in the equation
201: u_t = f(u,t), the user provides the discretized right-hand side
202: as a time-dependent matrix.
203: */
205: PetscCall(MatDuplicate(appctx.SEMop.stiff, MAT_COPY_VALUES, &appctx.SEMop.keptstiff));
207: /* attach the null space to the matrix, this probably is not needed but does no harm */
208: PetscCall(MatNullSpaceCreate(PETSC_COMM_WORLD, PETSC_TRUE, 0, NULL, &nsp));
209: PetscCall(MatSetNullSpace(appctx.SEMop.stiff, nsp));
210: PetscCall(MatSetNullSpace(appctx.SEMop.keptstiff, nsp));
211: PetscCall(MatNullSpaceTest(nsp, appctx.SEMop.stiff, NULL));
212: PetscCall(MatNullSpaceDestroy(&nsp));
213: /* attach the null space to the matrix, this probably is not needed but does no harm */
214: PetscCall(MatNullSpaceCreate(PETSC_COMM_WORLD, PETSC_TRUE, 0, NULL, &nsp));
215: PetscCall(MatSetNullSpace(appctx.SEMop.grad, nsp));
216: PetscCall(MatNullSpaceTest(nsp, appctx.SEMop.grad, NULL));
217: PetscCall(MatNullSpaceDestroy(&nsp));
219: /* Create the TS solver that solves the ODE and its adjoint; set its options */
220: PetscCall(TSCreate(PETSC_COMM_WORLD, &appctx.ts));
221: PetscCall(TSSetProblemType(appctx.ts, TS_NONLINEAR));
222: PetscCall(TSSetType(appctx.ts, TSRK));
223: PetscCall(TSSetDM(appctx.ts, appctx.da));
224: PetscCall(TSSetTime(appctx.ts, 0.0));
225: PetscCall(TSSetTimeStep(appctx.ts, appctx.initial_dt));
226: PetscCall(TSSetMaxSteps(appctx.ts, appctx.param.steps));
227: PetscCall(TSSetMaxTime(appctx.ts, appctx.param.Tend));
228: PetscCall(TSSetExactFinalTime(appctx.ts, TS_EXACTFINALTIME_MATCHSTEP));
229: PetscCall(TSSetTolerances(appctx.ts, 1e-7, NULL, 1e-7, NULL));
230: PetscCall(TSSetFromOptions(appctx.ts));
231: /* Need to save initial timestep user may have set with -ts_dt so it can be reset for each new TSSolve() */
232: PetscCall(TSGetTimeStep(appctx.ts, &appctx.initial_dt));
233: PetscCall(TSSetRHSFunction(appctx.ts, NULL, RHSFunction, &appctx));
234: PetscCall(TSSetRHSJacobian(appctx.ts, appctx.SEMop.stiff, appctx.SEMop.stiff, RHSJacobian, &appctx));
236: /* Set Objective and Initial conditions for the problem and compute Objective function (evolution of true_solution to final time */
237: PetscCall(InitialConditions(appctx.dat.ic, &appctx));
238: PetscCall(TrueSolution(appctx.dat.true_solution, &appctx));
239: PetscCall(ComputeObjective(appctx.param.Tend, appctx.dat.obj, &appctx));
241: PetscCall(TSSetSaveTrajectory(appctx.ts));
242: PetscCall(TSSetFromOptions(appctx.ts));
244: /* Create TAO solver and set desired solution method */
245: PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
246: PetscCall(TaoMonitorSet(tao, MonitorError, &appctx, NULL));
247: PetscCall(TaoSetType(tao, TAOBQNLS));
248: PetscCall(TaoSetSolution(tao, appctx.dat.ic));
249: /* Set routine for function and gradient evaluation */
250: PetscCall(TaoSetObjectiveAndGradient(tao, NULL, FormFunctionGradient, (void *)&appctx));
251: /* Check for any TAO command line options */
252: PetscCall(TaoSetTolerances(tao, 1e-8, PETSC_DEFAULT, PETSC_DEFAULT));
253: PetscCall(TaoSetFromOptions(tao));
254: PetscCall(TaoSolve(tao));
256: PetscCall(TaoDestroy(&tao));
257: PetscCall(MatDestroy(&appctx.SEMop.stiff));
258: PetscCall(MatDestroy(&appctx.SEMop.keptstiff));
259: PetscCall(MatDestroy(&appctx.SEMop.grad));
260: PetscCall(VecDestroy(&u));
261: PetscCall(VecDestroy(&appctx.dat.ic));
262: PetscCall(VecDestroy(&appctx.dat.true_solution));
263: PetscCall(VecDestroy(&appctx.dat.obj));
264: PetscCall(VecDestroy(&appctx.SEMop.grid));
265: PetscCall(VecDestroy(&appctx.SEMop.mass));
266: PetscCall(VecDestroy(&appctx.dat.curr_sol));
267: PetscCall(PetscFree2(appctx.SEMop.gll.nodes, appctx.SEMop.gll.weights));
268: PetscCall(DMDestroy(&appctx.da));
269: PetscCall(TSDestroy(&appctx.ts));
271: /*
272: Always call PetscFinalize() before exiting a program. This routine
273: - finalizes the PETSc libraries as well as MPI
274: - provides summary and diagnostic information if certain runtime
275: options are chosen (e.g., -log_view).
276: */
277: PetscCall(PetscFinalize());
278: return 0;
279: }
281: /* --------------------------------------------------------------------- */
282: /*
283: InitialConditions - Computes the initial conditions for the Tao optimization solve (these are also initial conditions for the first TSSolve()
285: The routine TrueSolution() computes the true solution for the Tao optimization solve which means they are the initial conditions for the objective function
287: Input Parameter:
288: u - uninitialized solution vector (global)
289: appctx - user-defined application context
291: Output Parameter:
292: u - vector with solution at initial time (global)
293: */
294: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
295: {
296: PetscScalar *s;
297: const PetscScalar *xg;
298: PetscInt i, xs, xn;
300: PetscFunctionBegin;
301: PetscCall(DMDAVecGetArray(appctx->da, u, &s));
302: PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
303: PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));
304: for (i = xs; i < xs + xn; i++) s[i] = 2.0 * appctx->param.mu * PETSC_PI * PetscSinScalar(PETSC_PI * xg[i]) / (2.0 + PetscCosScalar(PETSC_PI * xg[i])) + 0.25 * PetscExpReal(-4.0 * PetscPowReal(xg[i] - 2.0, 2.0));
305: PetscCall(DMDAVecRestoreArray(appctx->da, u, &s));
306: PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
307: PetscFunctionReturn(PETSC_SUCCESS);
308: }
310: /*
311: TrueSolution() computes the true solution for the Tao optimization solve which means they are the initial conditions for the objective function.
313: InitialConditions() computes the initial conditions for the beginning of the Tao iterations
315: Input Parameter:
316: u - uninitialized solution vector (global)
317: appctx - user-defined application context
319: Output Parameter:
320: u - vector with solution at initial time (global)
321: */
322: PetscErrorCode TrueSolution(Vec u, AppCtx *appctx)
323: {
324: PetscScalar *s;
325: const PetscScalar *xg;
326: PetscInt i, xs, xn;
328: PetscFunctionBegin;
329: PetscCall(DMDAVecGetArray(appctx->da, u, &s));
330: PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
331: PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));
332: for (i = xs; i < xs + xn; i++) s[i] = 2.0 * appctx->param.mu * PETSC_PI * PetscSinScalar(PETSC_PI * xg[i]) / (2.0 + PetscCosScalar(PETSC_PI * xg[i]));
333: PetscCall(DMDAVecRestoreArray(appctx->da, u, &s));
334: PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
335: PetscFunctionReturn(PETSC_SUCCESS);
336: }
337: /* --------------------------------------------------------------------- */
338: /*
339: Sets the desired profile for the final end time
341: Input Parameters:
342: t - final time
343: obj - vector storing the desired profile
344: appctx - user-defined application context
346: */
347: PetscErrorCode ComputeObjective(PetscReal t, Vec obj, AppCtx *appctx)
348: {
349: PetscScalar *s;
350: const PetscScalar *xg;
351: PetscInt i, xs, xn;
353: PetscFunctionBegin;
354: PetscCall(DMDAVecGetArray(appctx->da, obj, &s));
355: PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
356: PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));
357: for (i = xs; i < xs + xn; i++) {
358: s[i] = 2.0 * appctx->param.mu * PETSC_PI * PetscSinScalar(PETSC_PI * xg[i]) * PetscExpScalar(-PETSC_PI * PETSC_PI * t * appctx->param.mu) / (2.0 + PetscExpScalar(-PETSC_PI * PETSC_PI * t * appctx->param.mu) * PetscCosScalar(PETSC_PI * xg[i]));
359: }
360: PetscCall(DMDAVecRestoreArray(appctx->da, obj, &s));
361: PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
362: PetscFunctionReturn(PETSC_SUCCESS);
363: }
365: PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec globalin, Vec globalout, void *ctx)
366: {
367: AppCtx *appctx = (AppCtx *)ctx;
369: PetscFunctionBegin;
370: PetscCall(MatMult(appctx->SEMop.grad, globalin, globalout)); /* grad u */
371: PetscCall(VecPointwiseMult(globalout, globalin, globalout)); /* u grad u */
372: PetscCall(VecScale(globalout, -1.0));
373: PetscCall(MatMultAdd(appctx->SEMop.keptstiff, globalin, globalout, globalout));
374: PetscFunctionReturn(PETSC_SUCCESS);
375: }
377: /*
379: K is the discretiziation of the Laplacian
380: G is the discretization of the gradient
382: Computes Jacobian of K u + diag(u) G u which is given by
383: K + diag(u)G + diag(Gu)
384: */
385: PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec globalin, Mat A, Mat B, void *ctx)
386: {
387: AppCtx *appctx = (AppCtx *)ctx;
388: Vec Gglobalin;
390: PetscFunctionBegin;
391: /* A = diag(u) G */
393: PetscCall(MatCopy(appctx->SEMop.grad, A, SAME_NONZERO_PATTERN));
394: PetscCall(MatDiagonalScale(A, globalin, NULL));
396: /* A = A + diag(Gu) */
397: PetscCall(VecDuplicate(globalin, &Gglobalin));
398: PetscCall(MatMult(appctx->SEMop.grad, globalin, Gglobalin));
399: PetscCall(MatDiagonalSet(A, Gglobalin, ADD_VALUES));
400: PetscCall(VecDestroy(&Gglobalin));
402: /* A = K - A */
403: PetscCall(MatScale(A, -1.0));
404: PetscCall(MatAXPY(A, 1.0, appctx->SEMop.keptstiff, SAME_NONZERO_PATTERN));
405: PetscFunctionReturn(PETSC_SUCCESS);
406: }
408: /* --------------------------------------------------------------------- */
410: /*
411: RHSMatrixLaplacian - User-provided routine to compute the right-hand-side
412: matrix for the heat equation.
414: Input Parameters:
415: ts - the TS context
416: t - current time (ignored)
417: X - current solution (ignored)
418: dummy - optional user-defined context, as set by TSetRHSJacobian()
420: Output Parameters:
421: AA - Jacobian matrix
422: BB - optionally different matrix from which the preconditioner is built
423: str - flag indicating matrix structure
425: */
426: PetscErrorCode RHSMatrixLaplaciangllDM(TS ts, PetscReal t, Vec X, Mat A, Mat BB, void *ctx)
427: {
428: PetscReal **temp;
429: PetscReal vv;
430: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
431: PetscInt i, xs, xn, l, j;
432: PetscInt *rowsDM;
434: PetscFunctionBegin;
435: /*
436: Creates the element stiffness matrix for the given gll
437: */
438: PetscCall(PetscGaussLobattoLegendreElementLaplacianCreate(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
439: /* workaround for clang analyzer warning: Division by zero */
440: PetscCheck(appctx->param.N > 1, PETSC_COMM_WORLD, PETSC_ERR_ARG_WRONG, "Spectral element order should be > 1");
442: /* scale by the size of the element */
443: for (i = 0; i < appctx->param.N; i++) {
444: vv = -appctx->param.mu * 2.0 / appctx->param.Le;
445: for (j = 0; j < appctx->param.N; j++) temp[i][j] = temp[i][j] * vv;
446: }
448: PetscCall(MatSetOption(A, MAT_NEW_NONZERO_ALLOCATION_ERR, PETSC_FALSE));
449: PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));
451: xs = xs / (appctx->param.N - 1);
452: xn = xn / (appctx->param.N - 1);
454: PetscCall(PetscMalloc1(appctx->param.N, &rowsDM));
455: /*
456: loop over local elements
457: */
458: for (j = xs; j < xs + xn; j++) {
459: for (l = 0; l < appctx->param.N; l++) rowsDM[l] = 1 + (j - xs) * (appctx->param.N - 1) + l;
460: PetscCall(MatSetValuesLocal(A, appctx->param.N, rowsDM, appctx->param.N, rowsDM, &temp[0][0], ADD_VALUES));
461: }
462: PetscCall(PetscFree(rowsDM));
463: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
464: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
465: PetscCall(VecReciprocal(appctx->SEMop.mass));
466: PetscCall(MatDiagonalScale(A, appctx->SEMop.mass, 0));
467: PetscCall(VecReciprocal(appctx->SEMop.mass));
469: PetscCall(PetscGaussLobattoLegendreElementLaplacianDestroy(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
470: PetscFunctionReturn(PETSC_SUCCESS);
471: }
473: /*
474: RHSMatrixAdvection - User-provided routine to compute the right-hand-side
475: matrix for the Advection equation.
477: Input Parameters:
478: ts - the TS context
479: t - current time
480: global_in - global input vector
481: dummy - optional user-defined context, as set by TSetRHSJacobian()
483: Output Parameters:
484: AA - Jacobian matrix
485: BB - optionally different preconditioning matrix
486: str - flag indicating matrix structure
488: */
489: PetscErrorCode RHSMatrixAdvectiongllDM(TS ts, PetscReal t, Vec X, Mat A, Mat BB, void *ctx)
490: {
491: PetscReal **temp;
492: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
493: PetscInt xs, xn, l, j;
494: PetscInt *rowsDM;
496: PetscFunctionBegin;
497: /*
498: Creates the advection matrix for the given gll
499: */
500: PetscCall(PetscGaussLobattoLegendreElementAdvectionCreate(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
501: PetscCall(MatSetOption(A, MAT_NEW_NONZERO_ALLOCATION_ERR, PETSC_FALSE));
503: PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));
505: xs = xs / (appctx->param.N - 1);
506: xn = xn / (appctx->param.N - 1);
508: PetscCall(PetscMalloc1(appctx->param.N, &rowsDM));
509: for (j = xs; j < xs + xn; j++) {
510: for (l = 0; l < appctx->param.N; l++) rowsDM[l] = 1 + (j - xs) * (appctx->param.N - 1) + l;
511: PetscCall(MatSetValuesLocal(A, appctx->param.N, rowsDM, appctx->param.N, rowsDM, &temp[0][0], ADD_VALUES));
512: }
513: PetscCall(PetscFree(rowsDM));
514: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
515: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
517: PetscCall(VecReciprocal(appctx->SEMop.mass));
518: PetscCall(MatDiagonalScale(A, appctx->SEMop.mass, 0));
519: PetscCall(VecReciprocal(appctx->SEMop.mass));
520: PetscCall(PetscGaussLobattoLegendreElementAdvectionDestroy(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
521: PetscFunctionReturn(PETSC_SUCCESS);
522: }
523: /* ------------------------------------------------------------------ */
524: /*
525: FormFunctionGradient - Evaluates the function and corresponding gradient.
527: Input Parameters:
528: tao - the Tao context
529: IC - the input vector
530: ctx - optional user-defined context, as set when calling TaoSetObjectiveAndGradient()
532: Output Parameters:
533: f - the newly evaluated function
534: G - the newly evaluated gradient
536: Notes:
538: The forward equation is
539: M u_t = F(U)
540: which is converted to
541: u_t = M^{-1} F(u)
542: in the user code since TS has no direct way of providing a mass matrix. The Jacobian of this is
543: M^{-1} J
544: where J is the Jacobian of F. Now the adjoint equation is
545: M v_t = J^T v
546: but TSAdjoint does not solve this since it can only solve the transposed system for the
547: Jacobian the user provided. Hence TSAdjoint solves
548: w_t = J^T M^{-1} w (where w = M v)
549: since there is no way to indicate the mass matrix as a separate entity to TS. Thus one
550: must be careful in initializing the "adjoint equation" and using the result. This is
551: why
552: G = -2 M(u(T) - u_d)
553: below (instead of -2(u(T) - u_d) and why the result is
554: G = G/appctx->SEMop.mass (that is G = M^{-1}w)
555: below (instead of just the result of the "adjoint solve").
557: */
558: PetscErrorCode FormFunctionGradient(Tao tao, Vec IC, PetscReal *f, Vec G, void *ctx)
559: {
560: AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
561: Vec temp;
562: PetscInt its;
563: PetscReal ff, gnorm, cnorm, xdiff, errex;
564: TaoConvergedReason reason;
566: PetscFunctionBegin;
567: PetscCall(TSSetTime(appctx->ts, 0.0));
568: PetscCall(TSSetStepNumber(appctx->ts, 0));
569: PetscCall(TSSetTimeStep(appctx->ts, appctx->initial_dt));
570: PetscCall(VecCopy(IC, appctx->dat.curr_sol));
572: PetscCall(TSSolve(appctx->ts, appctx->dat.curr_sol));
574: PetscCall(VecWAXPY(G, -1.0, appctx->dat.curr_sol, appctx->dat.obj));
576: /*
577: Compute the L2-norm of the objective function, cost function is f
578: */
579: PetscCall(VecDuplicate(G, &temp));
580: PetscCall(VecPointwiseMult(temp, G, G));
581: PetscCall(VecDot(temp, appctx->SEMop.mass, f));
583: /* local error evaluation */
584: PetscCall(VecWAXPY(temp, -1.0, appctx->dat.ic, appctx->dat.true_solution));
585: PetscCall(VecPointwiseMult(temp, temp, temp));
586: /* for error evaluation */
587: PetscCall(VecDot(temp, appctx->SEMop.mass, &errex));
588: PetscCall(VecDestroy(&temp));
589: errex = PetscSqrtReal(errex);
591: /*
592: Compute initial conditions for the adjoint integration. See Notes above
593: */
595: PetscCall(VecScale(G, -2.0));
596: PetscCall(VecPointwiseMult(G, G, appctx->SEMop.mass));
597: PetscCall(TSSetCostGradients(appctx->ts, 1, &G, NULL));
598: PetscCall(TSAdjointSolve(appctx->ts));
599: PetscCall(VecPointwiseDivide(G, G, appctx->SEMop.mass));
601: PetscCall(TaoGetSolutionStatus(tao, &its, &ff, &gnorm, &cnorm, &xdiff, &reason));
602: PetscFunctionReturn(PETSC_SUCCESS);
603: }
605: PetscErrorCode MonitorError(Tao tao, void *ctx)
606: {
607: AppCtx *appctx = (AppCtx *)ctx;
608: Vec temp;
609: PetscReal nrm;
611: PetscFunctionBegin;
612: PetscCall(VecDuplicate(appctx->dat.ic, &temp));
613: PetscCall(VecWAXPY(temp, -1.0, appctx->dat.ic, appctx->dat.true_solution));
614: PetscCall(VecPointwiseMult(temp, temp, temp));
615: PetscCall(VecDot(temp, appctx->SEMop.mass, &nrm));
616: PetscCall(VecDestroy(&temp));
617: nrm = PetscSqrtReal(nrm);
618: PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Error for initial conditions %g\n", (double)nrm));
619: PetscFunctionReturn(PETSC_SUCCESS);
620: }
622: /*TEST
624: build:
625: requires: !complex
627: test:
628: args: -tao_max_it 5 -tao_gatol 1.e-4
629: requires: !single
631: test:
632: suffix: 2
633: nsize: 2
634: args: -tao_max_it 5 -tao_gatol 1.e-4
635: requires: !single
637: TEST*/