Actual source code: spectraladjointassimilation.c

  1: static char help[] = "Solves a simple data assimilation problem with one dimensional advection diffusion equation using TSAdjoint\n\n";

  3: /*

  5:     Not yet tested in parallel

  7: */

  9: /* ------------------------------------------------------------------------

 11:    This program uses the one-dimensional advection-diffusion equation),
 12:        u_t = mu*u_xx - a 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:   ------------------------------------------------------------------------- */

 22: /*
 23:    Include "petscts.h" so that we can use TS solvers.  Note that this file
 24:    automatically includes:
 25:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
 26:      petscmat.h  - matrices
 27:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 28:      petscviewer.h - viewers               petscpc.h   - preconditioners
 29:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
 30: */

 32: #include <petsctao.h>
 33: #include <petscts.h>
 34: #include <petscdt.h>
 35: #include <petscdraw.h>
 36: #include <petscdmda.h>

 38: /*
 39:    User-defined application context - contains data needed by the
 40:    application-provided call-back routines.
 41: */

 43: typedef struct {
 44:   PetscInt   n;       /* number of nodes */
 45:   PetscReal *nodes;   /* GLL nodes */
 46:   PetscReal *weights; /* GLL weights */
 47: } PetscGLL;

 49: typedef struct {
 50:   PetscInt  N;               /* grid points per elements*/
 51:   PetscInt  E;               /* number of elements */
 52:   PetscReal tol_L2, tol_max; /* error norms */
 53:   PetscInt  steps;           /* number of timesteps */
 54:   PetscReal Tend;            /* endtime */
 55:   PetscReal mu;              /* viscosity */
 56:   PetscReal a;               /* advection speed */
 57:   PetscReal L;               /* total length of domain */
 58:   PetscReal Le;
 59:   PetscReal Tadj;
 60: } PetscParam;

 62: typedef struct {
 63:   Vec reference; /* desired end state */
 64:   Vec grid;      /* total grid */
 65:   Vec grad;
 66:   Vec ic;
 67:   Vec curr_sol;
 68:   Vec joe;
 69:   Vec true_solution; /* actual initial conditions for the final solution */
 70: } PetscData;

 72: typedef struct {
 73:   Vec      grid;  /* total grid */
 74:   Vec      mass;  /* mass matrix for total integration */
 75:   Mat      stiff; /* stifness matrix */
 76:   Mat      advec;
 77:   Mat      keptstiff;
 78:   PetscGLL gll;
 79: } PetscSEMOperators;

 81: typedef struct {
 82:   DM                da; /* distributed array data structure */
 83:   PetscSEMOperators SEMop;
 84:   PetscParam        param;
 85:   PetscData         dat;
 86:   TS                ts;
 87:   PetscReal         initial_dt;
 88:   PetscReal        *solutioncoefficients;
 89:   PetscInt          ncoeff;
 90: } AppCtx;

 92: /*
 93:    User-defined routines
 94: */
 95: extern PetscErrorCode FormFunctionGradient(Tao, Vec, PetscReal *, Vec, void *);
 96: extern PetscErrorCode RHSLaplacian(TS, PetscReal, Vec, Mat, Mat, void *);
 97: extern PetscErrorCode RHSAdvection(TS, PetscReal, Vec, Mat, Mat, void *);
 98: extern PetscErrorCode InitialConditions(Vec, AppCtx *);
 99: extern PetscErrorCode ComputeReference(TS, PetscReal, Vec, AppCtx *);
100: extern PetscErrorCode MonitorError(Tao, void *);
101: extern PetscErrorCode MonitorDestroy(void **);
102: extern PetscErrorCode ComputeSolutionCoefficients(AppCtx *);
103: extern PetscErrorCode RHSFunction(TS, PetscReal, Vec, Vec, void *);
104: extern PetscErrorCode RHSJacobian(TS, PetscReal, Vec, Mat, Mat, void *);

106: int main(int argc, char **argv)
107: {
108:   AppCtx       appctx; /* user-defined application context */
109:   Tao          tao;
110:   Vec          u; /* approximate solution vector */
111:   PetscInt     i, xs, xm, ind, j, lenglob;
112:   PetscReal    x, *wrk_ptr1, *wrk_ptr2;
113:   MatNullSpace nsp;

115:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
116:      Initialize program and set problem parameters
117:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
118:   PetscFunctionBeginUser;
119:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));

121:   /*initialize parameters */
122:   appctx.param.N     = 10;      /* order of the spectral element */
123:   appctx.param.E     = 8;       /* number of elements */
124:   appctx.param.L     = 1.0;     /* length of the domain */
125:   appctx.param.mu    = 0.00001; /* diffusion coefficient */
126:   appctx.param.a     = 0.0;     /* advection speed */
127:   appctx.initial_dt  = 1e-4;
128:   appctx.param.steps = PETSC_INT_MAX;
129:   appctx.param.Tend  = 0.01;
130:   appctx.ncoeff      = 2;

132:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-N", &appctx.param.N, NULL));
133:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-E", &appctx.param.E, NULL));
134:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-ncoeff", &appctx.ncoeff, NULL));
135:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-Tend", &appctx.param.Tend, NULL));
136:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-mu", &appctx.param.mu, NULL));
137:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-a", &appctx.param.a, NULL));
138:   appctx.param.Le = appctx.param.L / appctx.param.E;

140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141:      Create GLL data structures
142:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143:   PetscCall(PetscMalloc2(appctx.param.N, &appctx.SEMop.gll.nodes, appctx.param.N, &appctx.SEMop.gll.weights));
144:   PetscCall(PetscDTGaussLobattoLegendreQuadrature(appctx.param.N, PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA, appctx.SEMop.gll.nodes, appctx.SEMop.gll.weights));
145:   appctx.SEMop.gll.n = appctx.param.N;
146:   lenglob            = appctx.param.E * (appctx.param.N - 1);

148:   /*
149:      Create distributed array (DMDA) to manage parallel grid and vectors
150:      and to set up the ghost point communication pattern.  There are E*(Nl-1)+1
151:      total grid values spread equally among all the processors, except first and last
152:   */

154:   PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, lenglob, 1, 1, NULL, &appctx.da));
155:   PetscCall(DMSetFromOptions(appctx.da));
156:   PetscCall(DMSetUp(appctx.da));

158:   /*
159:      Extract global and local vectors from DMDA; we use these to store the
160:      approximate solution.  Then duplicate these for remaining vectors that
161:      have the same types.
162:   */

164:   PetscCall(DMCreateGlobalVector(appctx.da, &u));
165:   PetscCall(VecDuplicate(u, &appctx.dat.ic));
166:   PetscCall(VecDuplicate(u, &appctx.dat.true_solution));
167:   PetscCall(VecDuplicate(u, &appctx.dat.reference));
168:   PetscCall(VecDuplicate(u, &appctx.SEMop.grid));
169:   PetscCall(VecDuplicate(u, &appctx.SEMop.mass));
170:   PetscCall(VecDuplicate(u, &appctx.dat.curr_sol));
171:   PetscCall(VecDuplicate(u, &appctx.dat.joe));

173:   PetscCall(DMDAGetCorners(appctx.da, &xs, NULL, NULL, &xm, NULL, NULL));
174:   PetscCall(DMDAVecGetArray(appctx.da, appctx.SEMop.grid, &wrk_ptr1));
175:   PetscCall(DMDAVecGetArray(appctx.da, appctx.SEMop.mass, &wrk_ptr2));

177:   /* Compute function over the locally owned part of the grid */

179:   xs = xs / (appctx.param.N - 1);
180:   xm = xm / (appctx.param.N - 1);

182:   /*
183:      Build total grid and mass over entire mesh (multi-elemental)
184:   */

186:   for (i = xs; i < xs + xm; i++) {
187:     for (j = 0; j < appctx.param.N - 1; j++) {
188:       x             = (appctx.param.Le / 2.0) * (appctx.SEMop.gll.nodes[j] + 1.0) + appctx.param.Le * i;
189:       ind           = i * (appctx.param.N - 1) + j;
190:       wrk_ptr1[ind] = x;
191:       wrk_ptr2[ind] = .5 * appctx.param.Le * appctx.SEMop.gll.weights[j];
192:       if (j == 0) wrk_ptr2[ind] += .5 * appctx.param.Le * appctx.SEMop.gll.weights[j];
193:     }
194:   }
195:   PetscCall(DMDAVecRestoreArray(appctx.da, appctx.SEMop.grid, &wrk_ptr1));
196:   PetscCall(DMDAVecRestoreArray(appctx.da, appctx.SEMop.mass, &wrk_ptr2));

198:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
199:    Create matrix data structure; set matrix evaluation routine.
200:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
201:   PetscCall(DMSetMatrixPreallocateOnly(appctx.da, PETSC_TRUE));
202:   PetscCall(DMCreateMatrix(appctx.da, &appctx.SEMop.stiff));
203:   PetscCall(DMCreateMatrix(appctx.da, &appctx.SEMop.advec));

205:   /*
206:    For linear problems with a time-dependent f(u,t) in the equation
207:    u_t = f(u,t), the user provides the discretized right-hand side
208:    as a time-dependent matrix.
209:    */
210:   PetscCall(RHSLaplacian(appctx.ts, 0.0, u, appctx.SEMop.stiff, appctx.SEMop.stiff, &appctx));
211:   PetscCall(RHSAdvection(appctx.ts, 0.0, u, appctx.SEMop.advec, appctx.SEMop.advec, &appctx));
212:   PetscCall(MatAXPY(appctx.SEMop.stiff, -1.0, appctx.SEMop.advec, DIFFERENT_NONZERO_PATTERN));
213:   PetscCall(MatDuplicate(appctx.SEMop.stiff, MAT_COPY_VALUES, &appctx.SEMop.keptstiff));

215:   /* attach the null space to the matrix, this probably is not needed but does no harm */
216:   PetscCall(MatNullSpaceCreate(PETSC_COMM_WORLD, PETSC_TRUE, 0, NULL, &nsp));
217:   PetscCall(MatSetNullSpace(appctx.SEMop.stiff, nsp));
218:   PetscCall(MatNullSpaceTest(nsp, appctx.SEMop.stiff, NULL));
219:   PetscCall(MatNullSpaceDestroy(&nsp));

221:   /* Create the TS solver that solves the ODE and its adjoint; set its options */
222:   PetscCall(TSCreate(PETSC_COMM_WORLD, &appctx.ts));
223:   PetscCall(TSSetSolutionFunction(appctx.ts, (PetscErrorCode (*)(TS, PetscReal, Vec, void *))ComputeReference, &appctx));
224:   PetscCall(TSSetProblemType(appctx.ts, TS_LINEAR));
225:   PetscCall(TSSetType(appctx.ts, TSRK));
226:   PetscCall(TSSetDM(appctx.ts, appctx.da));
227:   PetscCall(TSSetTime(appctx.ts, 0.0));
228:   PetscCall(TSSetTimeStep(appctx.ts, appctx.initial_dt));
229:   PetscCall(TSSetMaxSteps(appctx.ts, appctx.param.steps));
230:   PetscCall(TSSetMaxTime(appctx.ts, appctx.param.Tend));
231:   PetscCall(TSSetExactFinalTime(appctx.ts, TS_EXACTFINALTIME_MATCHSTEP));
232:   PetscCall(TSSetTolerances(appctx.ts, 1e-7, NULL, 1e-7, NULL));
233:   PetscCall(TSSetFromOptions(appctx.ts));
234:   /* Need to save initial timestep user may have set with -ts_dt so it can be reset for each new TSSolve() */
235:   PetscCall(TSGetTimeStep(appctx.ts, &appctx.initial_dt));
236:   PetscCall(TSSetRHSFunction(appctx.ts, NULL, TSComputeRHSFunctionLinear, &appctx));
237:   PetscCall(TSSetRHSJacobian(appctx.ts, appctx.SEMop.stiff, appctx.SEMop.stiff, TSComputeRHSJacobianConstant, &appctx));
238:   /*  PetscCall(TSSetRHSFunction(appctx.ts,NULL,RHSFunction,&appctx));
239:       PetscCall(TSSetRHSJacobian(appctx.ts,appctx.SEMop.stiff,appctx.SEMop.stiff,RHSJacobian,&appctx)); */

241:   /* Set random initial conditions as initial guess, compute analytic reference solution and analytic (true) initial conditions */
242:   PetscCall(ComputeSolutionCoefficients(&appctx));
243:   PetscCall(InitialConditions(appctx.dat.ic, &appctx));
244:   PetscCall(ComputeReference(appctx.ts, appctx.param.Tend, appctx.dat.reference, &appctx));
245:   PetscCall(ComputeReference(appctx.ts, 0.0, appctx.dat.true_solution, &appctx));

247:   /* Set up to save trajectory before TSSetFromOptions() so that TSTrajectory options can be captured */
248:   PetscCall(TSSetSaveTrajectory(appctx.ts));
249:   PetscCall(TSSetFromOptions(appctx.ts));

251:   /* Create TAO solver and set desired solution method  */
252:   PetscCall(TaoCreate(PETSC_COMM_WORLD, &tao));
253:   PetscCall(TaoMonitorSet(tao, MonitorError, &appctx, MonitorDestroy));
254:   PetscCall(TaoSetType(tao, TAOBQNLS));
255:   PetscCall(TaoSetSolution(tao, appctx.dat.ic));
256:   /* Set routine for function and gradient evaluation  */
257:   PetscCall(TaoSetObjectiveAndGradient(tao, NULL, FormFunctionGradient, (void *)&appctx));
258:   /* Check for any TAO command line options  */
259:   PetscCall(TaoSetTolerances(tao, 1e-8, PETSC_CURRENT, PETSC_CURRENT));
260:   PetscCall(TaoSetFromOptions(tao));
261:   PetscCall(TaoSolve(tao));

263:   PetscCall(TaoDestroy(&tao));
264:   PetscCall(PetscFree(appctx.solutioncoefficients));
265:   PetscCall(MatDestroy(&appctx.SEMop.advec));
266:   PetscCall(MatDestroy(&appctx.SEMop.stiff));
267:   PetscCall(MatDestroy(&appctx.SEMop.keptstiff));
268:   PetscCall(VecDestroy(&u));
269:   PetscCall(VecDestroy(&appctx.dat.ic));
270:   PetscCall(VecDestroy(&appctx.dat.joe));
271:   PetscCall(VecDestroy(&appctx.dat.true_solution));
272:   PetscCall(VecDestroy(&appctx.dat.reference));
273:   PetscCall(VecDestroy(&appctx.SEMop.grid));
274:   PetscCall(VecDestroy(&appctx.SEMop.mass));
275:   PetscCall(VecDestroy(&appctx.dat.curr_sol));
276:   PetscCall(PetscFree2(appctx.SEMop.gll.nodes, appctx.SEMop.gll.weights));
277:   PetscCall(DMDestroy(&appctx.da));
278:   PetscCall(TSDestroy(&appctx.ts));

280:   /*
281:      Always call PetscFinalize() before exiting a program.  This routine
282:        - finalizes the PETSc libraries as well as MPI
283:        - provides summary and diagnostic information if certain runtime
284:          options are chosen (e.g., -log_view).
285:   */
286:   PetscCall(PetscFinalize());
287:   return 0;
288: }

290: /*
291:     Computes the coefficients for the analytic solution to the PDE
292: */
293: PetscErrorCode ComputeSolutionCoefficients(AppCtx *appctx)
294: {
295:   PetscRandom rand;
296:   PetscInt    i;

298:   PetscFunctionBegin;
299:   PetscCall(PetscMalloc1(appctx->ncoeff, &appctx->solutioncoefficients));
300:   PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &rand));
301:   PetscCall(PetscRandomSetInterval(rand, .9, 1.0));
302:   for (i = 0; i < appctx->ncoeff; i++) PetscCall(PetscRandomGetValue(rand, &appctx->solutioncoefficients[i]));
303:   PetscCall(PetscRandomDestroy(&rand));
304:   PetscFunctionReturn(PETSC_SUCCESS);
305: }

307: /* --------------------------------------------------------------------- */
308: /*
309:    InitialConditions - Computes the (random) initial conditions for the Tao optimization solve (these are also initial conditions for the first TSSolve()

311:    Input Parameter:
312:    u - uninitialized solution vector (global)
313:    appctx - user-defined application context

315:    Output Parameter:
316:    u - vector with solution at initial time (global)
317: */
318: PetscErrorCode InitialConditions(Vec u, AppCtx *appctx)
319: {
320:   PetscScalar       *s;
321:   const PetscScalar *xg;
322:   PetscInt           i, j, lenglob;
323:   PetscReal          sum, val;
324:   PetscRandom        rand;

326:   PetscFunctionBegin;
327:   PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &rand));
328:   PetscCall(PetscRandomSetInterval(rand, .9, 1.0));
329:   PetscCall(DMDAVecGetArray(appctx->da, u, &s));
330:   PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
331:   lenglob = appctx->param.E * (appctx->param.N - 1);
332:   for (i = 0; i < lenglob; i++) {
333:     s[i] = 0;
334:     for (j = 0; j < appctx->ncoeff; j++) {
335:       PetscCall(PetscRandomGetValue(rand, &val));
336:       s[i] += val * PetscSinScalar(2 * (j + 1) * PETSC_PI * xg[i]);
337:     }
338:   }
339:   PetscCall(PetscRandomDestroy(&rand));
340:   PetscCall(DMDAVecRestoreArray(appctx->da, u, &s));
341:   PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
342:   /* make sure initial conditions do not contain the constant functions, since with periodic boundary conditions the constant functions introduce a null space */
343:   PetscCall(VecSum(u, &sum));
344:   PetscCall(VecShift(u, -sum / lenglob));
345:   PetscFunctionReturn(PETSC_SUCCESS);
346: }

348: /*
349:    TrueSolution() computes the true solution for the Tao optimization solve which means they are the initial conditions for the objective function.

351:              InitialConditions() computes the initial conditions for the beginning of the Tao iterations

353:    Input Parameter:
354:    u - uninitialized solution vector (global)
355:    appctx - user-defined application context

357:    Output Parameter:
358:    u - vector with solution at initial time (global)
359: */
360: PetscErrorCode TrueSolution(Vec u, AppCtx *appctx)
361: {
362:   PetscScalar       *s;
363:   const PetscScalar *xg;
364:   PetscInt           i, j, lenglob;
365:   PetscReal          sum;

367:   PetscFunctionBegin;
368:   PetscCall(DMDAVecGetArray(appctx->da, u, &s));
369:   PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
370:   lenglob = appctx->param.E * (appctx->param.N - 1);
371:   for (i = 0; i < lenglob; i++) {
372:     s[i] = 0;
373:     for (j = 0; j < appctx->ncoeff; j++) s[i] += appctx->solutioncoefficients[j] * PetscSinScalar(2 * (j + 1) * PETSC_PI * xg[i]);
374:   }
375:   PetscCall(DMDAVecRestoreArray(appctx->da, u, &s));
376:   PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
377:   /* make sure initial conditions do not contain the constant functions, since with periodic boundary conditions the constant functions introduce a null space */
378:   PetscCall(VecSum(u, &sum));
379:   PetscCall(VecShift(u, -sum / lenglob));
380:   PetscFunctionReturn(PETSC_SUCCESS);
381: }
382: /* --------------------------------------------------------------------- */
383: /*
384:    Sets the desired profile for the final end time

386:    Input Parameters:
387:    t - final time
388:    obj - vector storing the desired profile
389:    appctx - user-defined application context

391: */
392: PetscErrorCode ComputeReference(TS ts, PetscReal t, Vec obj, AppCtx *appctx)
393: {
394:   PetscScalar       *s, tc;
395:   const PetscScalar *xg;
396:   PetscInt           i, j, lenglob;

398:   PetscFunctionBegin;
399:   PetscCall(DMDAVecGetArray(appctx->da, obj, &s));
400:   PetscCall(DMDAVecGetArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
401:   lenglob = appctx->param.E * (appctx->param.N - 1);
402:   for (i = 0; i < lenglob; i++) {
403:     s[i] = 0;
404:     for (j = 0; j < appctx->ncoeff; j++) {
405:       tc = -appctx->param.mu * (j + 1) * (j + 1) * 4.0 * PETSC_PI * PETSC_PI * t;
406:       s[i] += appctx->solutioncoefficients[j] * PetscSinScalar(2 * (j + 1) * PETSC_PI * (xg[i] + appctx->param.a * t)) * PetscExpReal(tc);
407:     }
408:   }
409:   PetscCall(DMDAVecRestoreArray(appctx->da, obj, &s));
410:   PetscCall(DMDAVecRestoreArrayRead(appctx->da, appctx->SEMop.grid, (void *)&xg));
411:   PetscFunctionReturn(PETSC_SUCCESS);
412: }

414: PetscErrorCode RHSFunction(TS ts, PetscReal t, Vec globalin, Vec globalout, void *ctx)
415: {
416:   AppCtx *appctx = (AppCtx *)ctx;

418:   PetscFunctionBegin;
419:   PetscCall(MatMult(appctx->SEMop.keptstiff, globalin, globalout));
420:   PetscFunctionReturn(PETSC_SUCCESS);
421: }

423: PetscErrorCode RHSJacobian(TS ts, PetscReal t, Vec globalin, Mat A, Mat B, void *ctx)
424: {
425:   AppCtx *appctx = (AppCtx *)ctx;

427:   PetscFunctionBegin;
428:   PetscCall(MatCopy(appctx->SEMop.keptstiff, A, DIFFERENT_NONZERO_PATTERN));
429:   PetscFunctionReturn(PETSC_SUCCESS);
430: }

432: /* --------------------------------------------------------------------- */

434: /*
435:    RHSLaplacian -   matrix for diffusion

437:    Input Parameters:
438:    ts - the TS context
439:    t - current time  (ignored)
440:    X - current solution (ignored)
441:    dummy - optional user-defined context, as set by TSetRHSJacobian()

443:    Output Parameters:
444:    AA - Jacobian matrix
445:    BB - optionally different matrix from which the preconditioner is built
446:    str - flag indicating matrix structure

448:    Scales by the inverse of the mass matrix (perhaps that should be pulled out)

450: */
451: PetscErrorCode RHSLaplacian(TS ts, PetscReal t, Vec X, Mat A, Mat BB, void *ctx)
452: {
453:   PetscReal **temp;
454:   PetscReal   vv;
455:   AppCtx     *appctx = (AppCtx *)ctx; /* user-defined application context */
456:   PetscInt    i, xs, xn, l, j;
457:   PetscInt   *rowsDM;

459:   PetscFunctionBegin;
460:   /*
461:    Creates the element stiffness matrix for the given gll
462:    */
463:   PetscCall(PetscGaussLobattoLegendreElementLaplacianCreate(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));

465:   /* scale by the size of the element */
466:   for (i = 0; i < appctx->param.N; i++) {
467:     vv = -appctx->param.mu * 2.0 / appctx->param.Le;
468:     for (j = 0; j < appctx->param.N; j++) temp[i][j] = temp[i][j] * vv;
469:   }

471:   PetscCall(MatSetOption(A, MAT_NEW_NONZERO_ALLOCATION_ERR, PETSC_FALSE));
472:   PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));

474:   PetscCheck(appctx->param.N - 1 >= 1, PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Polynomial order must be at least 2");
475:   xs = xs / (appctx->param.N - 1);
476:   xn = xn / (appctx->param.N - 1);

478:   PetscCall(PetscMalloc1(appctx->param.N, &rowsDM));
479:   /*
480:    loop over local elements
481:    */
482:   for (j = xs; j < xs + xn; j++) {
483:     for (l = 0; l < appctx->param.N; l++) rowsDM[l] = 1 + (j - xs) * (appctx->param.N - 1) + l;
484:     PetscCall(MatSetValuesLocal(A, appctx->param.N, rowsDM, appctx->param.N, rowsDM, &temp[0][0], ADD_VALUES));
485:   }
486:   PetscCall(PetscFree(rowsDM));
487:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
488:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
489:   PetscCall(VecReciprocal(appctx->SEMop.mass));
490:   PetscCall(MatDiagonalScale(A, appctx->SEMop.mass, 0));
491:   PetscCall(VecReciprocal(appctx->SEMop.mass));

493:   PetscCall(PetscGaussLobattoLegendreElementLaplacianDestroy(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
494:   PetscFunctionReturn(PETSC_SUCCESS);
495: }

497: /*
498:     Almost identical to Laplacian

500:     Note that the element matrix is NOT scaled by the size of element like the Laplacian term.
501:  */
502: PetscErrorCode RHSAdvection(TS ts, PetscReal t, Vec X, Mat A, Mat BB, void *ctx)
503: {
504:   PetscReal **temp;
505:   PetscReal   vv;
506:   AppCtx     *appctx = (AppCtx *)ctx; /* user-defined application context */
507:   PetscInt    i, xs, xn, l, j;
508:   PetscInt   *rowsDM;

510:   PetscFunctionBegin;
511:   /*
512:    Creates the element stiffness matrix for the given gll
513:    */
514:   PetscCall(PetscGaussLobattoLegendreElementAdvectionCreate(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));

516:   /* scale by the size of the element */
517:   for (i = 0; i < appctx->param.N; i++) {
518:     vv = -appctx->param.a;
519:     for (j = 0; j < appctx->param.N; j++) temp[i][j] = temp[i][j] * vv;
520:   }

522:   PetscCall(MatSetOption(A, MAT_NEW_NONZERO_ALLOCATION_ERR, PETSC_FALSE));
523:   PetscCall(DMDAGetCorners(appctx->da, &xs, NULL, NULL, &xn, NULL, NULL));

525:   PetscCheck(appctx->param.N - 1 >= 1, PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Polynomial order must be at least 2");
526:   xs = xs / (appctx->param.N - 1);
527:   xn = xn / (appctx->param.N - 1);

529:   PetscCall(PetscMalloc1(appctx->param.N, &rowsDM));
530:   /*
531:    loop over local elements
532:    */
533:   for (j = xs; j < xs + xn; j++) {
534:     for (l = 0; l < appctx->param.N; l++) rowsDM[l] = 1 + (j - xs) * (appctx->param.N - 1) + l;
535:     PetscCall(MatSetValuesLocal(A, appctx->param.N, rowsDM, appctx->param.N, rowsDM, &temp[0][0], ADD_VALUES));
536:   }
537:   PetscCall(PetscFree(rowsDM));
538:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
539:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
540:   PetscCall(VecReciprocal(appctx->SEMop.mass));
541:   PetscCall(MatDiagonalScale(A, appctx->SEMop.mass, 0));
542:   PetscCall(VecReciprocal(appctx->SEMop.mass));

544:   PetscCall(PetscGaussLobattoLegendreElementAdvectionDestroy(appctx->SEMop.gll.n, appctx->SEMop.gll.nodes, appctx->SEMop.gll.weights, &temp));
545:   PetscFunctionReturn(PETSC_SUCCESS);
546: }

548: /* ------------------------------------------------------------------ */
549: /*
550:    FormFunctionGradient - Evaluates the function and corresponding gradient.

552:    Input Parameters:
553:    tao - the Tao context
554:    ic   - the input vector
555:    ctx - optional user-defined context, as set when calling TaoSetObjectiveAndGradient()

557:    Output Parameters:
558:    f   - the newly evaluated function
559:    G   - the newly evaluated gradient

561:    Notes:

563:           The forward equation is
564:               M u_t = F(U)
565:           which is converted to
566:                 u_t = M^{-1} F(u)
567:           in the user code since TS has no direct way of providing a mass matrix. The Jacobian of this is
568:                  M^{-1} J
569:           where J is the Jacobian of F. Now the adjoint equation is
570:                 M v_t = J^T v
571:           but TSAdjoint does not solve this since it can only solve the transposed system for the
572:           Jacobian the user provided. Hence TSAdjoint solves
573:                  w_t = J^T M^{-1} w  (where w = M v)
574:           since there is no way to indicate the mass matrix as a separate entity to TS. Thus one
575:           must be careful in initializing the "adjoint equation" and using the result. This is
576:           why
577:               G = -2 M(u(T) - u_d)
578:           below (instead of -2(u(T) - u_d)

580: */
581: PetscErrorCode FormFunctionGradient(Tao tao, Vec ic, PetscReal *f, Vec G, void *ctx)
582: {
583:   AppCtx *appctx = (AppCtx *)ctx; /* user-defined application context */
584:   Vec     temp;

586:   PetscFunctionBegin;
587:   PetscCall(TSSetTime(appctx->ts, 0.0));
588:   PetscCall(TSSetStepNumber(appctx->ts, 0));
589:   PetscCall(TSSetTimeStep(appctx->ts, appctx->initial_dt));
590:   PetscCall(VecCopy(ic, appctx->dat.curr_sol));

592:   PetscCall(TSSolve(appctx->ts, appctx->dat.curr_sol));
593:   PetscCall(VecCopy(appctx->dat.curr_sol, appctx->dat.joe));

595:   /*     Compute the difference between the current ODE solution and target ODE solution */
596:   PetscCall(VecWAXPY(G, -1.0, appctx->dat.curr_sol, appctx->dat.reference));

598:   /*     Compute the objective/cost function   */
599:   PetscCall(VecDuplicate(G, &temp));
600:   PetscCall(VecPointwiseMult(temp, G, G));
601:   PetscCall(VecDot(temp, appctx->SEMop.mass, f));
602:   PetscCall(VecDestroy(&temp));

604:   /*     Compute initial conditions for the adjoint integration. See Notes above  */
605:   PetscCall(VecScale(G, -2.0));
606:   PetscCall(VecPointwiseMult(G, G, appctx->SEMop.mass));
607:   PetscCall(TSSetCostGradients(appctx->ts, 1, &G, NULL));

609:   PetscCall(TSAdjointSolve(appctx->ts));
610:   /* PetscCall(VecPointwiseDivide(G,G,appctx->SEMop.mass));*/
611:   PetscFunctionReturn(PETSC_SUCCESS);
612: }

614: PetscErrorCode MonitorError(Tao tao, void *ctx)
615: {
616:   AppCtx   *appctx = (AppCtx *)ctx;
617:   Vec       temp, grad;
618:   PetscReal nrm;
619:   PetscInt  its;
620:   PetscReal fct, gnorm;

622:   PetscFunctionBegin;
623:   PetscCall(VecDuplicate(appctx->dat.ic, &temp));
624:   PetscCall(VecWAXPY(temp, -1.0, appctx->dat.ic, appctx->dat.true_solution));
625:   PetscCall(VecPointwiseMult(temp, temp, temp));
626:   PetscCall(VecDot(temp, appctx->SEMop.mass, &nrm));
627:   nrm = PetscSqrtReal(nrm);
628:   PetscCall(TaoGetGradient(tao, &grad, NULL, NULL));
629:   PetscCall(VecPointwiseMult(temp, temp, temp));
630:   PetscCall(VecDot(temp, appctx->SEMop.mass, &gnorm));
631:   gnorm = PetscSqrtReal(gnorm);
632:   PetscCall(VecDestroy(&temp));
633:   PetscCall(TaoGetIterationNumber(tao, &its));
634:   PetscCall(TaoGetSolutionStatus(tao, NULL, &fct, NULL, NULL, NULL, NULL));
635:   if (!its) {
636:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%% Iteration Error Objective Gradient-norm\n"));
637:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "history = [\n"));
638:   }
639:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "%3" PetscInt_FMT " %g %g %g\n", its, (double)nrm, (double)fct, (double)gnorm));
640:   PetscFunctionReturn(PETSC_SUCCESS);
641: }

643: PetscErrorCode MonitorDestroy(void **ctx)
644: {
645:   PetscFunctionBegin;
646:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "];\n"));
647:   PetscFunctionReturn(PETSC_SUCCESS);
648: }

650: /*TEST

652:    build:
653:      requires: !complex

655:    test:
656:      requires: !single
657:      args: -ts_adapt_dt_max 3.e-3 -E 10 -N 8 -ncoeff 5 -tao_bqnls_mat_lmvm_scale_type none

659:    test:
660:      suffix: cn
661:      requires: !single
662:      args: -ts_type cn -ts_dt .003 -pc_type lu -E 10 -N 8 -ncoeff 5 -tao_bqnls_mat_lmvm_scale_type none

664:    test:
665:      suffix: 2
666:      requires: !single
667:      args: -ts_adapt_dt_max 3.e-3 -E 10 -N 8 -ncoeff 5 -a .1 -tao_bqnls_mat_lmvm_scale_type none

669: TEST*/