Actual source code: spectraladjointassimilation.c

petsc-master 2020-08-10
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  2: static char help[] ="Solves a simple data assimilation problem with one dimensional advection diffusion equation using TSAdjoint\n\n";

  4: /*

  6:     Not yet tested in parallel

  8: */
  9: /*
 10:    Concepts: TS^time-dependent linear problems
 11:    Concepts: TS^heat equation
 12:    Concepts: TS^diffusion equation
 13:    Concepts: adjoints
 14:    Processors: n
 15: */

 17: /* ------------------------------------------------------------------------

 19:    This program uses the one-dimensional advection-diffusion equation),
 20:        u_t = mu*u_xx - a u_x,
 21:    on the domain 0 <= x <= 1, with periodic boundary conditions

 23:    to demonstrate solving a data assimilation problem of finding the initial conditions
 24:    to produce a given solution at a fixed time.

 26:    The operators are discretized with the spectral element method

 28:   ------------------------------------------------------------------------- */

 30: /*
 31:    Include "petscts.h" so that we can use TS solvers.  Note that this file
 32:    automatically includes:
 33:      petscsys.h       - base PETSc routines   petscvec.h  - vectors
 34:      petscmat.h  - matrices
 35:      petscis.h     - index sets            petscksp.h  - Krylov subspace methods
 36:      petscviewer.h - viewers               petscpc.h   - preconditioners
 37:      petscksp.h   - linear solvers        petscsnes.h - nonlinear solvers
 38: */

 40:  #include <petsctao.h>
 41:  #include <petscts.h>
 42:  #include <petscdt.h>
 43:  #include <petscdraw.h>
 44:  #include <petscdmda.h>

 46: /*
 47:    User-defined Section 1.5 Writing Application Codes with PETSc context - contains data needed by the
 48:    Section 1.5 Writing Application Codes with PETSc-provided call-back routines.
 49: */

 51: typedef struct {
 52:   PetscInt  n;                /* number of nodes */
 53:   PetscReal *nodes;           /* GLL nodes */
 54:   PetscReal *weights;         /* GLL weights */
 55: } PetscGLL;

 57: typedef struct {
 58:   PetscInt    N;             /* grid points per elements*/
 59:   PetscInt    E;              /* number of elements */
 60:   PetscReal   tol_L2,tol_max; /* error norms */
 61:   PetscInt    steps;          /* number of timesteps */
 62:   PetscReal   Tend;           /* endtime */
 63:   PetscReal   mu;             /* viscosity */
 64:   PetscReal   a;              /* advection speed */
 65:   PetscReal   L;              /* total length of domain */
 66:   PetscReal   Le;
 67:   PetscReal   Tadj;
 68: } PetscParam;

 70: typedef struct {
 71:   Vec         reference;               /* desired end state */
 72:   Vec         grid;              /* total grid */
 73:   Vec         grad;
 74:   Vec         ic;
 75:   Vec         curr_sol;
 76:   Vec         joe;
 77:   Vec         true_solution;     /* actual initial conditions for the final solution */
 78: } PetscData;

 80: typedef struct {
 81:   Vec         grid;              /* total grid */
 82:   Vec         mass;              /* mass matrix for total integration */
 83:   Mat         stiff;             /* stifness matrix */
 84:   Mat         advec;
 85:   Mat         keptstiff;
 86:   PetscGLL    gll;
 87: } PetscSEMOperators;

 89: typedef struct {
 90:   DM                da;                /* distributed array data structure */
 91:   PetscSEMOperators SEMop;
 92:   PetscParam        param;
 93:   PetscData         dat;
 94:   TS                ts;
 95:   PetscReal         initial_dt;
 96:   PetscReal         *solutioncoefficients;
 97:   PetscInt          ncoeff;
 98: } AppCtx;

100: /*
101:    User-defined routines
102: */
103: extern PetscErrorCode FormFunctionGradient(Tao,Vec,PetscReal*,Vec,void*);
104: extern PetscErrorCode RHSLaplacian(TS,PetscReal,Vec,Mat,Mat,void*);
105: extern PetscErrorCode RHSAdvection(TS,PetscReal,Vec,Mat,Mat,void*);
106: extern PetscErrorCode InitialConditions(Vec,AppCtx*);
107: extern PetscErrorCode ComputeReference(TS,PetscReal,Vec,AppCtx*);
108: extern PetscErrorCode MonitorError(Tao,void*);
109: extern PetscErrorCode MonitorDestroy(void**);
110: extern PetscErrorCode ComputeSolutionCoefficients(AppCtx*);
111: extern PetscErrorCode RHSFunction(TS,PetscReal,Vec,Vec,void*);
112: extern PetscErrorCode RHSJacobian(TS,PetscReal,Vec,Mat,Mat,void*);

114: int main(int argc,char **argv)
115: {
116:   AppCtx         appctx;                 /* user-defined Section 1.5 Writing Application Codes with PETSc context */
117:   Tao            tao;
118:   Vec            u;                      /* approximate solution vector */
120:   PetscInt       i, xs, xm, ind, j, lenglob;
121:   PetscReal      x, *wrk_ptr1, *wrk_ptr2;
122:   MatNullSpace   nsp;

124:    /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
125:      Initialize program and set problem parameters
126:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

129:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;

131:   /*initialize parameters */
132:   appctx.param.N    = 10;  /* order of the spectral element */
133:   appctx.param.E    = 8;  /* number of elements */
134:   appctx.param.L    = 1.0;  /* length of the domain */
135:   appctx.param.mu   = 0.00001; /* diffusion coefficient */
136:   appctx.param.a    = 0.0;     /* advection speed */
137:   appctx.initial_dt = 1e-4;
138:   appctx.param.steps = PETSC_MAX_INT;
139:   appctx.param.Tend  = 0.01;
140:   appctx.ncoeff      = 2;

142:   PetscOptionsGetInt(NULL,NULL,"-N",&appctx.param.N,NULL);
143:   PetscOptionsGetInt(NULL,NULL,"-E",&appctx.param.E,NULL);
144:   PetscOptionsGetInt(NULL,NULL,"-ncoeff",&appctx.ncoeff,NULL);
145:   PetscOptionsGetReal(NULL,NULL,"-Tend",&appctx.param.Tend,NULL);
146:   PetscOptionsGetReal(NULL,NULL,"-mu",&appctx.param.mu,NULL);
147:   PetscOptionsGetReal(NULL,NULL,"-a",&appctx.param.a,NULL);
148:   appctx.param.Le = appctx.param.L/appctx.param.E;


151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152:      Create GLL data structures
153:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
154:   PetscMalloc2(appctx.param.N,&appctx.SEMop.gll.nodes,appctx.param.N,&appctx.SEMop.gll.weights);
155:   PetscDTGaussLobattoLegendreQuadrature(appctx.param.N,PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA,appctx.SEMop.gll.nodes,appctx.SEMop.gll.weights);
156:   appctx.SEMop.gll.n = appctx.param.N;
157:   lenglob  = appctx.param.E*(appctx.param.N-1);

159:   /*
160:      Create distributed array (DMDA) to manage parallel grid and vectors
161:      and to set up the ghost point communication pattern.  There are E*(Nl-1)+1
162:      total grid values spread equally among all the processors, except first and last
163:   */

165:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_PERIODIC,lenglob,1,1,NULL,&appctx.da);
166:   DMSetFromOptions(appctx.da);
167:   DMSetUp(appctx.da);

169:   /*
170:      Extract global and local vectors from DMDA; we use these to store the
171:      approximate solution.  Then duplicate these for remaining vectors that
172:      have the same types.
173:   */

175:   DMCreateGlobalVector(appctx.da,&u);
176:   VecDuplicate(u,&appctx.dat.ic);
177:   VecDuplicate(u,&appctx.dat.true_solution);
178:   VecDuplicate(u,&appctx.dat.reference);
179:   VecDuplicate(u,&appctx.SEMop.grid);
180:   VecDuplicate(u,&appctx.SEMop.mass);
181:   VecDuplicate(u,&appctx.dat.curr_sol);
182:   VecDuplicate(u,&appctx.dat.joe);

184:   DMDAGetCorners(appctx.da,&xs,NULL,NULL,&xm,NULL,NULL);
185:   DMDAVecGetArray(appctx.da,appctx.SEMop.grid,&wrk_ptr1);
186:   DMDAVecGetArray(appctx.da,appctx.SEMop.mass,&wrk_ptr2);

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

190:     xs=xs/(appctx.param.N-1);
191:     xm=xm/(appctx.param.N-1);

193:   /*
194:      Build total grid and mass over entire mesh (multi-elemental)
195:   */

197:   for (i=xs; i<xs+xm; i++) {
198:     for (j=0; j<appctx.param.N-1; j++) {
199:       x = (appctx.param.Le/2.0)*(appctx.SEMop.gll.nodes[j]+1.0)+appctx.param.Le*i;
200:       ind=i*(appctx.param.N-1)+j;
201:       wrk_ptr1[ind]=x;
202:       wrk_ptr2[ind]=.5*appctx.param.Le*appctx.SEMop.gll.weights[j];
203:       if (j==0) wrk_ptr2[ind]+=.5*appctx.param.Le*appctx.SEMop.gll.weights[j];
204:     }
205:   }
206:   DMDAVecRestoreArray(appctx.da,appctx.SEMop.grid,&wrk_ptr1);
207:   DMDAVecRestoreArray(appctx.da,appctx.SEMop.mass,&wrk_ptr2);


210:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
211:    Create matrix data structure; set matrix evaluation routine.
212:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
213:   DMSetMatrixPreallocateOnly(appctx.da, PETSC_TRUE);
214:   DMCreateMatrix(appctx.da,&appctx.SEMop.stiff);
215:   DMCreateMatrix(appctx.da,&appctx.SEMop.advec);

217:   /*
218:    For linear problems with a time-dependent f(u,t) in the equation
219:    u_t = f(u,t), the user provides the discretized right-hand-side
220:    as a time-dependent matrix.
221:    */
222:   RHSLaplacian(appctx.ts,0.0,u,appctx.SEMop.stiff,appctx.SEMop.stiff,&appctx);
223:   RHSAdvection(appctx.ts,0.0,u,appctx.SEMop.advec,appctx.SEMop.advec,&appctx);
224:   MatAXPY(appctx.SEMop.stiff,-1.0,appctx.SEMop.advec,DIFFERENT_NONZERO_PATTERN);
225:   MatDuplicate(appctx.SEMop.stiff,MAT_COPY_VALUES,&appctx.SEMop.keptstiff);

227:   /* attach the null space to the matrix, this probably is not needed but does no harm */
228:   MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_TRUE,0,NULL,&nsp);
229:   MatSetNullSpace(appctx.SEMop.stiff,nsp);
230:   MatNullSpaceTest(nsp,appctx.SEMop.stiff,NULL);
231:   MatNullSpaceDestroy(&nsp);

233:   /* Create the TS solver that solves the ODE and its adjoint; set its options */
234:   TSCreate(PETSC_COMM_WORLD,&appctx.ts);
235:   TSSetSolutionFunction(appctx.ts,(PetscErrorCode (*)(TS,PetscReal,Vec, void *))ComputeReference,&appctx);
236:   TSSetProblemType(appctx.ts,TS_LINEAR);
237:   TSSetType(appctx.ts,TSRK);
238:   TSSetDM(appctx.ts,appctx.da);
239:   TSSetTime(appctx.ts,0.0);
240:   TSSetTimeStep(appctx.ts,appctx.initial_dt);
241:   TSSetMaxSteps(appctx.ts,appctx.param.steps);
242:   TSSetMaxTime(appctx.ts,appctx.param.Tend);
243:   TSSetExactFinalTime(appctx.ts,TS_EXACTFINALTIME_MATCHSTEP);
244:   TSSetTolerances(appctx.ts,1e-7,NULL,1e-7,NULL);
245:   TSSetFromOptions(appctx.ts);
246:   /* Need to save initial timestep user may have set with -ts_dt so it can be reset for each new TSSolve() */
247:   TSGetTimeStep(appctx.ts,&appctx.initial_dt);
248:   TSSetRHSFunction(appctx.ts,NULL,TSComputeRHSFunctionLinear,&appctx);
249:   TSSetRHSJacobian(appctx.ts,appctx.SEMop.stiff,appctx.SEMop.stiff,TSComputeRHSJacobianConstant,&appctx);
250:   /*  TSSetRHSFunction(appctx.ts,NULL,RHSFunction,&appctx);
251:       TSSetRHSJacobian(appctx.ts,appctx.SEMop.stiff,appctx.SEMop.stiff,RHSJacobian,&appctx); */

253:   /* Set random initial conditions as initial guess, compute analytic reference solution and analytic (true) initial conditions */
254:   ComputeSolutionCoefficients(&appctx);
255:   InitialConditions(appctx.dat.ic,&appctx);
256:   ComputeReference(appctx.ts,appctx.param.Tend,appctx.dat.reference,&appctx);
257:   ComputeReference(appctx.ts,0.0,appctx.dat.true_solution,&appctx);

259:   /* Set up to save trajectory before TSSetFromOptions() so that TSTrajectory options can be captured */
260:   TSSetSaveTrajectory(appctx.ts);
261:   TSSetFromOptions(appctx.ts);

263:   /* Create TAO solver and set desired solution method  */
264:   TaoCreate(PETSC_COMM_WORLD,&tao);
265:   TaoSetMonitor(tao,MonitorError,&appctx,MonitorDestroy);
266:   TaoSetType(tao,TAOBQNLS);
267:   TaoSetInitialVector(tao,appctx.dat.ic);
268:   /* Set routine for function and gradient evaluation  */
269:   TaoSetObjectiveAndGradientRoutine(tao,FormFunctionGradient,(void *)&appctx);
270:   /* Check for any TAO command line options  */
271:   TaoSetTolerances(tao,1e-8,PETSC_DEFAULT,PETSC_DEFAULT);
272:   TaoSetFromOptions(tao);
273:   TaoSolve(tao);

275:   TaoDestroy(&tao);
276:   PetscFree(appctx.solutioncoefficients);
277:   MatDestroy(&appctx.SEMop.advec);
278:   MatDestroy(&appctx.SEMop.stiff);
279:   MatDestroy(&appctx.SEMop.keptstiff);
280:   VecDestroy(&u);
281:   VecDestroy(&appctx.dat.ic);
282:   VecDestroy(&appctx.dat.joe);
283:   VecDestroy(&appctx.dat.true_solution);
284:   VecDestroy(&appctx.dat.reference);
285:   VecDestroy(&appctx.SEMop.grid);
286:   VecDestroy(&appctx.SEMop.mass);
287:   VecDestroy(&appctx.dat.curr_sol);
288:   PetscFree2(appctx.SEMop.gll.nodes,appctx.SEMop.gll.weights);
289:   DMDestroy(&appctx.da);
290:   TSDestroy(&appctx.ts);

292:   /*
293:      Always call PetscFinalize() before exiting a program.  This routine
294:        - finalizes the PETSc libraries as well as MPI
295:        - provides summary and diagnostic information if certain runtime
296:          options are chosen (e.g., -log_summary).
297:   */
298:     PetscFinalize();
299:     return ierr;
300: }

302: /*
303:     Computes the coefficients for the analytic solution to the PDE
304: */
305: PetscErrorCode ComputeSolutionCoefficients(AppCtx *appctx)
306: {
307:   PetscErrorCode    ierr;
308:   PetscRandom       rand;
309:   PetscInt          i;

312:   PetscMalloc1(appctx->ncoeff,&appctx->solutioncoefficients);
313:   PetscRandomCreate(PETSC_COMM_WORLD,&rand);
314:   PetscRandomSetInterval(rand,.9,1.0);
315:   for (i=0; i<appctx->ncoeff; i++) {
316:     PetscRandomGetValue(rand,&appctx->solutioncoefficients[i]);
317:   }
318:   PetscRandomDestroy(&rand);
319:   return(0);
320: }

322: /* --------------------------------------------------------------------- */
323: /*
324:    InitialConditions - Computes the (random) initial conditions for the Tao optimization solve (these are also initial conditions for the first TSSolve()

326:    Input Parameter:
327:    u - uninitialized solution vector (global)
328:    appctx - user-defined Section 1.5 Writing Application Codes with PETSc context

330:    Output Parameter:
331:    u - vector with solution at initial time (global)
332: */
333: PetscErrorCode InitialConditions(Vec u,AppCtx *appctx)
334: {
335:   PetscScalar       *s;
336:   const PetscScalar *xg;
337:   PetscErrorCode    ierr;
338:   PetscInt          i,j,lenglob;
339:   PetscReal         sum,val;
340:   PetscRandom       rand;

343:   PetscRandomCreate(PETSC_COMM_WORLD,&rand);
344:   PetscRandomSetInterval(rand,.9,1.0);
345:   DMDAVecGetArray(appctx->da,u,&s);
346:   DMDAVecGetArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
347:   lenglob  = appctx->param.E*(appctx->param.N-1);
348:   for (i=0; i<lenglob; i++) {
349:     s[i]= 0;
350:     for (j=0; j<appctx->ncoeff; j++) {
351:        PetscRandomGetValue(rand,&val);
352:       s[i] += val*PetscSinScalar(2*(j+1)*PETSC_PI*xg[i]);
353:     }
354:   }
355:   PetscRandomDestroy(&rand);
356:   DMDAVecRestoreArray(appctx->da,u,&s);
357:   DMDAVecRestoreArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
358:   /* make sure initial conditions do not contain the constant functions, since with periodic boundary conditions the constant functions introduce a null space */
359:   VecSum(u,&sum);
360:   VecShift(u,-sum/lenglob);
361:   return(0);
362: }


365: /*
366:    TrueSolution() computes the true solution for the Tao optimization solve which means they are the initial conditions for the objective function.

368:              InitialConditions() computes the initial conditions for the begining of the Tao iterations

370:    Input Parameter:
371:    u - uninitialized solution vector (global)
372:    appctx - user-defined Section 1.5 Writing Application Codes with PETSc context

374:    Output Parameter:
375:    u - vector with solution at initial time (global)
376: */
377: PetscErrorCode TrueSolution(Vec u,AppCtx *appctx)
378: {
379:   PetscScalar       *s;
380:   const PetscScalar *xg;
381:   PetscErrorCode    ierr;
382:   PetscInt          i,j,lenglob;
383:   PetscReal         sum;

386:   DMDAVecGetArray(appctx->da,u,&s);
387:   DMDAVecGetArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
388:   lenglob  = appctx->param.E*(appctx->param.N-1);
389:   for (i=0; i<lenglob; i++) {
390:     s[i]= 0;
391:     for (j=0; j<appctx->ncoeff; j++) {
392:       s[i] += appctx->solutioncoefficients[j]*PetscSinScalar(2*(j+1)*PETSC_PI*xg[i]);
393:     }
394:   }
395:   DMDAVecRestoreArray(appctx->da,u,&s);
396:   DMDAVecRestoreArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
397:   /* make sure initial conditions do not contain the constant functions, since with periodic boundary conditions the constant functions introduce a null space */
398:   VecSum(u,&sum);
399:   VecShift(u,-sum/lenglob);
400:   return(0);
401: }
402: /* --------------------------------------------------------------------- */
403: /*
404:    Sets the desired profile for the final end time

406:    Input Parameters:
407:    t - final time
408:    obj - vector storing the desired profile
409:    appctx - user-defined Section 1.5 Writing Application Codes with PETSc context

411: */
412: PetscErrorCode ComputeReference(TS ts,PetscReal t,Vec obj,AppCtx *appctx)
413: {
414:   PetscScalar       *s,tc;
415:   const PetscScalar *xg;
416:   PetscErrorCode    ierr;
417:   PetscInt          i, j,lenglob;

420:   DMDAVecGetArray(appctx->da,obj,&s);
421:   DMDAVecGetArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
422:   lenglob  = appctx->param.E*(appctx->param.N-1);
423:   for (i=0; i<lenglob; i++) {
424:     s[i]= 0;
425:     for (j=0; j<appctx->ncoeff; j++) {
426:       tc    = -appctx->param.mu*(j+1)*(j+1)*4.0*PETSC_PI*PETSC_PI*t;
427:       s[i] += appctx->solutioncoefficients[j]*PetscSinScalar(2*(j+1)*PETSC_PI*(xg[i] + appctx->param.a*t))*PetscExpReal(tc);
428:     }
429:   }
430:   DMDAVecRestoreArray(appctx->da,obj,&s);
431:   DMDAVecRestoreArrayRead(appctx->da,appctx->SEMop.grid,(void*)&xg);
432:   return(0);
433: }

435: PetscErrorCode RHSFunction(TS ts,PetscReal t,Vec globalin,Vec globalout,void *ctx)
436: {
438:   AppCtx          *appctx = (AppCtx*)ctx;

441:   MatMult(appctx->SEMop.keptstiff,globalin,globalout);
442:   return(0);
443: }

445: PetscErrorCode RHSJacobian(TS ts,PetscReal t,Vec globalin,Mat A, Mat B,void *ctx)
446: {
448:   AppCtx         *appctx = (AppCtx*)ctx;

451:   MatCopy(appctx->SEMop.keptstiff,A,DIFFERENT_NONZERO_PATTERN);
452:   return(0);
453: }

455: /* --------------------------------------------------------------------- */

457: /*
458:    RHSLaplacian -   matrix for diffusion

460:    Input Parameters:
461:    ts - the TS context
462:    t - current time  (ignored)
463:    X - current solution (ignored)
464:    dummy - optional user-defined context, as set by TSetRHSJacobian()

466:    Output Parameters:
467:    AA - Jacobian matrix
468:    BB - optionally different matrix from which the preconditioner is built
469:    str - flag indicating matrix structure

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

473: */
474: PetscErrorCode RHSLaplacian(TS ts,PetscReal t,Vec X,Mat A,Mat BB,void *ctx)
475: {
476:   PetscReal      **temp;
477:   PetscReal      vv;
478:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined Section 1.5 Writing Application Codes with PETSc context */
480:   PetscInt       i,xs,xn,l,j;
481:   PetscInt       *rowsDM;

484:   /*
485:    Creates the element stiffness matrix for the given gll
486:    */
487:   PetscGaussLobattoLegendreElementLaplacianCreate(appctx->SEMop.gll.n,appctx->SEMop.gll.nodes,appctx->SEMop.gll.weights,&temp);

489:   /* scale by the size of the element */
490:   for (i=0; i<appctx->param.N; i++) {
491:     vv=-appctx->param.mu*2.0/appctx->param.Le;
492:     for (j=0; j<appctx->param.N; j++) temp[i][j]=temp[i][j]*vv;
493:   }

495:   MatSetOption(A,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_FALSE);
496:   DMDAGetCorners(appctx->da,&xs,NULL,NULL,&xn,NULL,NULL);

498:   if (appctx->param.N-1 < 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_ARG_OUTOFRANGE,"Polynomial order must be at least 2");
499:   xs   = xs/(appctx->param.N-1);
500:   xn   = xn/(appctx->param.N-1);

502:   PetscMalloc1(appctx->param.N,&rowsDM);
503:   /*
504:    loop over local elements
505:    */
506:   for (j=xs; j<xs+xn; j++) {
507:     for (l=0; l<appctx->param.N; l++) {
508:       rowsDM[l] = 1+(j-xs)*(appctx->param.N-1)+l;
509:     }
510:     MatSetValuesLocal(A,appctx->param.N,rowsDM,appctx->param.N,rowsDM,&temp[0][0],ADD_VALUES);
511:   }
512:   PetscFree(rowsDM);
513:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
514:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
515:   VecReciprocal(appctx->SEMop.mass);
516:   MatDiagonalScale(A,appctx->SEMop.mass,0);
517:   VecReciprocal(appctx->SEMop.mass);

519:   PetscGaussLobattoLegendreElementLaplacianDestroy(appctx->SEMop.gll.n,appctx->SEMop.gll.nodes,appctx->SEMop.gll.weights,&temp);
520:   return(0);
521: }

523: /*
524:     Almost identical to Laplacian

526:     Note that the element matrix is NOT scaled by the size of element like the Laplacian term.
527:  */
528: PetscErrorCode RHSAdvection(TS ts,PetscReal t,Vec X,Mat A,Mat BB,void *ctx)
529: {
530:   PetscReal      **temp;
531:   PetscReal      vv;
532:   AppCtx         *appctx = (AppCtx*)ctx;     /* user-defined Section 1.5 Writing Application Codes with PETSc context */
534:   PetscInt       i,xs,xn,l,j;
535:   PetscInt       *rowsDM;

538:   /*
539:    Creates the element stiffness matrix for the given gll
540:    */
541:   PetscGaussLobattoLegendreElementAdvectionCreate(appctx->SEMop.gll.n,appctx->SEMop.gll.nodes,appctx->SEMop.gll.weights,&temp);

543:   /* scale by the size of the element */
544:   for (i=0; i<appctx->param.N; i++) {
545:     vv = -appctx->param.a;
546:     for (j=0; j<appctx->param.N; j++) temp[i][j]=temp[i][j]*vv;
547:   }

549:   MatSetOption(A,MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_FALSE);
550:   DMDAGetCorners(appctx->da,&xs,NULL,NULL,&xn,NULL,NULL);

552:   if (appctx->param.N-1 < 1) SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_ARG_OUTOFRANGE,"Polynomial order must be at least 2");
553:   xs   = xs/(appctx->param.N-1);
554:   xn   = xn/(appctx->param.N-1);

556:   PetscMalloc1(appctx->param.N,&rowsDM);
557:   /*
558:    loop over local elements
559:    */
560:   for (j=xs; j<xs+xn; j++) {
561:     for (l=0; l<appctx->param.N; l++) {
562:       rowsDM[l] = 1+(j-xs)*(appctx->param.N-1)+l;
563:     }
564:     MatSetValuesLocal(A,appctx->param.N,rowsDM,appctx->param.N,rowsDM,&temp[0][0],ADD_VALUES);
565:   }
566:   PetscFree(rowsDM);
567:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
568:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
569:   VecReciprocal(appctx->SEMop.mass);
570:   MatDiagonalScale(A,appctx->SEMop.mass,0);
571:   VecReciprocal(appctx->SEMop.mass);

573:   PetscGaussLobattoLegendreElementAdvectionDestroy(appctx->SEMop.gll.n,appctx->SEMop.gll.nodes,appctx->SEMop.gll.weights,&temp);
574:   return(0);
575: }

577: /* ------------------------------------------------------------------ */
578: /*
579:    FormFunctionGradient - Evaluates the function and corresponding gradient.

581:    Input Parameters:
582:    tao - the Tao context
583:    ic   - the input vector
584:    ctx - optional user-defined context, as set when calling TaoSetObjectiveAndGradientRoutine()

586:    Output Parameters:
587:    f   - the newly evaluated function
588:    G   - the newly evaluated gradient

590:    Notes:

592:           The forward equation is
593:               M u_t = F(U)
594:           which is converted to
595:                 u_t = M^{-1} F(u)
596:           in the user code since TS has no direct way of providing a mass matrix. The Jacobian of this is
597:                  M^{-1} J
598:           where J is the Jacobian of F. Now the adjoint equation is
599:                 M v_t = J^T v
600:           but TSAdjoint does not solve this since it can only solve the transposed system for the
601:           Jacobian the user provided. Hence TSAdjoint solves
602:                  w_t = J^T M^{-1} w  (where w = M v)
603:           since there is no way to indicate the mass matrix as a separate entitity to TS. Thus one
604:           must be careful in initializing the "adjoint equation" and using the result. This is
605:           why
606:               G = -2 M(u(T) - u_d)
607:           below (instead of -2(u(T) - u_d)


610: */
611: PetscErrorCode FormFunctionGradient(Tao tao,Vec ic,PetscReal *f,Vec G,void *ctx)
612: {
613:   AppCtx           *appctx = (AppCtx*)ctx;     /* user-defined Section 1.5 Writing Application Codes with PETSc context */
614:   PetscErrorCode    ierr;
615:   Vec               temp;

618:   TSSetTime(appctx->ts,0.0);
619:   TSSetStepNumber(appctx->ts,0);
620:   TSSetTimeStep(appctx->ts,appctx->initial_dt);
621:   VecCopy(ic,appctx->dat.curr_sol);

623:   TSSolve(appctx->ts,appctx->dat.curr_sol);
624:   VecCopy(appctx->dat.curr_sol,appctx->dat.joe);

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

629:   /*     Compute the objective/cost function   */
630:   VecDuplicate(G,&temp);
631:   VecPointwiseMult(temp,G,G);
632:   VecDot(temp,appctx->SEMop.mass,f);
633:   VecDestroy(&temp);

635:   /*     Compute initial conditions for the adjoint integration. See Notes above  */
636:   VecScale(G, -2.0);
637:   VecPointwiseMult(G,G,appctx->SEMop.mass);
638:   TSSetCostGradients(appctx->ts,1,&G,NULL);

640:   TSAdjointSolve(appctx->ts);
641:   /* VecPointwiseDivide(G,G,appctx->SEMop.mass);*/
642:   return(0);
643: }

645: PetscErrorCode MonitorError(Tao tao,void *ctx)
646: {
647:   AppCtx         *appctx = (AppCtx*)ctx;
648:   Vec            temp,grad;
649:   PetscReal      nrm;
651:   PetscInt       its;
652:   PetscReal      fct,gnorm;

655:   VecDuplicate(appctx->dat.ic,&temp);
656:   VecWAXPY(temp,-1.0,appctx->dat.ic,appctx->dat.true_solution);
657:   VecPointwiseMult(temp,temp,temp);
658:   VecDot(temp,appctx->SEMop.mass,&nrm);
659:   nrm   = PetscSqrtReal(nrm);
660:   TaoGetGradientVector(tao,&grad);
661:   VecPointwiseMult(temp,temp,temp);
662:   VecDot(temp,appctx->SEMop.mass,&gnorm);
663:   gnorm = PetscSqrtReal(gnorm);
664:   VecDestroy(&temp);
665:   TaoGetIterationNumber(tao,&its);
666:   TaoGetObjective(tao,&fct);
667:   if (!its) {
668:     PetscPrintf(PETSC_COMM_WORLD,"%% Iteration Error Objective Gradient-norm\n");
669:     PetscPrintf(PETSC_COMM_WORLD,"history = [\n");
670:   }
671:   PetscPrintf(PETSC_COMM_WORLD,"%3D %g %g %g\n",its,(double)nrm,(double)fct,(double)gnorm);
672:   return(0);
673: }

675: PetscErrorCode MonitorDestroy(void **ctx)
676: {

680:   PetscPrintf(PETSC_COMM_WORLD,"];\n");
681:   return(0);
682: }


685: /*TEST

687:    build:
688:      requires: !complex

690:    test:
691:      requires: !single
692:      args:  -ts_adapt_dt_max 3.e-3 -E 10 -N 8 -ncoeff 5 -tao_bqnls_mat_lmvm_scale_type none

694:    test:
695:      suffix: cn
696:      requires: !single
697:      args:  -ts_type cn -ts_dt .003 -pc_type lu -E 10 -N 8 -ncoeff 5 -tao_bqnls_mat_lmvm_scale_type none

699:    test:
700:      suffix: 2
701:      requires: !single
702:      args:  -ts_adapt_dt_max 3.e-3 -E 10 -N 8 -ncoeff 5  -a .1 -tao_bqnls_mat_lmvm_scale_type none


705: TEST*/