Actual source code: ex67.c


  2: static char help[] = "Krylov methods to solve u''  = f in parallel with periodic boundary conditions,\n\
  3:                       with a singular, inconsistent system.\n\n";

  5: /*T
  6:    Concepts: KSP^basic parallel example
  7:    Concepts: periodic boundary conditions
  8:    Processors: n
  9: T*/

 11: /*

 13:    This tests solving singular inconsistent systems with GMRES

 15:    Default: Solves a symmetric system
 16:    -symmetric false: Solves a non-symmetric system where nullspace(A) != nullspace(A')

 18:   -ksp_pc_side left or right

 20:    See the KSPSolve() for a discussion of when right preconditioning with nullspace(A) != nullspace(A') can fail to produce the
 21:    norm minimizing solution.

 23:    Note that though this example does solve the system with right preconditioning and nullspace(A) != nullspace(A') it does not produce the
 24:    norm minimizing solution, that is the computed solution is not orthogonal to the nullspace(A).

 26:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 27:    Include "petscksp.h" so that we can use KSP solvers.  Note that this
 28:    file automatically includes:
 29:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 30:      petscmat.h - matrices
 31:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 32:      petscviewer.h - viewers               petscpc.h  - preconditioners
 33:      petscksp.h   - linear solvers
 34: */

 36: #include <petscdm.h>
 37: #include <petscdmda.h>
 38: #include <petscsnes.h>
 39: #include <petsc/private/kspimpl.h>

 41: PetscBool symmetric = PETSC_TRUE;

 43: PetscErrorCode FormMatrix(Mat,void*);
 44: PetscErrorCode FormRightHandSide(Vec,void*);

 46: int main(int argc,char **argv)
 47: {
 48:   KSP            ksp;
 49:   Mat            J;
 50:   DM             da;
 51:   Vec            x,r;              /* vectors */
 52:   PetscInt       M = 10;
 53:   MatNullSpace   constants,nconstants;

 55:   PetscInitialize(&argc,&argv,(char*)0,help);
 56:   PetscOptionsGetInt(NULL,NULL,"-M",&M,NULL);
 57:   PetscOptionsGetBool(NULL,NULL,"-symmetric",&symmetric,NULL);

 59:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 60:      Create linear solver context
 61:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 63:   KSPCreate(PETSC_COMM_WORLD,&ksp);

 65:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 66:      Create vector data structures; set function evaluation routine
 67:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 69:   /*
 70:      Create distributed array (DMDA) to manage parallel grid and vectors
 71:   */
 72:   DMDACreate1d(PETSC_COMM_WORLD,DM_BOUNDARY_PERIODIC,M,1,2,NULL,&da);
 73:   DMSetFromOptions(da);
 74:   DMSetUp(da);

 76:   /*
 77:      Extract global and local vectors from DMDA; then duplicate for remaining
 78:      vectors that are the same types
 79:   */
 80:   DMCreateGlobalVector(da,&x);
 81:   VecDuplicate(x,&r);

 83:   /*
 84:      Set function evaluation routine and vector.  Whenever the nonlinear
 85:      solver needs to compute the nonlinear function, it will call this
 86:      routine.
 87:       - Note that the final routine argument is the user-defined
 88:         context that provides application-specific data for the
 89:         function evaluation routine.
 90:   */
 91:   FormRightHandSide(r,da);

 93:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 94:      Create matrix data structure;
 95:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 96:   DMCreateMatrix(da,&J);
 97:   MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_TRUE,0,NULL,&constants);
 98:   if (symmetric) {
 99:     MatSetOption(J,MAT_SYMMETRY_ETERNAL,PETSC_TRUE);
100:     MatSetOption(J,MAT_SYMMETRIC,PETSC_TRUE);
101:   } else {
102:     Vec         n;
103:     PetscInt    zero = 0;
104:     PetscScalar zeros = 0.0;
105:     VecDuplicate(x,&n);
106:     /* the nullspace(A') is the constant vector but with a zero in the very first entry; hence nullspace(A') != nullspace(A) */
107:     VecSet(n,1.0);
108:     VecSetValues(n,1,&zero,&zeros,INSERT_VALUES);
109:     VecAssemblyBegin(n);
110:     VecAssemblyEnd(n);
111:     VecNormalize(n,NULL);
112:     MatNullSpaceCreate(PETSC_COMM_WORLD,PETSC_FALSE,1,&n,&nconstants);
113:     MatSetTransposeNullSpace(J,nconstants);
114:     MatNullSpaceDestroy(&nconstants);
115:     VecDestroy(&n);
116:   }
117:   MatSetNullSpace(J,constants);
118:   FormMatrix(J,da);

120:   KSPSetOperators(ksp,J,J);

122:   KSPSetFromOptions(ksp);
123:   KSPSolve(ksp,r,x);
124:   KSPSolveTranspose(ksp,r,x);

126:   VecDestroy(&x);
127:   VecDestroy(&r);
128:   MatDestroy(&J);
129:   MatNullSpaceDestroy(&constants);
130:   KSPDestroy(&ksp);
131:   DMDestroy(&da);
132:   PetscFinalize();
133:   return 0;
134: }

136: /*

138:      This intentionally includes something in the right hand side that is not in the range of the matrix A.
139:      Since MatSetNullSpace() is called and the matrix is symmetric; the Krylov method automatically removes this
140:      portion of the right hand side before solving the linear system.
141: */
142: PetscErrorCode FormRightHandSide(Vec f,void *ctx)
143: {
144:   DM             da    = (DM) ctx;
145:   PetscScalar    *ff;
146:   PetscInt       i,M,xs,xm;
147:   PetscReal      h;

150:   DMDAVecGetArray(da,f,&ff);

152:   /*
153:      Get local grid boundaries (for 1-dimensional DMDA):
154:        xs, xm  - starting grid index, width of local grid (no ghost points)
155:   */
156:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
157:   DMDAGetInfo(da,NULL,&M,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);

159:   /*
160:      Compute function over locally owned part of the grid
161:      Note the [i-1] and [i+1] will automatically access the ghost points from other processes or the periodic points.
162:   */
163:   h = 1.0/M;
164:   for (i=xs; i<xs+xm; i++) ff[i] = - PetscSinReal(2.0*PETSC_PI*i*h) + 1.0;

166:   /*
167:      Restore vectors
168:   */
169:   DMDAVecRestoreArray(da,f,&ff);
170:   return 0;
171: }
172: /* ------------------------------------------------------------------- */
173: PetscErrorCode FormMatrix(Mat jac,void *ctx)
174: {
175:   PetscScalar    A[3];
176:   PetscInt       i,M,xs,xm;
177:   DM             da = (DM) ctx;
178:   MatStencil     row,cols[3];
179:   PetscReal      h;

182:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

184:   /*
185:     Get range of locally owned matrix
186:   */
187:   DMDAGetInfo(da,NULL,&M,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL,NULL);

189:   MatZeroEntries(jac);
190:   h = 1.0/M;
191:   /* because of periodic boundary conditions we can simply loop over all local nodes and access to the left and right */
192:   if (symmetric) {
193:     for (i=xs; i<xs+xm; i++) {
194:       row.i = i;
195:       cols[0].i = i - 1;
196:       cols[1].i = i;
197:       cols[2].i = i + 1;
198:       A[0] = A[2] = 1.0/(h*h); A[1] = -2.0/(h*h);
199:       MatSetValuesStencil(jac,1,&row,3,cols,A,ADD_VALUES);
200:     }
201:   } else {
202:     MatStencil  *acols;
203:     PetscScalar *avals;

205:     /* only works for one process */
206:     MatSetOption(jac,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_FALSE);
207:     row.i = 0;
208:     PetscMalloc1(M,&acols);
209:     PetscMalloc1(M,&avals);
210:     for (i=0; i<M; i++) {
211:       acols[i].i = i;
212:       avals[i]  = (i % 2) ? 1 : -1;
213:     }
214:     MatSetValuesStencil(jac,1,&row,M,acols,avals,ADD_VALUES);
215:     PetscFree(acols);
216:     PetscFree(avals);
217:     row.i = 1;
218:     cols[0].i = - 1;
219:     cols[1].i = 1;
220:     cols[2].i = 1 + 1;
221:     A[0] = A[2] = 1.0/(h*h); A[1] = -2.0/(h*h);
222:     MatSetValuesStencil(jac,1,&row,3,cols,A,ADD_VALUES);
223:     for (i=2; i<xs+xm-1; i++) {
224:       row.i = i;
225:       cols[0].i = i - 1;
226:       cols[1].i = i;
227:       cols[2].i = i + 1;
228:       A[0] = A[2] = 1.0/(h*h); A[1] = -2.0/(h*h);
229:       MatSetValuesStencil(jac,1,&row,3,cols,A,ADD_VALUES);
230:     }
231:     row.i = M - 1 ;
232:     cols[0].i = M-2;
233:     cols[1].i = M-1;
234:     cols[2].i = M+1;
235:     A[0] = A[2] = 1.0/(h*h); A[1] = -2.0/(h*h);
236:     MatSetValuesStencil(jac,1,&row,3,cols,A,ADD_VALUES);
237:   }
238:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
239:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
240:   return 0;
241: }

243: /*TEST

245:    test:
246:       suffix: nonsymmetric_left
247:       args: -symmetric false -ksp_view -ksp_converged_reason -pc_type jacobi -mat_no_inode -ksp_monitor_true_residual -ksp_rtol 1.e-14 -ksp_max_it 12 -ksp_pc_side left
248:       filter: sed 's/ATOL/RTOL/g'
249:       requires: !single

251:    test:
252:       suffix: nonsymmetric_right
253:       args: -symmetric false -ksp_view -ksp_converged_reason -pc_type jacobi -mat_no_inode -ksp_monitor_true_residual -ksp_rtol 1.e-14 -ksp_max_it 12 -ksp_pc_side right
254:       filter: sed 's/ATOL/RTOL/g'
255:       requires: !single

257:    test:
258:       suffix: symmetric_left
259:       args: -ksp_view -ksp_converged_reason -pc_type sor -mat_no_inode -ksp_monitor_true_residual -ksp_rtol 1.e-14 -ksp_max_it 12 -ksp_pc_side left
260:       requires: !single

262:    test:
263:       suffix: symmetric_right
264:       args: -ksp_view -ksp_converged_reason -pc_type sor -mat_no_inode -ksp_monitor_true_residual -ksp_rtol 1.e-14 -ksp_max_it 12 -ksp_pc_side right
265:       filter: sed 's/ATOL/RTOL/g'
266:       requires: !single

268:    test:
269:       suffix: transpose_asm
270:       args: -symmetric false -ksp_monitor -ksp_view -pc_type asm -sub_pc_type lu -sub_pc_factor_zeropivot 1.e-33 -ksp_converged_reason
271:       filter: sed 's/ATOL/RTOL/g'

273: TEST*/