Actual source code: schurm.c

  1: #include <../src/ksp/ksp/utils/schurm/schurm.h>

  3: const char *const MatSchurComplementAinvTypes[] = {"DIAG", "LUMP", "BLOCKDIAG", "FULL", "MatSchurComplementAinvType", "MAT_SCHUR_COMPLEMENT_AINV_", NULL};

  5: PetscErrorCode MatCreateVecs_SchurComplement(Mat N, Vec *right, Vec *left)
  6: {
  7:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

  9:   PetscFunctionBegin;
 10:   if (Na->D) {
 11:     PetscCall(MatCreateVecs(Na->D, right, left));
 12:     PetscFunctionReturn(PETSC_SUCCESS);
 13:   }
 14:   if (right) PetscCall(MatCreateVecs(Na->B, right, NULL));
 15:   if (left) PetscCall(MatCreateVecs(Na->C, NULL, left));
 16:   PetscFunctionReturn(PETSC_SUCCESS);
 17: }

 19: PetscErrorCode MatView_SchurComplement(Mat N, PetscViewer viewer)
 20: {
 21:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

 23:   PetscFunctionBegin;
 24:   PetscCall(PetscViewerASCIIPrintf(viewer, "Schur complement A11 - A10 inv(A00) A01\n"));
 25:   if (Na->D) {
 26:     PetscCall(PetscViewerASCIIPrintf(viewer, "A11\n"));
 27:     PetscCall(PetscViewerASCIIPushTab(viewer));
 28:     PetscCall(MatView(Na->D, viewer));
 29:     PetscCall(PetscViewerASCIIPopTab(viewer));
 30:   } else {
 31:     PetscCall(PetscViewerASCIIPrintf(viewer, "A11 = 0\n"));
 32:   }
 33:   PetscCall(PetscViewerASCIIPrintf(viewer, "A10\n"));
 34:   PetscCall(PetscViewerASCIIPushTab(viewer));
 35:   PetscCall(MatView(Na->C, viewer));
 36:   PetscCall(PetscViewerASCIIPopTab(viewer));
 37:   PetscCall(PetscViewerASCIIPrintf(viewer, "KSP solver for A00 block viewable with the additional option -%sksp_view\n", ((PetscObject)Na->ksp)->prefix ? ((PetscObject)Na->ksp)->prefix : NULL));
 38:   PetscCall(PetscViewerASCIIPrintf(viewer, "A01\n"));
 39:   PetscCall(PetscViewerASCIIPushTab(viewer));
 40:   PetscCall(MatView(Na->B, viewer));
 41:   PetscCall(PetscViewerASCIIPopTab(viewer));
 42:   PetscFunctionReturn(PETSC_SUCCESS);
 43: }

 45: /*
 46:            A11^T - A01^T ksptrans(A00,Ap00) A10^T
 47: */
 48: PetscErrorCode MatMultTranspose_SchurComplement(Mat N, Vec x, Vec y)
 49: {
 50:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

 52:   PetscFunctionBegin;
 53:   if (!Na->work1) PetscCall(MatCreateVecs(Na->A, &Na->work1, NULL));
 54:   if (!Na->work2) PetscCall(MatCreateVecs(Na->A, &Na->work2, NULL));
 55:   PetscCall(MatMultTranspose(Na->C, x, Na->work1));
 56:   PetscCall(KSPSolveTranspose(Na->ksp, Na->work1, Na->work2));
 57:   PetscCall(MatMultTranspose(Na->B, Na->work2, y));
 58:   PetscCall(VecScale(y, -1.0));
 59:   if (Na->D) PetscCall(MatMultTransposeAdd(Na->D, x, y, y));
 60:   PetscFunctionReturn(PETSC_SUCCESS);
 61: }

 63: /*
 64:            A11 - A10 ksp(A00,Ap00) A01
 65: */
 66: PetscErrorCode MatMult_SchurComplement(Mat N, Vec x, Vec y)
 67: {
 68:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

 70:   PetscFunctionBegin;
 71:   if (!Na->work1) PetscCall(MatCreateVecs(Na->A, &Na->work1, NULL));
 72:   if (!Na->work2) PetscCall(MatCreateVecs(Na->A, &Na->work2, NULL));
 73:   PetscCall(MatMult(Na->B, x, Na->work1));
 74:   PetscCall(KSPSolve(Na->ksp, Na->work1, Na->work2));
 75:   PetscCall(MatMult(Na->C, Na->work2, y));
 76:   PetscCall(VecScale(y, -1.0));
 77:   if (Na->D) PetscCall(MatMultAdd(Na->D, x, y, y));
 78:   PetscFunctionReturn(PETSC_SUCCESS);
 79: }

 81: /*
 82:            A11 - A10 ksp(A00,Ap00) A01
 83: */
 84: PetscErrorCode MatMultAdd_SchurComplement(Mat N, Vec x, Vec y, Vec z)
 85: {
 86:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

 88:   PetscFunctionBegin;
 89:   if (!Na->work1) PetscCall(MatCreateVecs(Na->A, &Na->work1, NULL));
 90:   if (!Na->work2) PetscCall(MatCreateVecs(Na->A, &Na->work2, NULL));
 91:   PetscCall(MatMult(Na->B, x, Na->work1));
 92:   PetscCall(KSPSolve(Na->ksp, Na->work1, Na->work2));
 93:   if (y == z) {
 94:     PetscCall(VecScale(Na->work2, -1.0));
 95:     PetscCall(MatMultAdd(Na->C, Na->work2, z, z));
 96:   } else {
 97:     PetscCall(MatMult(Na->C, Na->work2, z));
 98:     PetscCall(VecAYPX(z, -1.0, y));
 99:   }
100:   if (Na->D) PetscCall(MatMultAdd(Na->D, x, z, z));
101:   PetscFunctionReturn(PETSC_SUCCESS);
102: }

104: PetscErrorCode MatSetFromOptions_SchurComplement(Mat N, PetscOptionItems PetscOptionsObject)
105: {
106:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

108:   PetscFunctionBegin;
109:   PetscOptionsHeadBegin(PetscOptionsObject, "MatSchurComplementOptions");
110:   Na->ainvtype = MAT_SCHUR_COMPLEMENT_AINV_DIAG;
111:   PetscCall(PetscOptionsEnum("-mat_schur_complement_ainv_type", "Type of approximation for DIAGFORM(A00) used when assembling Sp = A11 - A10 inv(DIAGFORM(A00)) A01", "MatSchurComplementSetAinvType", MatSchurComplementAinvTypes, (PetscEnum)Na->ainvtype,
112:                              (PetscEnum *)&Na->ainvtype, NULL));
113:   PetscOptionsHeadEnd();
114:   PetscCall(KSPSetFromOptions(Na->ksp));
115:   PetscFunctionReturn(PETSC_SUCCESS);
116: }

118: PetscErrorCode MatDestroy_SchurComplement(Mat N)
119: {
120:   Mat_SchurComplement *Na = (Mat_SchurComplement *)N->data;

122:   PetscFunctionBegin;
123:   PetscCall(MatDestroy(&Na->A));
124:   PetscCall(MatDestroy(&Na->Ap));
125:   PetscCall(MatDestroy(&Na->B));
126:   PetscCall(MatDestroy(&Na->C));
127:   PetscCall(MatDestroy(&Na->D));
128:   PetscCall(VecDestroy(&Na->work1));
129:   PetscCall(VecDestroy(&Na->work2));
130:   PetscCall(KSPDestroy(&Na->ksp));
131:   PetscCall(PetscFree(N->data));
132:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_schurcomplement_seqdense_C", NULL));
133:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_schurcomplement_mpidense_C", NULL));
134:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_anytype_C", NULL));
135:   PetscFunctionReturn(PETSC_SUCCESS);
136: }

138: /*@
139:   MatCreateSchurComplement - Creates a new `Mat` that behaves like the Schur complement of a matrix

141:   Collective

143:   Input Parameters:
144: + A00  - the upper-left block of the original matrix $A = [A00 A01; A10 A11]$
145: . Ap00 - matrix from which the preconditioner is constructed for use in $ksp(A00,Ap00)$ to approximate the action of $A00^{-1}$
146: . A01  - the upper-right block of the original matrix $A = [A00 A01; A10 A11]$
147: . A10  - the lower-left block of the original matrix $A = [A00 A01; A10 A11]$
148: - A11  - (optional) the lower-right block of the original matrix $A = [A00 A01; A10 A11]$

150:   Output Parameter:
151: . S - the matrix that behaves as the Schur complement $S = A11 - A10 ksp(A00,Ap00) A01$

153:   Level: intermediate

155:   Notes:
156:   The Schur complement is NOT explicitly formed! Rather, this function returns a virtual Schur complement
157:   that can compute the matrix-vector product by using formula $S = A11 - A10 A^{-1} A01$
158:   for Schur complement `S` and a `KSP` solver to approximate the action of $A^{-1}$.

160:   All four matrices must have the same MPI communicator.

162:   `A00` and  `A11` must be square matrices.

164:   `MatGetSchurComplement()` takes as arguments the index sets for the submatrices and returns both the virtual Schur complement (what this returns) plus
165:   a sparse approximation to the Schur complement (useful for building a preconditioner for the Schur complement) which can be obtained from this
166:   matrix with `MatSchurComplementGetPmat()`

168:   Developer Notes:
169:   The API that includes `MatGetSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementGetPmat()` should be refactored to
170:   remove redundancy and be clearer and simpler.

172: .seealso: [](ch_ksp), `MatCreateNormal()`, `MatMult()`, `MatCreate()`, `MatSchurComplementGetKSP()`, `MatSchurComplementUpdateSubMatrices()`, `MatCreateTranspose()`, `MatGetSchurComplement()`,
173:           `MatSchurComplementGetPmat()`, `MatSchurComplementSetSubMatrices()`
174: @*/
175: PetscErrorCode MatCreateSchurComplement(Mat A00, Mat Ap00, Mat A01, Mat A10, Mat A11, Mat *S)
176: {
177:   PetscFunctionBegin;
178:   PetscCall(KSPInitializePackage());
179:   PetscCall(MatCreate(PetscObjectComm((PetscObject)A00), S));
180:   PetscCall(MatSetType(*S, MATSCHURCOMPLEMENT));
181:   PetscCall(MatSchurComplementSetSubMatrices(*S, A00, Ap00, A01, A10, A11));
182:   PetscFunctionReturn(PETSC_SUCCESS);
183: }

185: /*@
186:   MatSchurComplementSetSubMatrices - Sets the matrices that define the Schur complement

188:   Collective

190:   Input Parameters:
191: + S    - matrix obtained with `MatSetType`(S,`MATSCHURCOMPLEMENT`)
192: . A00  - the upper-left block of the original matrix $A = [A00 A01; A10 A11]$
193: . Ap00 - matrix from which the preconditioner is constructed for use in $ksp(A00,Ap00)$ to approximate the action of $A00^{-1}$
194: . A01  - the upper-right block of the original matrix $A = [A00 A01; A10 A11]$
195: . A10  - the lower-left block of the original matrix $A = [A00 A01; A10 A11]$
196: - A11  - (optional) the lower-right block of the original matrix $A = [A00 A01; A10 A11]$

198:   Level: intermediate

200:   Notes:
201:   The Schur complement is NOT explicitly formed! Rather, this
202:   object performs the matrix-vector product of the Schur complement by using formula $S = A11 - A10 ksp(A00,Ap00) A01$

204:   All four matrices must have the same MPI communicator.

206:   `A00` and `A11` must be square matrices.

208:   This is to be used in the context of code such as
209: .vb
210:      MatSetType(S,MATSCHURCOMPLEMENT);
211:      MatSchurComplementSetSubMatrices(S,...);
212: .ve
213:   while `MatSchurComplementUpdateSubMatrices()` should only be called after `MatCreateSchurComplement()` or `MatSchurComplementSetSubMatrices()`

215: .seealso: [](ch_ksp), `Mat`, `MatCreateNormal()`, `MatMult()`, `MatCreate()`, `MatSchurComplementGetKSP()`, `MatSchurComplementUpdateSubMatrices()`, `MatCreateTranspose()`, `MatCreateSchurComplement()`, `MatGetSchurComplement()`
216: @*/
217: PetscErrorCode MatSchurComplementSetSubMatrices(Mat S, Mat A00, Mat Ap00, Mat A01, Mat A10, Mat A11)
218: {
219:   Mat_SchurComplement *Na = (Mat_SchurComplement *)S->data;
220:   PetscBool            isschur;

222:   PetscFunctionBegin;
223:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
224:   if (!isschur) PetscFunctionReturn(PETSC_SUCCESS);
225:   PetscCheck(!S->assembled, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONGSTATE, "Use MatSchurComplementUpdateSubMatrices() for already used matrix");
230:   PetscCheckSameComm(A00, 2, Ap00, 3);
231:   PetscCheckSameComm(A00, 2, A01, 4);
232:   PetscCheckSameComm(A00, 2, A10, 5);
233:   PetscCheck(A00->rmap->n == A00->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A00 %" PetscInt_FMT " do not equal local columns %" PetscInt_FMT, A00->rmap->n, A00->cmap->n);
234:   PetscCheck(A00->rmap->n == Ap00->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A00 %" PetscInt_FMT " do not equal local rows of Ap00 %" PetscInt_FMT, A00->rmap->n, Ap00->rmap->n);
235:   PetscCheck(Ap00->rmap->n == Ap00->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of Ap00 %" PetscInt_FMT " do not equal local columns %" PetscInt_FMT, Ap00->rmap->n, Ap00->cmap->n);
236:   PetscCheck(A00->cmap->n == A01->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local columns of A00 %" PetscInt_FMT " do not equal local rows of A01 %" PetscInt_FMT, A00->cmap->n, A01->rmap->n);
237:   PetscCheck(A10->cmap->n == A00->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local columns of A10 %" PetscInt_FMT " do not equal local rows of A00 %" PetscInt_FMT, A10->cmap->n, A00->rmap->n);
238:   if (A11) {
240:     PetscCheckSameComm(A00, 2, A11, 6);
241:     PetscCheck(A10->rmap->n == A11->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A10 %" PetscInt_FMT " do not equal local rows A11 %" PetscInt_FMT, A10->rmap->n, A11->rmap->n);
242:   }

244:   PetscCall(MatSetSizes(S, A10->rmap->n, A01->cmap->n, A10->rmap->N, A01->cmap->N));
245:   PetscCall(PetscObjectReference((PetscObject)A00));
246:   PetscCall(PetscObjectReference((PetscObject)Ap00));
247:   PetscCall(PetscObjectReference((PetscObject)A01));
248:   PetscCall(PetscObjectReference((PetscObject)A10));
249:   PetscCall(PetscObjectReference((PetscObject)A11));
250:   Na->A  = A00;
251:   Na->Ap = Ap00;
252:   Na->B  = A01;
253:   Na->C  = A10;
254:   Na->D  = A11;
255:   PetscCall(MatSetUp(S));
256:   PetscCall(KSPSetOperators(Na->ksp, A00, Ap00));
257:   S->assembled = PETSC_TRUE;
258:   PetscFunctionReturn(PETSC_SUCCESS);
259: }

261: /*@
262:   MatSchurComplementGetKSP - Gets the `KSP` object that is used to solve with `A00` in the Schur complement matrix $S = A11 - A10 ksp(A00,Ap00) A01$

264:   Not Collective

266:   Input Parameter:
267: . S - matrix obtained with `MatCreateSchurComplement()` (or equivalent) and implementing the action of $ A11 - A10 ksp(A00,Ap00) A01 $

269:   Output Parameter:
270: . ksp - the linear solver object

272:   Options Database Key:
273: . -fieldsplit_<splitname_0>_XXX - sets `KSP` and `PC` options for the 0-split solver inside the Schur complement used in `PCFIELDSPLIT`; default <splitname_0> is 0.

275:   Level: intermediate

277: .seealso: [](ch_ksp), `Mat`, `MatSchurComplementSetKSP()`, `MatCreateSchurComplement()`, `MatCreateNormal()`, `MatMult()`, `MatCreate()`
278: @*/
279: PetscErrorCode MatSchurComplementGetKSP(Mat S, KSP *ksp)
280: {
281:   Mat_SchurComplement *Na;
282:   PetscBool            isschur;

284:   PetscFunctionBegin;
286:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
287:   PetscCheck(isschur, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONG, "Not for type %s", ((PetscObject)S)->type_name);
288:   PetscAssertPointer(ksp, 2);
289:   Na   = (Mat_SchurComplement *)S->data;
290:   *ksp = Na->ksp;
291:   PetscFunctionReturn(PETSC_SUCCESS);
292: }

294: /*@
295:   MatSchurComplementSetKSP - Sets the `KSP` object that is used to solve with `A00` in the Schur complement matrix $ S = A11 - A10 ksp(A00,Ap00) A01$

297:   Not Collective

299:   Input Parameters:
300: + S   - matrix created with `MatCreateSchurComplement()`
301: - ksp - the linear solver object

303:   Level: developer

305:   Developer Notes:
306:   This is used in `PCFIELDSPLIT` to reuse the 0-split `KSP` to implement $ksp(A00,Ap00)$ in `S`.
307:   The `KSP` operators are overwritten with `A00` and `Ap00` currently set in `S`.

309: .seealso: [](ch_ksp), `Mat`, `MatSchurComplementGetKSP()`, `MatCreateSchurComplement()`, `MatCreateNormal()`, `MatMult()`, `MatCreate()`, `MATSCHURCOMPLEMENT`
310: @*/
311: PetscErrorCode MatSchurComplementSetKSP(Mat S, KSP ksp)
312: {
313:   Mat_SchurComplement *Na;
314:   PetscBool            isschur;

316:   PetscFunctionBegin;
318:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
319:   if (!isschur) PetscFunctionReturn(PETSC_SUCCESS);
321:   Na = (Mat_SchurComplement *)S->data;
322:   PetscCall(PetscObjectReference((PetscObject)ksp));
323:   PetscCall(KSPDestroy(&Na->ksp));
324:   Na->ksp = ksp;
325:   PetscCall(KSPSetOperators(Na->ksp, Na->A, Na->Ap));
326:   PetscFunctionReturn(PETSC_SUCCESS);
327: }

329: /*@
330:   MatSchurComplementUpdateSubMatrices - Updates the Schur complement matrix object with new submatrices

332:   Collective

334:   Input Parameters:
335: + S    - matrix obtained with `MatCreateSchurComplement()` (or `MatSchurSetSubMatrices()`) and implementing the action of $A11 - A10 ksp(A00,Ap00) A01$
336: . A00  - the upper-left block of the original matrix $A = [A00 A01; A10 A11]$
337: . Ap00 - matrix from which the preconditioner is constructed for use in $ksp(A00,Ap00)$ to approximate the action of $A00^{-1}$
338: . A01  - the upper-right block of the original matrix $A = [A00 A01; A10 A11]$
339: . A10  - the lower-left block of the original matrix $A = [A00 A01; A10 A11]$
340: - A11  - (optional) the lower-right block of the original matrix $A = [A00 A01; A10 A11]$

342:   Level: intermediate

344:   Notes:
345:   All four matrices must have the same MPI communicator

347:   `A00` and  `A11` must be square matrices

349:   All of the matrices provided must have the same sizes as was used with `MatCreateSchurComplement()` or `MatSchurComplementSetSubMatrices()`
350:   though they need not be the same matrices.

352:   This can only be called after `MatCreateSchurComplement()` or `MatSchurComplementSetSubMatrices()`, it cannot be called immediately after `MatSetType`(S,`MATSCHURCOMPLEMENT`);

354:   Developer Notes:
355:   This code is almost identical to `MatSchurComplementSetSubMatrices()`. The API should be refactored.

357: .seealso: [](ch_ksp), `Mat`, `MatCreateNormal()`, `MatMult()`, `MatCreate()`, `MatSchurComplementGetKSP()`, `MatCreateSchurComplement()`
358: @*/
359: PetscErrorCode MatSchurComplementUpdateSubMatrices(Mat S, Mat A00, Mat Ap00, Mat A01, Mat A10, Mat A11)
360: {
361:   Mat_SchurComplement *Na = (Mat_SchurComplement *)S->data;
362:   PetscBool            isschur;

364:   PetscFunctionBegin;
366:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
367:   if (!isschur) PetscFunctionReturn(PETSC_SUCCESS);
368:   PetscCheck(S->assembled, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONGSTATE, "Use MatSchurComplementSetSubMatrices() for a new matrix");
373:   PetscCheckSameComm(A00, 2, Ap00, 3);
374:   PetscCheckSameComm(A00, 2, A01, 4);
375:   PetscCheckSameComm(A00, 2, A10, 5);
376:   PetscCheck(A00->rmap->n == A00->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A00 %" PetscInt_FMT " do not equal local columns %" PetscInt_FMT, A00->rmap->n, A00->cmap->n);
377:   PetscCheck(A00->rmap->n == Ap00->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A00 %" PetscInt_FMT " do not equal local rows of Ap00 %" PetscInt_FMT, A00->rmap->n, Ap00->rmap->n);
378:   PetscCheck(Ap00->rmap->n == Ap00->cmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of Ap00 %" PetscInt_FMT " do not equal local columns %" PetscInt_FMT, Ap00->rmap->n, Ap00->cmap->n);
379:   PetscCheck(A00->cmap->n == A01->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local columns of A00 %" PetscInt_FMT " do not equal local rows of A01 %" PetscInt_FMT, A00->cmap->n, A01->rmap->n);
380:   PetscCheck(A10->cmap->n == A00->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local columns of A10 %" PetscInt_FMT " do not equal local rows of A00 %" PetscInt_FMT, A10->cmap->n, A00->rmap->n);
381:   if (A11) {
383:     PetscCheckSameComm(A00, 2, A11, 6);
384:     PetscCheck(A10->rmap->n == A11->rmap->n, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Local rows of A10 %" PetscInt_FMT " do not equal local rows A11 %" PetscInt_FMT, A10->rmap->n, A11->rmap->n);
385:   }

387:   PetscCall(PetscObjectReference((PetscObject)A00));
388:   PetscCall(PetscObjectReference((PetscObject)Ap00));
389:   PetscCall(PetscObjectReference((PetscObject)A01));
390:   PetscCall(PetscObjectReference((PetscObject)A10));
391:   if (A11) PetscCall(PetscObjectReference((PetscObject)A11));

393:   PetscCall(MatDestroy(&Na->A));
394:   PetscCall(MatDestroy(&Na->Ap));
395:   PetscCall(MatDestroy(&Na->B));
396:   PetscCall(MatDestroy(&Na->C));
397:   PetscCall(MatDestroy(&Na->D));

399:   Na->A  = A00;
400:   Na->Ap = Ap00;
401:   Na->B  = A01;
402:   Na->C  = A10;
403:   Na->D  = A11;

405:   PetscCall(KSPSetOperators(Na->ksp, A00, Ap00));
406:   PetscFunctionReturn(PETSC_SUCCESS);
407: }

409: /*@
410:   MatSchurComplementGetSubMatrices - Get the individual submatrices in the Schur complement

412:   Collective

414:   Input Parameter:
415: . S - matrix obtained with `MatCreateSchurComplement()` (or equivalent) and implementing the action of $A11 - A10 ksp(A00,Ap00) A01$

417:   Output Parameters:
418: + A00  - the upper-left block of the original matrix $A = [A00 A01; A10 A11]$
419: . Ap00 - matrix from which the preconditioner is constructed for use in $ksp(A00,Ap00)$ to approximate the action of $A^{-1}$
420: . A01  - the upper-right block of the original matrix $A = [A00 A01; A10 A11]$
421: . A10  - the lower-left block of the original matrix $A = [A00 A01; A10 A11]$
422: - A11  - (optional) the lower-right block of the original matrix $A = [A00 A01; A10 A11]$

424:   Level: intermediate

426:   Note:
427:   Use `NULL` for any unneeded output argument.

429:   The reference counts of the submatrices are not increased before they are returned and the matrices should not be modified or destroyed.

431: .seealso: [](ch_ksp), `MatCreateNormal()`, `MatMult()`, `MatCreate()`, `MatSchurComplementGetKSP()`, `MatCreateSchurComplement()`, `MatSchurComplementUpdateSubMatrices()`
432: @*/
433: PetscErrorCode MatSchurComplementGetSubMatrices(Mat S, Mat *A00, Mat *Ap00, Mat *A01, Mat *A10, Mat *A11)
434: {
435:   Mat_SchurComplement *Na = (Mat_SchurComplement *)S->data;
436:   PetscBool            flg;

438:   PetscFunctionBegin;
440:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &flg));
441:   PetscCheck(flg, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONG, "Not for type %s", ((PetscObject)S)->type_name);
442:   if (A00) *A00 = Na->A;
443:   if (Ap00) *Ap00 = Na->Ap;
444:   if (A01) *A01 = Na->B;
445:   if (A10) *A10 = Na->C;
446:   if (A11) *A11 = Na->D;
447:   PetscFunctionReturn(PETSC_SUCCESS);
448: }

450: #include <petsc/private/kspimpl.h>

452: /*@
453:   MatSchurComplementComputeExplicitOperator - Compute the Schur complement matrix explicitly

455:   Collective

457:   Input Parameter:
458: . A - the matrix obtained with `MatCreateSchurComplement()`

460:   Output Parameter:
461: . S - the Schur complement matrix

463:   Level: advanced

465:   Notes:
466:   This can be expensive when `S` is large, so it is mainly for testing

468:   Use `MatSchurComplementGetPmat()` to get a sparse approximation for the Schur complement suitable for use in building a preconditioner

470:   `S` will automatically have the same prefix as `A` appended by `explicit_operator_`,
471:   there are three options available: `-fieldsplit_1_explicit_operator_mat_type`,
472:   `-fieldsplit_1_explicit_operator_mat_symmetric`, and `-fieldsplit_1_explicit_operator_mat_hermitian`

474:   Developer Note:
475:   The three aforementioned should not be parsed and used in this routine, but rather in `MatSetFromOptions()`

477: .seealso: [](ch_ksp), `MatCreateSchurComplement()`, `MatSchurComplementUpdateSubMatrices()`, `MatSchurComplementGetPmat()`
478: @*/
479: PetscErrorCode MatSchurComplementComputeExplicitOperator(Mat A, Mat *S)
480: {
481:   Mat       P, B, C, D, E = NULL, Bd, AinvBd, sub = NULL;
482:   MatType   mtype;
483:   VecType   vtype;
484:   KSP       ksp;
485:   PetscInt  n, N, m, M;
486:   PetscBool flg = PETSC_FALSE, set, symm;
487:   char      prefix[256], type[256];

489:   PetscFunctionBegin;
490:   PetscAssertPointer(S, 2);
492:   PetscCall(PetscObjectQuery((PetscObject)A, "AinvB", (PetscObject *)&AinvBd));
493:   set = (PetscBool)(AinvBd != NULL);
494:   if (set && AinvBd->cmap->N == -1) PetscFunctionReturn(PETSC_SUCCESS); // early bail out if composed Mat is uninitialized
495:   PetscCall(MatSchurComplementGetSubMatrices(A, &P, NULL, &B, &C, &D));
496:   PetscCall(MatGetVecType(B, &vtype));
497:   PetscCall(MatGetLocalSize(B, &m, &n));
498:   PetscCall(MatSchurComplementGetKSP(A, &ksp));
499:   PetscCall(KSPSetUp(ksp));
500:   if (set) {
501:     PetscCheck(AinvBd->cmap->N >= A->cmap->N, PetscObjectComm((PetscObject)A), PETSC_ERR_ARG_SIZ, "Composed Mat should have at least as many columns as the Schur complement (%" PetscInt_FMT " >= %" PetscInt_FMT ")", AinvBd->cmap->N, A->cmap->N);
502:     PetscCall(MatGetType(AinvBd, &mtype));
503:     if (AinvBd->cmap->N > A->cmap->N) {
504:       Mat s[2];

506:       PetscCall(MatDuplicate(AinvBd, MAT_DO_NOT_COPY_VALUES, &Bd));
507:       PetscCall(MatDenseGetSubMatrix(Bd, PETSC_DECIDE, PETSC_DECIDE, A->cmap->N, AinvBd->cmap->N, s));
508:       PetscCall(MatDenseGetSubMatrix(AinvBd, PETSC_DECIDE, PETSC_DECIDE, A->cmap->N, AinvBd->cmap->N, s + 1));
509:       PetscCall(MatCopy(s[1], s[0], SAME_NONZERO_PATTERN)); // copy the last columns of the composed Mat, which are likely the input columns of PCApply_FieldSplit_Schur()
510:       PetscCall(MatDenseRestoreSubMatrix(AinvBd, s + 1));
511:       PetscCall(MatDenseRestoreSubMatrix(Bd, s));
512:       PetscCall(MatDenseGetSubMatrix(Bd, PETSC_DECIDE, PETSC_DECIDE, 0, A->cmap->N, &sub));
513:       PetscCall(MatConvert(B, mtype, MAT_REUSE_MATRIX, &sub)); // copy A01 into the first columns of the block of RHS of KSPMatSolve()
514:       PetscCall(MatDenseRestoreSubMatrix(Bd, &sub));
515:     } else PetscCall(MatConvert(B, mtype, MAT_INITIAL_MATRIX, &Bd));
516:   } else {
517:     PetscCall(MatGetSize(B, &M, &N));
518:     PetscCall(MatCreateDenseFromVecType(PetscObjectComm((PetscObject)A), vtype, m, n, M, N, -1, NULL, &AinvBd));
519:     PetscCall(MatGetType(AinvBd, &mtype));
520:     PetscCall(MatConvert(B, mtype, MAT_INITIAL_MATRIX, &Bd));
521:   }
522:   PetscCall(KSPMatSolve(ksp, Bd, AinvBd));
523:   if (set && AinvBd->cmap->N > A->cmap->N) {
524:     Mat          AinvB;
525:     PetscScalar *v;
526:     PetscBool    match;

528:     PetscCall(PetscObjectTypeCompareAny((PetscObject)AinvBd, &match, MATSEQDENSECUDA, MATMPIDENSECUDA, ""));
529:     if (match) {
530: #if PetscDefined(HAVE_CUDA)
531:       PetscCall(MatDenseCUDAGetArrayWrite(AinvBd, &v));
532:       PetscCall(MatCreateDenseCUDA(PetscObjectComm((PetscObject)A), AinvBd->rmap->n, A->cmap->n, AinvBd->rmap->N, A->cmap->N, v, &AinvB));
533:       PetscCall(MatDenseCUDAReplaceArray(AinvB, v));
534:       PetscCall(MatDenseCUDARestoreArrayWrite(AinvBd, &v));
535: #endif
536:     } else {
537:       PetscCall(PetscObjectTypeCompareAny((PetscObject)AinvBd, &match, MATSEQDENSEHIP, MATMPIDENSEHIP, ""));
538:       if (match) {
539: #if PetscDefined(HAVE_HIP)
540:         PetscCall(MatDenseHIPGetArrayWrite(AinvBd, &v));
541:         PetscCall(MatCreateDenseHIP(PetscObjectComm((PetscObject)A), AinvBd->rmap->n, A->cmap->n, AinvBd->rmap->N, A->cmap->N, v, &AinvB));
542:         PetscCall(MatDenseHIPReplaceArray(AinvB, v));
543:         PetscCall(MatDenseHIPRestoreArrayWrite(AinvBd, &v));
544: #endif
545:       } else {
546:         PetscCall(MatDenseGetArrayWrite(AinvBd, &v)); // no easy way to resize a Mat, so create a new one with the same data pointer
547:         PetscCall(MatCreateDense(PetscObjectComm((PetscObject)A), AinvBd->rmap->n, A->cmap->n, AinvBd->rmap->N, A->cmap->N, v, &AinvB));
548:         PetscCall(MatDenseReplaceArray(AinvB, v)); // let MatDestroy() free the data pointer
549:         PetscCall(MatDenseRestoreArrayWrite(AinvBd, &v));
550:       }
551:     }
552:     PetscCall(MatHeaderReplace(AinvBd, &AinvB)); // replace the input composed Mat with just A00^-1 A01 (trailing columns are removed)
553:   }
554:   PetscCall(MatDestroy(&Bd));
555:   if (!set) PetscCall(MatFilter(AinvBd, PETSC_SMALL, PETSC_FALSE, PETSC_FALSE));
556:   if (D && !*S) {
557:     PetscCall(MatGetLocalSize(D, &m, &n));
558:     PetscCall(MatGetSize(D, &M, &N));
559:     PetscCall(MatCreateDenseFromVecType(PetscObjectComm((PetscObject)A), vtype, m, n, M, N, -1, NULL, S));
560:   } else if (*S) {
561:     PetscCall(MatGetType(AinvBd, &mtype));
562:     PetscCall(MatSetType(*S, mtype));
563:   }
564:   PetscCall(MatMatMult(C, AinvBd, *S ? MAT_REUSE_MATRIX : MAT_INITIAL_MATRIX, PETSC_DETERMINE, S));
565:   if (!set) PetscCall(MatDestroy(&AinvBd));
566:   else {
567:     PetscCall(MatScale(AinvBd, -1.0));
568:     PetscCall(MatFilter(AinvBd, PETSC_MACHINE_EPSILON, PETSC_FALSE, PETSC_FALSE));
569:     PetscCall(MatFilter(*S, PETSC_MACHINE_EPSILON, PETSC_FALSE, PETSC_FALSE));
570:   }
571:   if (D) {
572:     PetscCall(PetscObjectTypeCompareAny((PetscObject)D, &flg, MATSEQSBAIJ, MATMPISBAIJ, ""));
573:     if (flg) {
574:       PetscCall(MatIsSymmetricKnown(A, &set, &symm));
575:       if (!set || !symm) PetscCall(MatConvert(D, MATBAIJ, MAT_INITIAL_MATRIX, &E)); /* convert the (1,1) block to nonsymmetric storage for MatAXPY() */
576:     }
577:     PetscCall(MatAXPY(*S, -1.0, E ? E : D, DIFFERENT_NONZERO_PATTERN));        /* calls Mat[Get|Restore]RowUpperTriangular(), so only the upper triangular part is valid with symmetric storage */
578:     if (!E && flg) PetscCall(MatConvert(*S, MATSBAIJ, MAT_INPLACE_MATRIX, S)); /* if A is symmetric and the (1,1) block is a MatSBAIJ, return S as a MatSBAIJ since the lower triangular part is invalid */
579:   }
580:   PetscCall(MatDestroy(&E));
581:   PetscCall(MatScale(*S, -1.0));
582:   PetscCall(PetscSNPrintf(prefix, sizeof(prefix), "%sexplicit_operator_", ((PetscObject)A)->prefix ? ((PetscObject)A)->prefix : ""));
583:   PetscCall(MatSetOptionsPrefix(*S, prefix));
584:   PetscObjectOptionsBegin((PetscObject)*S);
585:   PetscCall(PetscOptionsFList("-mat_type", "Matrix type", "MatSetType", MatList, !E && flg ? MATSBAIJ : mtype, type, 256, &set));
586:   if (set) PetscCall(MatConvert(*S, type, MAT_INPLACE_MATRIX, S));
587:   flg = PETSC_FALSE;
588:   PetscCall(PetscOptionsBool("-mat_symmetric", "Sets the MAT_SYMMETRIC option", "MatSetOption", flg, &flg, &set));
589:   if (set) PetscCall(MatSetOption(*S, MAT_SYMMETRIC, flg));
590:   if (PetscDefined(USE_COMPLEX)) {
591:     flg = PETSC_FALSE;
592:     PetscCall(PetscOptionsBool("-mat_hermitian", "Sets the MAT_HERMITIAN option", "MatSetOption", flg, &flg, &set));
593:     if (set) PetscCall(MatSetOption(*S, MAT_HERMITIAN, flg));
594:   }
595:   PetscOptionsEnd();
596:   PetscFunctionReturn(PETSC_SUCCESS);
597: }

599: /* Developer Notes:
600:     This should be implemented with a MatCreate_SchurComplement() as that is the standard design for new Mat classes. */
601: PetscErrorCode MatGetSchurComplement_Basic(Mat mat, IS isrow0, IS iscol0, IS isrow1, IS iscol1, MatReuse mreuse, Mat *S, MatSchurComplementAinvType ainvtype, MatReuse preuse, Mat *Sp)
602: {
603:   Mat      A = NULL, Ap = NULL, B = NULL, C = NULL, D = NULL;
604:   MatReuse reuse;

606:   PetscFunctionBegin;
616:   if (mreuse == MAT_IGNORE_MATRIX && preuse == MAT_IGNORE_MATRIX) PetscFunctionReturn(PETSC_SUCCESS);

620:   PetscCheck(!mat->factortype, PetscObjectComm((PetscObject)mat), PETSC_ERR_ARG_WRONGSTATE, "Not for factored matrix");

622:   reuse = MAT_INITIAL_MATRIX;
623:   if (mreuse == MAT_REUSE_MATRIX) {
624:     PetscCall(MatSchurComplementGetSubMatrices(*S, &A, &Ap, &B, &C, &D));
625:     PetscCheck(A && Ap && B && C, PetscObjectComm((PetscObject)mat), PETSC_ERR_ARG_WRONGSTATE, "Attempting to reuse matrix but Schur complement matrices unset");
626:     PetscCheck(A == Ap, PetscObjectComm((PetscObject)mat), PETSC_ERR_ARG_WRONGSTATE, "Matrix for constructing the preconditioner does not match operator");
627:     PetscCall(MatDestroy(&Ap)); /* get rid of extra reference */
628:     reuse = MAT_REUSE_MATRIX;
629:   }
630:   PetscCall(MatCreateSubMatrix(mat, isrow0, iscol0, reuse, &A));
631:   PetscCall(MatCreateSubMatrix(mat, isrow0, iscol1, reuse, &B));
632:   PetscCall(MatCreateSubMatrix(mat, isrow1, iscol0, reuse, &C));
633:   PetscCall(MatCreateSubMatrix(mat, isrow1, iscol1, reuse, &D));
634:   switch (mreuse) {
635:   case MAT_INITIAL_MATRIX:
636:     PetscCall(MatCreateSchurComplement(A, A, B, C, D, S));
637:     break;
638:   case MAT_REUSE_MATRIX:
639:     PetscCall(MatSchurComplementUpdateSubMatrices(*S, A, A, B, C, D));
640:     break;
641:   default:
642:     PetscCheck(mreuse == MAT_IGNORE_MATRIX, PetscObjectComm((PetscObject)mat), PETSC_ERR_SUP, "Unrecognized value of mreuse %d", (int)mreuse);
643:   }
644:   if (preuse != MAT_IGNORE_MATRIX) PetscCall(MatCreateSchurComplementPmat(A, B, C, D, ainvtype, preuse, Sp));
645:   PetscCall(MatDestroy(&A));
646:   PetscCall(MatDestroy(&B));
647:   PetscCall(MatDestroy(&C));
648:   PetscCall(MatDestroy(&D));
649:   PetscFunctionReturn(PETSC_SUCCESS);
650: }

652: /*@
653:   MatGetSchurComplement - Obtain the Schur complement from eliminating part of the matrix in another part.

655:   Collective

657:   Input Parameters:
658: + A        - matrix in which the complement is to be taken
659: . isrow0   - rows to eliminate
660: . iscol0   - columns to eliminate, (isrow0,iscol0) should be square and nonsingular
661: . isrow1   - rows in which the Schur complement is formed
662: . iscol1   - columns in which the Schur complement is formed
663: . mreuse   - `MAT_INITIAL_MATRIX` or `MAT_REUSE_MATRIX`, use `MAT_IGNORE_MATRIX` to put nothing in `S`
664: . ainvtype - the type of approximation used for the inverse of the (0,0) block used in forming `Sp`:
665:              `MAT_SCHUR_COMPLEMENT_AINV_DIAG`, `MAT_SCHUR_COMPLEMENT_AINV_LUMP`, `MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG`, or `MAT_SCHUR_COMPLEMENT_AINV_FULL`
666: - preuse   - `MAT_INITIAL_MATRIX` or `MAT_REUSE_MATRIX`, use `MAT_IGNORE_MATRIX` to put nothing in `Sp`

668:   Output Parameters:
669: + S  - exact Schur complement, often of type `MATSCHURCOMPLEMENT` which is difficult to use for preconditioning
670: - Sp - approximate Schur complement from which a preconditioner can be built $A11 - A10 inv(DIAGFORM(A00)) A01$

672:   Level: advanced

674:   Notes:
675:   Since the real Schur complement is usually dense, providing a good approximation to `Sp` usually requires
676:   application-specific information.

678:   Sometimes users would like to provide problem-specific data in the Schur complement, usually only for special row
679:   and column index sets.  In that case, the user should call `PetscObjectComposeFunction()` on the *S matrix and pass mreuse of `MAT_REUSE_MATRIX` to set
680:   "MatGetSchurComplement_C" to their function.  If their function needs to fall back to the default implementation, it
681:   should call `MatGetSchurComplement_Basic()`.

683:   `MatCreateSchurComplement()` takes as arguments the four submatrices and returns the virtual Schur complement (what this function returns in S).

685:   `MatSchurComplementGetPmat()` takes the virtual Schur complement and returns an explicit approximate Schur complement (what this returns in Sp).

687:   In other words calling `MatCreateSchurComplement()` followed by `MatSchurComplementGetPmat()` produces the same output as this function but with slightly different
688:   inputs. The actually submatrices of the original block matrix instead of index sets to the submatrices.

690:   Developer Notes:
691:   The API that includes `MatGetSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementGetPmat()` should be refactored to
692:   remove redundancy and be clearer and simpler.

694: .seealso: [](ch_ksp), `MatCreateSubMatrix()`, `PCFIELDSPLIT`, `MatCreateSchurComplement()`, `MatSchurComplementAinvType`
695: @*/
696: PetscErrorCode MatGetSchurComplement(Mat A, IS isrow0, IS iscol0, IS isrow1, IS iscol1, MatReuse mreuse, Mat *S, MatSchurComplementAinvType ainvtype, MatReuse preuse, Mat *Sp)
697: {
698:   PetscErrorCode (*f)(Mat, IS, IS, IS, IS, MatReuse, Mat *, MatReuse, Mat *) = NULL;

700:   PetscFunctionBegin;
712:   PetscCheck(!A->factortype, PetscObjectComm((PetscObject)A), PETSC_ERR_ARG_WRONGSTATE, "Not for factored matrix");
713:   if (mreuse == MAT_REUSE_MATRIX) { /* This is the only situation, in which we can demand that the user pass a non-NULL pointer to non-garbage in S. */
714:     PetscCall(PetscObjectQueryFunction((PetscObject)*S, "MatGetSchurComplement_C", &f));
715:   }
716:   if (f) PetscCall((*f)(A, isrow0, iscol0, isrow1, iscol1, mreuse, S, preuse, Sp));
717:   else PetscCall(MatGetSchurComplement_Basic(A, isrow0, iscol0, isrow1, iscol1, mreuse, S, ainvtype, preuse, Sp));
718:   PetscFunctionReturn(PETSC_SUCCESS);
719: }

721: /*@
722:   MatSchurComplementSetAinvType - set the type of approximation used for the inverse of the (0,0) block used in forming `Sp` in `MatSchurComplementGetPmat()`

724:   Not Collective

726:   Input Parameters:
727: + S        - matrix obtained with `MatCreateSchurComplement()` (or equivalent) and implementing the action of $A11 - A10 ksp(A00,Ap00) A01$
728: - ainvtype - type of approximation to be used to form approximate Schur complement $Sp = A11 - A10 inv(DIAGFORM(A00)) A01$:
729:              `MAT_SCHUR_COMPLEMENT_AINV_DIAG`, `MAT_SCHUR_COMPLEMENT_AINV_LUMP`, `MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG`, or `MAT_SCHUR_COMPLEMENT_AINV_FULL`

731:   Options Database Key:
732: . -mat_schur_complement_ainv_type diag | lump | blockdiag | full - set schur complement type

734:   Level: advanced

736: .seealso: [](ch_ksp), `MatSchurComplementAinvType`, `MatCreateSchurComplement()`, `MatGetSchurComplement()`, `MatSchurComplementGetPmat()`, `MatSchurComplementGetAinvType()`
737: @*/
738: PetscErrorCode MatSchurComplementSetAinvType(Mat S, MatSchurComplementAinvType ainvtype)
739: {
740:   PetscBool            isschur;
741:   Mat_SchurComplement *schur;

743:   PetscFunctionBegin;
745:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
746:   if (!isschur) PetscFunctionReturn(PETSC_SUCCESS);
748:   schur = (Mat_SchurComplement *)S->data;
749:   PetscCheck(ainvtype == MAT_SCHUR_COMPLEMENT_AINV_DIAG || ainvtype == MAT_SCHUR_COMPLEMENT_AINV_LUMP || ainvtype == MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG || ainvtype == MAT_SCHUR_COMPLEMENT_AINV_FULL, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Unknown MatSchurComplementAinvType: %d", (int)ainvtype);
750:   schur->ainvtype = ainvtype;
751:   PetscFunctionReturn(PETSC_SUCCESS);
752: }

754: /*@
755:   MatSchurComplementGetAinvType - get the type of approximation for the inverse of the (0,0) block used in forming `Sp` in `MatSchurComplementGetPmat()`

757:   Not Collective

759:   Input Parameter:
760: . S - matrix obtained with `MatCreateSchurComplement()` (or equivalent) and implementing the action of $A11 - A10 ksp(A00,Ap00) A01$

762:   Output Parameter:
763: . ainvtype - type of approximation used to form approximate Schur complement Sp = A11 - A10 inv(DIAGFORM(A00)) A01:
764:              `MAT_SCHUR_COMPLEMENT_AINV_DIAG`, `MAT_SCHUR_COMPLEMENT_AINV_LUMP`, `MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG`, or `MAT_SCHUR_COMPLEMENT_AINV_FULL`

766:   Level: advanced

768: .seealso: [](ch_ksp), `MatSchurComplementAinvType`, `MatCreateSchurComplement()`, `MatGetSchurComplement()`, `MatSchurComplementGetPmat()`, `MatSchurComplementSetAinvType()`
769: @*/
770: PetscErrorCode MatSchurComplementGetAinvType(Mat S, MatSchurComplementAinvType *ainvtype)
771: {
772:   PetscBool            isschur;
773:   Mat_SchurComplement *schur;

775:   PetscFunctionBegin;
777:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
778:   PetscCheck(isschur, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONG, "Not for type %s", ((PetscObject)S)->type_name);
779:   schur = (Mat_SchurComplement *)S->data;
780:   if (ainvtype) *ainvtype = schur->ainvtype;
781:   PetscFunctionReturn(PETSC_SUCCESS);
782: }

784: /*@
785:   MatCreateSchurComplementPmat - create a matrix for preconditioning the Schur complement by explicitly assembling the sparse matrix
786:   $Sp = A11 - A10 inv(DIAGFORM(A00)) A01$

788:   Collective

790:   Input Parameters:
791: + A00      - the upper-left part of the original matrix $A = [A00 A01; A10 A11]$
792: . A01      - (optional) the upper-right part of the original matrix $A = [A00 A01; A10 A11]$
793: . A10      - (optional) the lower-left part of the original matrix $A = [A00 A01; A10 A11]$
794: . A11      - (optional) the lower-right part of the original matrix $A = [A00 A01; A10 A11]$
795: . ainvtype - type of approximation for DIAGFORM(A00) used when forming $Sp = A11 - A10 inv(DIAGFORM(A00)) A01$. See `MatSchurComplementAinvType`.
796: - preuse   - `MAT_INITIAL_MATRIX` for a new `Sp`, or `MAT_REUSE_MATRIX` to reuse an existing `Sp`, or `MAT_IGNORE_MATRIX` to put nothing in `Sp`

798:   Output Parameter:
799: . Sp - approximate Schur complement suitable for constructing a preconditioner for the true Schur complement $S = A11 - A10 inv(A00) A01$

801:   Level: advanced

803: .seealso: [](ch_ksp), `MatCreateSchurComplement()`, `MatGetSchurComplement()`, `MatSchurComplementGetPmat()`, `MatSchurComplementAinvType`
804: @*/
805: PetscErrorCode MatCreateSchurComplementPmat(Mat A00, Mat A01, Mat A10, Mat A11, MatSchurComplementAinvType ainvtype, MatReuse preuse, Mat *Sp)
806: {
807:   PetscInt N00;

809:   PetscFunctionBegin;
810:   /* Use an appropriate approximate inverse of A00 to form A11 - A10 inv(DIAGFORM(A00)) A01; a NULL A01, A10 or A11 indicates a zero matrix. */
811:   /* TODO: Perhaps should create an appropriately-sized zero matrix of the same type as A00? */
813:   if (preuse == MAT_IGNORE_MATRIX) PetscFunctionReturn(PETSC_SUCCESS);

815:   /* A zero size A00 or empty A01 or A10 imply S = A11. */
816:   PetscCall(MatGetSize(A00, &N00, NULL));
817:   if (!A01 || !A10 || !N00) {
818:     if (preuse == MAT_INITIAL_MATRIX) {
819:       PetscCall(MatDuplicate(A11, MAT_COPY_VALUES, Sp));
820:     } else { /* MAT_REUSE_MATRIX */
821:       /* TODO: when can we pass SAME_NONZERO_PATTERN? */
822:       PetscCall(MatCopy(A11, *Sp, DIFFERENT_NONZERO_PATTERN));
823:     }
824:   } else {
825:     Mat       AdB, T;
826:     Vec       diag;
827:     PetscBool flg;

829:     if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_LUMP || ainvtype == MAT_SCHUR_COMPLEMENT_AINV_DIAG) {
830:       PetscCall(PetscObjectTypeCompare((PetscObject)A01, MATTRANSPOSEVIRTUAL, &flg));
831:       if (flg) {
832:         PetscCall(MatTransposeGetMat(A01, &T));
833:         PetscCall(MatTranspose(T, MAT_INITIAL_MATRIX, &AdB));
834:       } else {
835:         PetscCall(PetscObjectTypeCompare((PetscObject)A01, MATHERMITIANTRANSPOSEVIRTUAL, &flg));
836:         if (flg) {
837:           PetscCall(MatHermitianTransposeGetMat(A01, &T));
838:           PetscCall(MatHermitianTranspose(T, MAT_INITIAL_MATRIX, &AdB));
839:         }
840:       }
841:       if (!flg) PetscCall(MatDuplicate(A01, MAT_COPY_VALUES, &AdB));
842:       else {
843:         PetscScalar shift, scale;

845:         PetscCall(MatShellGetScalingShifts(A01, &shift, &scale, (Vec *)MAT_SHELL_NOT_ALLOWED, (Vec *)MAT_SHELL_NOT_ALLOWED, (Vec *)MAT_SHELL_NOT_ALLOWED, (Mat *)MAT_SHELL_NOT_ALLOWED, (IS *)MAT_SHELL_NOT_ALLOWED, (IS *)MAT_SHELL_NOT_ALLOWED));
846:         PetscCall(MatShift(AdB, shift));
847:         PetscCall(MatScale(AdB, scale));
848:       }
849:       PetscCall(MatCreateVecs(A00, &diag, NULL));
850:       if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_LUMP) {
851:         PetscCall(MatGetRowSum(A00, diag));
852:       } else {
853:         PetscCall(MatGetDiagonal(A00, diag));
854:       }
855:       PetscCall(VecReciprocal(diag));
856:       PetscCall(MatDiagonalScale(AdB, diag, NULL));
857:       PetscCall(VecDestroy(&diag));
858:     } else if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG) {
859:       Mat      A00_inv;
860:       MatType  type;
861:       MPI_Comm comm;

863:       PetscCall(PetscObjectGetComm((PetscObject)A00, &comm));
864:       PetscCall(MatGetType(A00, &type));
865:       PetscCall(MatCreate(comm, &A00_inv));
866:       PetscCall(MatSetType(A00_inv, type));
867:       PetscCall(MatInvertBlockDiagonalMat(A00, A00_inv));
868:       PetscCall(MatMatMult(A00_inv, A01, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &AdB));
869:       PetscCall(MatDestroy(&A00_inv));
870:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Unknown MatSchurComplementAinvType: %d", ainvtype);
871:     /* Cannot really reuse Sp in MatMatMult() because of MatAYPX() -->
872:          MatAXPY() --> MatHeaderReplace() --> MatDestroy_XXX_MatMatMult()  */
873:     if (preuse == MAT_REUSE_MATRIX) PetscCall(MatDestroy(Sp));
874:     PetscCall(MatMatMult(A10, AdB, MAT_INITIAL_MATRIX, PETSC_DETERMINE, Sp));
875:     PetscCall(MatScale(*Sp, -1.0));
876:     if (A11) { /* TODO: when can we pass SAME_NONZERO_PATTERN? */
877:       PetscCall(MatAXPY(*Sp, 1.0, A11, DIFFERENT_NONZERO_PATTERN));
878:     }
879:     PetscCall(MatDestroy(&AdB));
880:   }
881:   PetscFunctionReturn(PETSC_SUCCESS);
882: }

884: static PetscErrorCode MatSchurComplementGetPmat_Basic(Mat S, MatReuse preuse, Mat *Sp)
885: {
886:   Mat                  A, B, C, D;
887:   Mat_SchurComplement *schur = (Mat_SchurComplement *)S->data;
888:   MatNullSpace         sp;

890:   PetscFunctionBegin;
891:   if (preuse == MAT_IGNORE_MATRIX) PetscFunctionReturn(PETSC_SUCCESS);
892:   PetscCall(MatSchurComplementGetSubMatrices(S, &A, NULL, &B, &C, &D));
893:   PetscCheck(A, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONGSTATE, "Schur complement component matrices unset");
894:   if (schur->ainvtype != MAT_SCHUR_COMPLEMENT_AINV_FULL) PetscCall(MatCreateSchurComplementPmat(A, B, C, D, schur->ainvtype, preuse, Sp));
895:   else {
896:     if (preuse == MAT_REUSE_MATRIX) PetscCall(MatDestroy(Sp));
897:     PetscCall(MatSchurComplementComputeExplicitOperator(S, Sp));
898:   }
899:   /* If the Schur complement has a nullspace, then Sp nullspace contains it, independently of the ainv type */
900:   PetscCall(MatGetNullSpace(S, &sp));
901:   if (sp) PetscCall(MatSetNullSpace(*Sp, sp));
902:   PetscCall(MatGetTransposeNullSpace(S, &sp));
903:   if (sp) PetscCall(MatSetTransposeNullSpace(*Sp, sp));
904:   PetscFunctionReturn(PETSC_SUCCESS);
905: }

907: /*@
908:   MatSchurComplementGetPmat - Obtain a matrix for preconditioning the Schur complement by assembling $Sp = A11 - A10 inv(DIAGFORM(A00)) A01$

910:   Collective

912:   Input Parameters:
913: + S      - matrix obtained with MatCreateSchurComplement() (or equivalent) that implements the action of $A11 - A10 ksp(A00,Ap00) A01$
914: - preuse - `MAT_INITIAL_MATRIX` for a new `Sp`, or `MAT_REUSE_MATRIX` to reuse an existing `Sp`, or `MAT_IGNORE_MATRIX` to put nothing in `Sp`

916:   Output Parameter:
917: . Sp - approximate Schur complement suitable for preconditioning the exact Schur complement $S = A11 - A10 inv(A00) A01$

919:   Level: advanced

921:   Notes:
922:   The approximation of `Sp` depends on the argument passed to `MatSchurComplementSetAinvType()`
923:   `MAT_SCHUR_COMPLEMENT_AINV_DIAG`, `MAT_SCHUR_COMPLEMENT_AINV_LUMP`, `MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG`, or `MAT_SCHUR_COMPLEMENT_AINV_FULL`
924:   -mat_schur_complement_ainv_type <diag,lump,blockdiag,full>

926:   Sometimes users would like to provide problem-specific data in the Schur complement, usually only
927:   for special row and column index sets.  In that case, the user should call `PetscObjectComposeFunction()` to set
928:   "MatSchurComplementGetPmat_C" to their function.  If their function needs to fall back to the default implementation,
929:   it should call `MatSchurComplementGetPmat_Basic()`.

931:   Developer Notes:
932:   The API that includes `MatGetSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementGetPmat()` should be refactored to
933:   remove redundancy and be clearer and simpler.

935:   This routine should be called `MatSchurComplementCreatePmat()`

937: .seealso: [](ch_ksp), `MatCreateSubMatrix()`, `PCFIELDSPLIT`, `MatGetSchurComplement()`, `MatCreateSchurComplement()`, `MatSchurComplementSetAinvType()`
938: @*/
939: PetscErrorCode MatSchurComplementGetPmat(Mat S, MatReuse preuse, Mat *Sp)
940: {
941:   PetscErrorCode (*f)(Mat, MatReuse, Mat *);

943:   PetscFunctionBegin;
947:   if (preuse != MAT_IGNORE_MATRIX) {
948:     PetscAssertPointer(Sp, 3);
949:     if (preuse == MAT_INITIAL_MATRIX) *Sp = NULL;
951:   }
952:   PetscCheck(!S->factortype, PetscObjectComm((PetscObject)S), PETSC_ERR_ARG_WRONGSTATE, "Not for factored matrix");

954:   PetscCall(PetscObjectQueryFunction((PetscObject)S, "MatSchurComplementGetPmat_C", &f));
955:   if (f) PetscCall((*f)(S, preuse, Sp));
956:   else PetscCall(MatSchurComplementGetPmat_Basic(S, preuse, Sp));
957:   PetscFunctionReturn(PETSC_SUCCESS);
958: }

960: static PetscErrorCode MatProductNumeric_SchurComplement_Dense(Mat C)
961: {
962:   Mat_Product         *product = C->product;
963:   Mat_SchurComplement *Na      = (Mat_SchurComplement *)product->A->data;
964:   Mat                  work1, work2;
965:   PetscScalar         *v;
966:   PetscInt             lda;

968:   PetscFunctionBegin;
969:   PetscCall(MatMatMult(Na->B, product->B, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &work1));
970:   PetscCall(MatDuplicate(work1, MAT_DO_NOT_COPY_VALUES, &work2));
971:   PetscCall(KSPMatSolve(Na->ksp, work1, work2));
972:   PetscCall(MatDestroy(&work1));
973:   PetscCall(MatDenseGetArrayWrite(C, &v));
974:   PetscCall(MatDenseGetLDA(C, &lda));
975:   PetscCall(MatCreateDense(PetscObjectComm((PetscObject)C), C->rmap->n, C->cmap->n, C->rmap->N, C->cmap->N, v, &work1));
976:   PetscCall(MatDenseSetLDA(work1, lda));
977:   PetscCall(MatMatMult(Na->C, work2, MAT_REUSE_MATRIX, PETSC_DETERMINE, &work1));
978:   PetscCall(MatDenseRestoreArrayWrite(C, &v));
979:   PetscCall(MatDestroy(&work2));
980:   PetscCall(MatDestroy(&work1));
981:   if (Na->D) {
982:     PetscCall(MatMatMult(Na->D, product->B, MAT_INITIAL_MATRIX, PETSC_DETERMINE, &work1));
983:     PetscCall(MatAYPX(C, -1.0, work1, SAME_NONZERO_PATTERN));
984:     PetscCall(MatDestroy(&work1));
985:   } else PetscCall(MatScale(C, -1.0));
986:   PetscFunctionReturn(PETSC_SUCCESS);
987: }

989: static PetscErrorCode MatProductSymbolic_SchurComplement_Dense(Mat C)
990: {
991:   Mat_Product *product = C->product;
992:   Mat          A = product->A, B = product->B;
993:   PetscInt     m = A->rmap->n, n = B->cmap->n, M = A->rmap->N, N = B->cmap->N;
994:   PetscBool    flg;

996:   PetscFunctionBegin;
997:   PetscCall(MatSetSizes(C, m, n, M, N));
998:   PetscCall(PetscObjectBaseTypeCompareAny((PetscObject)C, &flg, MATSEQDENSE, MATMPIDENSE, ""));
999:   if (!flg) {
1000:     PetscCall(MatSetType(C, ((PetscObject)B)->type_name));
1001:     C->ops->productsymbolic = MatProductSymbolic_SchurComplement_Dense;
1002:   }
1003:   PetscCall(MatSetUp(C));
1004:   C->ops->productnumeric = MatProductNumeric_SchurComplement_Dense;
1005:   PetscFunctionReturn(PETSC_SUCCESS);
1006: }

1008: static PetscErrorCode MatProductSetFromOptions_SchurComplement_Dense(Mat C)
1009: {
1010:   Mat_Product *product = C->product;

1012:   PetscFunctionBegin;
1013:   if (product->type != MATPRODUCT_AB) PetscFunctionReturn(PETSC_SUCCESS);
1014:   C->ops->productsymbolic = MatProductSymbolic_SchurComplement_Dense;
1015:   PetscFunctionReturn(PETSC_SUCCESS);
1016: }

1018: static PetscErrorCode MatProductNumeric_SchurComplement_Any(Mat C)
1019: {
1020:   Mat_SchurComplement *Na = (Mat_SchurComplement *)C->data;

1022:   PetscFunctionBegin;
1023:   if (Na->D && Na->D->product) PetscCall(MatProductNumeric(Na->D));
1024:   if (Na->B->product) PetscCall(MatProductNumeric(Na->B));
1025:   if (Na->C->product) PetscCall(MatProductNumeric(Na->C));
1026:   C->assembled = PETSC_TRUE;
1027:   PetscFunctionReturn(PETSC_SUCCESS);
1028: }

1030: static PetscErrorCode MatProductSymbolic_SchurComplement_Any(Mat C)
1031: {
1032:   Mat_SchurComplement *Na = (Mat_SchurComplement *)C->data;

1034:   PetscFunctionBegin;
1035:   if (Na->D && Na->D->product) PetscCall(MatProductSymbolic(Na->D));
1036:   if (Na->B->product) PetscCall(MatProductSymbolic(Na->B));
1037:   if (Na->C->product) PetscCall(MatProductSymbolic(Na->C));
1038:   C->ops->productnumeric = MatProductNumeric_SchurComplement_Any;
1039:   C->preallocated        = PETSC_TRUE;
1040:   C->assembled           = PETSC_FALSE;
1041:   PetscFunctionReturn(PETSC_SUCCESS);
1042: }

1044: static PetscErrorCode MatProductSetFromOptions_SchurComplement_Any(Mat C)
1045: {
1046:   Mat_Product         *product = C->product;
1047:   Mat_SchurComplement *Na, *Ca;
1048:   Mat                  B = product->B, S = product->A, pB = NULL, pC = NULL, pD = NULL;
1049:   KSP                  ksp;
1050:   PetscInt             m = PETSC_DECIDE, n = PETSC_DECIDE, M = PETSC_DECIDE, N = PETSC_DECIDE;
1051:   MatProductType       pbtype = MATPRODUCT_UNSPECIFIED, pctype = MATPRODUCT_UNSPECIFIED;
1052:   PetscBool            isschur;

1054:   PetscFunctionBegin;
1055:   if (product->type == MATPRODUCT_ABC || product->type == MATPRODUCT_AtB) PetscFunctionReturn(PETSC_SUCCESS);
1056:   /* A * S not yet supported (should be easy though) */
1057:   PetscCall(PetscObjectTypeCompare((PetscObject)S, MATSCHURCOMPLEMENT, &isschur));
1058:   if (!isschur) PetscFunctionReturn(PETSC_SUCCESS);

1060:   Na = (Mat_SchurComplement *)S->data;
1061:   if (Na->D) {
1062:     PetscCall(MatProductCreate(Na->D, B, NULL, &pD));
1063:     PetscCall(MatProductSetType(pD, product->type));
1064:     PetscCall(MatProductSetFromOptions(pD));
1065:   }
1066:   if (pD && !pD->ops->productsymbolic) {
1067:     PetscCall(MatDestroy(&pD));
1068:     PetscFunctionReturn(PETSC_SUCCESS);
1069:   }

1071:   /* S = A11 - A10 M A01 */
1072:   switch (product->type) {
1073:   case MATPRODUCT_AB: /* A11 B - A10 * M * A01 * B */
1074:     pbtype = product->type;
1075:     PetscCall(PetscObjectReference((PetscObject)Na->C));
1076:     pC = Na->C;
1077:     m  = S->rmap->n;
1078:     M  = S->rmap->N;
1079:     n  = B->cmap->n;
1080:     N  = B->cmap->N;
1081:     break;
1082:   case MATPRODUCT_ABt: /* A11 B^t - A10 * M * A01 * B^t */
1083:     pbtype = product->type;
1084:     PetscCall(PetscObjectReference((PetscObject)Na->C));
1085:     pC = Na->C;
1086:     m  = S->rmap->n;
1087:     M  = S->rmap->N;
1088:     n  = B->rmap->n;
1089:     N  = B->rmap->N;
1090:     break;
1091:   case MATPRODUCT_PtAP: /* Pt A11 P - Pt * A10 * M * A01 * P */
1092:     pbtype = MATPRODUCT_AB;
1093:     pctype = MATPRODUCT_AtB;
1094:     m      = B->cmap->n;
1095:     M      = B->cmap->N;
1096:     n      = B->cmap->n;
1097:     N      = B->cmap->N;
1098:     break;
1099:   case MATPRODUCT_RARt: /* R A11 Rt - R * A10 * M * A01 * Rt */
1100:     pbtype = MATPRODUCT_ABt;
1101:     pctype = MATPRODUCT_AB;
1102:     m      = B->rmap->n;
1103:     M      = B->rmap->N;
1104:     n      = B->rmap->n;
1105:     N      = B->rmap->N;
1106:     break;
1107:   default:
1108:     break;
1109:   }
1110:   PetscCall(MatProductCreate(Na->B, B, NULL, &pB));
1111:   PetscCall(MatProductSetType(pB, pbtype));
1112:   PetscCall(MatProductSetFromOptions(pB));
1113:   if (!pB->ops->productsymbolic) {
1114:     PetscCall(MatDestroy(&pB));
1115:     PetscFunctionReturn(PETSC_SUCCESS);
1116:   }
1117:   if (pC == NULL) { /* Some work can in principle be saved here if we recognize symmetry */
1118:     PetscCall(MatProductCreate(B, Na->C, NULL, &pC));
1119:     PetscCall(MatProductSetType(pC, pctype));
1120:     PetscCall(MatProductSetFromOptions(pC));
1121:     if (!pC->ops->productsymbolic) {
1122:       PetscCall(MatDestroy(&pC));
1123:       PetscFunctionReturn(PETSC_SUCCESS);
1124:     }
1125:   }
1126:   PetscCall(MatSetType(C, MATSCHURCOMPLEMENT));
1127:   PetscCall(MatSetSizes(C, m, n, M, N));
1128:   PetscCall(PetscLayoutSetUp(C->rmap));
1129:   PetscCall(PetscLayoutSetUp(C->cmap));
1130:   PetscCall(PetscObjectReference((PetscObject)Na->A));
1131:   PetscCall(PetscObjectReference((PetscObject)Na->Ap));
1132:   Ca                      = (Mat_SchurComplement *)C->data;
1133:   Ca->A                   = Na->A;
1134:   Ca->Ap                  = Na->Ap;
1135:   Ca->B                   = pB;
1136:   Ca->C                   = pC;
1137:   Ca->D                   = pD;
1138:   C->ops->productsymbolic = MatProductSymbolic_SchurComplement_Any;
1139:   PetscCall(MatSchurComplementGetKSP(S, &ksp));
1140:   PetscCall(MatSchurComplementSetKSP(C, ksp));
1141:   PetscFunctionReturn(PETSC_SUCCESS);
1142: }

1144: /*MC
1145:   MATSCHURCOMPLEMENT -  "schurcomplement" - Matrix type that behaves like the Schur complement of a matrix.

1147:   Level: intermediate

1149: .seealso: [](ch_matrices), `Mat`, `MatCreate()`, `MatType`, `MatCreateSchurComplement()`, `MatSchurComplementComputeExplicitOperator()`,
1150:           `MatSchurComplementGetSubMatrices()`, `MatSchurComplementGetKSP()`
1151: M*/
1152: PETSC_EXTERN PetscErrorCode MatCreate_SchurComplement(Mat N)
1153: {
1154:   Mat_SchurComplement *Na;

1156:   PetscFunctionBegin;
1157:   PetscCall(PetscNew(&Na));
1158:   N->data = (void *)Na;

1160:   N->ops->destroy        = MatDestroy_SchurComplement;
1161:   N->ops->getvecs        = MatCreateVecs_SchurComplement;
1162:   N->ops->view           = MatView_SchurComplement;
1163:   N->ops->mult           = MatMult_SchurComplement;
1164:   N->ops->multtranspose  = MatMultTranspose_SchurComplement;
1165:   N->ops->multadd        = MatMultAdd_SchurComplement;
1166:   N->ops->setfromoptions = MatSetFromOptions_SchurComplement;
1167:   N->assembled           = PETSC_FALSE;
1168:   N->preallocated        = PETSC_FALSE;

1170:   PetscCall(KSPCreate(PetscObjectComm((PetscObject)N), &Na->ksp));
1171:   PetscCall(PetscObjectChangeTypeName((PetscObject)N, MATSCHURCOMPLEMENT));
1172:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_schurcomplement_seqdense_C", MatProductSetFromOptions_SchurComplement_Dense));
1173:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_schurcomplement_mpidense_C", MatProductSetFromOptions_SchurComplement_Dense));
1174:   PetscCall(PetscObjectComposeFunction((PetscObject)N, "MatProductSetFromOptions_anytype_C", MatProductSetFromOptions_SchurComplement_Any));
1175:   PetscFunctionReturn(PETSC_SUCCESS);
1176: }