Date: 2021-03-03T22:27:38+00:00

Actual source code: schurm.c
petsc-3.14.5 2021-03-03
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      1: #include <../src/ksp/ksp/utils/schurm/schurm.h>

      3: const char *const MatSchurComplementAinvTypes[] = {"DIAG","LUMP","BLOCKDIAG","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;
      8:   PetscErrorCode      ierr;

     11:   if (Na->D) {
     12:     MatCreateVecs(Na->D,right,left);
     13:     return(0);
     14:   }
     15:   if (right) {
     16:     MatCreateVecs(Na->B,right,NULL);
     17:   }
     18:   if (left) {
     19:     MatCreateVecs(Na->C,NULL,left);
     20:   }
     21:   return(0);
     22: }

     24: PetscErrorCode MatView_SchurComplement(Mat N,PetscViewer viewer)
     25: {
     26:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
     27:   PetscErrorCode      ierr;

     30:   PetscViewerASCIIPrintf(viewer,"Schur complement A11 - A10 inv(A00) A01\n");
     31:   if (Na->D) {
     32:     PetscViewerASCIIPrintf(viewer,"A11\n");
     33:     PetscViewerASCIIPushTab(viewer);
     34:     MatView(Na->D,viewer);
     35:     PetscViewerASCIIPopTab(viewer);
     36:   } else {
     37:     PetscViewerASCIIPrintf(viewer,"A11 = 0\n");
     38:   }
     39:   PetscViewerASCIIPrintf(viewer,"A10\n");
     40:   PetscViewerASCIIPushTab(viewer);
     41:   MatView(Na->C,viewer);
     42:   PetscViewerASCIIPopTab(viewer);
     43:   PetscViewerASCIIPrintf(viewer,"KSP of A00\n");
     44:   PetscViewerASCIIPushTab(viewer);
     45:   KSPView(Na->ksp,viewer);
     46:   PetscViewerASCIIPopTab(viewer);
     47:   PetscViewerASCIIPrintf(viewer,"A01\n");
     48:   PetscViewerASCIIPushTab(viewer);
     49:   MatView(Na->B,viewer);
     50:   PetscViewerASCIIPopTab(viewer);
     51:   return(0);
     52: }

     54: /*
     55:            A11^T - A01^T ksptrans(A00,Ap00) A10^T
     56: */
     57: PetscErrorCode MatMultTranspose_SchurComplement(Mat N,Vec x,Vec y)
     58: {
     59:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
     60:   PetscErrorCode      ierr;

     63:   if (!Na->work1) {MatCreateVecs(Na->A,&Na->work1,NULL);}
     64:   if (!Na->work2) {MatCreateVecs(Na->A,&Na->work2,NULL);}
     65:   MatMultTranspose(Na->C,x,Na->work1);
     66:   KSPSolveTranspose(Na->ksp,Na->work1,Na->work2);
     67:   MatMultTranspose(Na->B,Na->work2,y);
     68:   VecScale(y,-1.0);
     69:   if (Na->D) {
     70:     MatMultTransposeAdd(Na->D,x,y,y);
     71:   }
     72:   return(0);
     73: }

     75: /*
     76:            A11 - A10 ksp(A00,Ap00) A01
     77: */
     78: PetscErrorCode MatMult_SchurComplement(Mat N,Vec x,Vec y)
     79: {
     80:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
     81:   PetscErrorCode      ierr;

     84:   if (!Na->work1) {MatCreateVecs(Na->A,&Na->work1,NULL);}
     85:   if (!Na->work2) {MatCreateVecs(Na->A,&Na->work2,NULL);}
     86:   MatMult(Na->B,x,Na->work1);
     87:   KSPSolve(Na->ksp,Na->work1,Na->work2);
     88:   MatMult(Na->C,Na->work2,y);
     89:   VecScale(y,-1.0);
     90:   if (Na->D) {
     91:     MatMultAdd(Na->D,x,y,y);
     92:   }
     93:   return(0);
     94: }

     96: /*
     97:            A11 - A10 ksp(A00,Ap00) A01
     98: */
     99: PetscErrorCode MatMultAdd_SchurComplement(Mat N,Vec x,Vec y,Vec z)
    100: {
    101:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
    102:   PetscErrorCode      ierr;

    105:   if (!Na->work1) {MatCreateVecs(Na->A,&Na->work1,NULL);}
    106:   if (!Na->work2) {MatCreateVecs(Na->A,&Na->work2,NULL);}
    107:   MatMult(Na->B,x,Na->work1);
    108:   KSPSolve(Na->ksp,Na->work1,Na->work2);
    109:   if (y == z) {
    110:     VecScale(Na->work2,-1.0);
    111:     MatMultAdd(Na->C,Na->work2,z,z);
    112:   } else {
    113:     MatMult(Na->C,Na->work2,z);
    114:     VecAYPX(z,-1.0,y);
    115:   }
    116:   if (Na->D) {
    117:     MatMultAdd(Na->D,x,z,z);
    118:   }
    119:   return(0);
    120: }

    122: PetscErrorCode MatSetFromOptions_SchurComplement(PetscOptionItems *PetscOptionsObject,Mat N)
    123: {
    124:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
    125:   PetscErrorCode      ierr;

    128:   PetscOptionsHead(PetscOptionsObject,"MatSchurComplementOptions");
    129:   Na->ainvtype = MAT_SCHUR_COMPLEMENT_AINV_DIAG;
    130:   PetscOptionsEnum("-mat_schur_complement_ainv_type","Type of approximation for inv(A00) used when assembling Sp = A11 - A10 inv(A00) A01","MatSchurComplementSetAinvType",MatSchurComplementAinvTypes,(PetscEnum)Na->ainvtype,(PetscEnum*)&Na->ainvtype,NULL);
    131:   PetscOptionsTail();
    132:   KSPSetFromOptions(Na->ksp);
    133:   return(0);
    134: }

    136: PetscErrorCode MatDestroy_SchurComplement(Mat N)
    137: {
    138:   Mat_SchurComplement *Na = (Mat_SchurComplement*)N->data;
    139:   PetscErrorCode      ierr;

    142:   MatDestroy(&Na->A);
    143:   MatDestroy(&Na->Ap);
    144:   MatDestroy(&Na->B);
    145:   MatDestroy(&Na->C);
    146:   MatDestroy(&Na->D);
    147:   VecDestroy(&Na->work1);
    148:   VecDestroy(&Na->work2);
    149:   KSPDestroy(&Na->ksp);
    150:   PetscFree(N->data);
    151:   return(0);
    152: }

    154: /*@
    155:       MatCreateSchurComplement - Creates a new matrix object that behaves like the Schur complement of a matrix

    157:    Collective on A00

    159:    Input Parameters:
    160: +   A00,A01,A10,A11  - the four parts of the original matrix A = [A00 A01; A10 A11] (A11 is optional)
    161: -   Ap00             - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A^{-1}

    163:    Output Parameter:
    164: .   S - the matrix that the Schur complement S = A11 - A10 ksp(A00,Ap00) A01

    166:    Level: intermediate

    168:    Notes:
    169:     The Schur complement is NOT actually formed! Rather, this
    170:           object performs the matrix-vector product by using formula S = A11 - A10 A^{-1} A01
    171:           for Schur complement S and a KSP solver to approximate the action of A^{-1}.

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

    175:           A00 and  A11 must be square matrices.

    177:           MatGetSchurComplement() takes as arguments the index sets for the submatrices and returns both the virtual Schur complement (what this returns) plus
    178:           a sparse approximation to the true Schur complement (useful for building a preconditioner for the Schur complement).

    180:           MatSchurComplementGetPmat() can be called on the output of this function to generate an explicit approximation to the Schur complement.

    182:     Developer Notes:
    183:     The API that includes MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementGetPmat() should be refactored to
    184:     remove redundancy and be clearer and simplier.

    187: .seealso: MatCreateNormal(), MatMult(), MatCreate(), MatSchurComplementGetKSP(), MatSchurComplementUpdateSubMatrices(), MatCreateTranspose(), MatGetSchurComplement(),
    188:           MatSchurComplementGetPmat()

    190: @*/
    191: PetscErrorCode  MatCreateSchurComplement(Mat A00,Mat Ap00,Mat A01,Mat A10,Mat A11,Mat *S)
    192: {

    196:   KSPInitializePackage();
    197:   MatCreate(PetscObjectComm((PetscObject)A00),S);
    198:   MatSetType(*S,MATSCHURCOMPLEMENT);
    199:   MatSchurComplementSetSubMatrices(*S,A00,Ap00,A01,A10,A11);
    200:   return(0);
    201: }

    203: /*@
    204:       MatSchurComplementSetSubMatrices - Sets the matrices that define the Schur complement

    206:    Collective on S

    208:    Input Parameter:
    209: +   S                - matrix obtained with MatCreateSchurComplement (or equivalent) and implementing the action of A11 - A10 ksp(A00,Ap00) A01
    210: .   A00,A01,A10,A11  - the four parts of A = [A00 A01; A10 A11] (A11 is optional)
    211: -   Ap00             - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A^{-1}.

    213:    Level: intermediate

    215:    Notes:
    216:     The Schur complement is NOT actually formed! Rather, this
    217:           object performs the matrix-vector product by using formula S = A11 - A10 A^{-1} A01
    218:           for Schur complement S and a KSP solver to approximate the action of A^{-1}.

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

    222:           A00 and  A11 must be square matrices.

    224: .seealso: MatCreateNormal(), MatMult(), MatCreate(), MatSchurComplementGetKSP(), MatSchurComplementUpdateSubMatrices(), MatCreateTranspose(), MatCreateSchurComplement(), MatGetSchurComplement()

    226: @*/
    227: PetscErrorCode  MatSchurComplementSetSubMatrices(Mat S,Mat A00,Mat Ap00,Mat A01,Mat A10,Mat A11)
    228: {
    229:   PetscErrorCode      ierr;
    230:   Mat_SchurComplement *Na = (Mat_SchurComplement*)S->data;
    231:   PetscBool           isschur;

    234:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    235:   if (!isschur) return(0);
    236:   if (S->assembled) SETERRQ(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONGSTATE,"Use MatSchurComplementUpdateSubMatrices() for already used matrix");
    244:   if (A00->rmap->n != A00->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A00 %D do not equal local columns %D",A00->rmap->n,A00->cmap->n);
    245:   if (A00->rmap->n != Ap00->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A00 %D do not equal local rows of Ap00 %D",A00->rmap->n,Ap00->rmap->n);
    246:   if (Ap00->rmap->n != Ap00->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of Ap00 %D do not equal local columns %D",Ap00->rmap->n,Ap00->cmap->n);
    247:   if (A00->cmap->n != A01->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local columns of A00 %D do not equal local rows of A01 %D",A00->cmap->n,A01->rmap->n);
    248:   if (A10->cmap->n != A00->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local columns of A10 %D do not equal local rows of A00 %D",A10->cmap->n,A00->rmap->n);
    249:   if (A11) {
    252:     if (A10->rmap->n != A11->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A10 %D do not equal local rows A11 %D",A10->rmap->n,A11->rmap->n);
    253:   }

    255:   MatSetSizes(S,A10->rmap->n,A01->cmap->n,A10->rmap->N,A01->cmap->N);
    256:   PetscObjectReference((PetscObject)A00);
    257:   PetscObjectReference((PetscObject)Ap00);
    258:   PetscObjectReference((PetscObject)A01);
    259:   PetscObjectReference((PetscObject)A10);
    260:   Na->A  = A00;
    261:   Na->Ap = Ap00;
    262:   Na->B  = A01;
    263:   Na->C  = A10;
    264:   Na->D  = A11;
    265:   if (A11) {
    266:     PetscObjectReference((PetscObject)A11);
    267:   }
    268:   MatSetUp(S);
    269:   KSPSetOperators(Na->ksp,A00,Ap00);
    270:   S->assembled = PETSC_TRUE;
    271:   return(0);
    272: }

    274: /*@
    275:   MatSchurComplementGetKSP - Gets the KSP object that is used to invert A00 in the Schur complement matrix S = A11 - A10 ksp(A00,Ap00) A01

    277:   Not Collective

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

    282:   Output Parameter:
    283: . ksp - the linear solver object

    285:   Options Database:
    286: . -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.

    288:   Level: intermediate

    290: .seealso: MatSchurComplementSetKSP(), MatCreateSchurComplement(), MatCreateNormal(), MatMult(), MatCreate()
    291: @*/
    292: PetscErrorCode MatSchurComplementGetKSP(Mat S, KSP *ksp)
    293: {
    294:   Mat_SchurComplement *Na;
    295:   PetscBool           isschur;
    296:   PetscErrorCode      ierr;

    300:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    301:   if (!isschur) SETERRQ1(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONG,"Not for type %s",((PetscObject)S)->type_name);
    303:   Na   = (Mat_SchurComplement*) S->data;
    304:   *ksp = Na->ksp;
    305:   return(0);
    306: }

    308: /*@
    309:   MatSchurComplementSetKSP - Sets the KSP object that is used to invert A00 in the Schur complement matrix S = A11 - A10 ksp(A00,Ap00) A01

    311:   Not Collective

    313:   Input Parameters:
    314: + S   - matrix created with MatCreateSchurComplement()
    315: - ksp - the linear solver object

    317:   Level: developer

    319:   Developer Notes:
    320:     This is used in PCFieldSplit to reuse the 0-split KSP to implement ksp(A00,Ap00) in S.
    321:     The KSP operators are overwritten with A00 and Ap00 currently set in S.

    323: .seealso: MatSchurComplementGetKSP(), MatCreateSchurComplement(), MatCreateNormal(), MatMult(), MatCreate(), MATSCHURCOMPLEMENT
    324: @*/
    325: PetscErrorCode MatSchurComplementSetKSP(Mat S, KSP ksp)
    326: {
    327:   Mat_SchurComplement *Na;
    328:   PetscErrorCode      ierr;
    329:   PetscBool           isschur;

    333:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    334:   if (!isschur) return(0);
    336:   Na      = (Mat_SchurComplement*) S->data;
    337:   PetscObjectReference((PetscObject)ksp);
    338:   KSPDestroy(&Na->ksp);
    339:   Na->ksp = ksp;
    340:   KSPSetOperators(Na->ksp, Na->A, Na->Ap);
    341:   return(0);
    342: }

    344: /*@
    345:       MatSchurComplementUpdateSubMatrices - Updates the Schur complement matrix object with new submatrices

    347:    Collective on S

    349:    Input Parameters:
    350: +   S                - matrix obtained with MatCreateSchurComplement() (or equivalent) and implementing the action of A11 - A10 ksp(A00,Ap00) A01
    351: .   A00,A01,A10,A11  - the four parts of A = [A00 A01; A10 A11] (A11 is optional)
    352: -   Ap00             - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A^{-1}.

    354:    Level: intermediate

    356:    Notes:
    357:     All four matrices must have the same MPI communicator

    359:           A00 and  A11 must be square matrices

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

    364: .seealso: MatCreateNormal(), MatMult(), MatCreate(), MatSchurComplementGetKSP(), MatCreateSchurComplement()

    366: @*/
    367: PetscErrorCode  MatSchurComplementUpdateSubMatrices(Mat S,Mat A00,Mat Ap00,Mat A01,Mat A10,Mat A11)
    368: {
    369:   PetscErrorCode      ierr;
    370:   Mat_SchurComplement *Na = (Mat_SchurComplement*)S->data;
    371:   PetscBool           isschur;

    375:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    376:   if (!isschur) return(0);
    377:   if (!S->assembled) SETERRQ(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONGSTATE,"Use MatSchurComplementSetSubMatrices() for a new matrix");
    385:   if (A00->rmap->n != A00->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A00 %D do not equal local columns %D",A00->rmap->n,A00->cmap->n);
    386:   if (A00->rmap->n != Ap00->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A00 %D do not equal local rows of Ap00 %D",A00->rmap->n,Ap00->rmap->n);
    387:   if (Ap00->rmap->n != Ap00->cmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of Ap00 %D do not equal local columns %D",Ap00->rmap->n,Ap00->cmap->n);
    388:   if (A00->cmap->n != A01->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local columns of A00 %D do not equal local rows of A01 %D",A00->cmap->n,A01->rmap->n);
    389:   if (A10->cmap->n != A00->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local columns of A10 %D do not equal local rows of A00 %D",A10->cmap->n,A00->rmap->n);
    390:   if (A11) {
    393:     if (A10->rmap->n != A11->rmap->n) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_SIZ,"Local rows of A10 %D do not equal local rows A11 %D",A10->rmap->n,A11->rmap->n);
    394:   }

    396:   PetscObjectReference((PetscObject)A00);
    397:   PetscObjectReference((PetscObject)Ap00);
    398:   PetscObjectReference((PetscObject)A01);
    399:   PetscObjectReference((PetscObject)A10);
    400:   if (A11) {
    401:     PetscObjectReference((PetscObject)A11);
    402:   }

    404:   MatDestroy(&Na->A);
    405:   MatDestroy(&Na->Ap);
    406:   MatDestroy(&Na->B);
    407:   MatDestroy(&Na->C);
    408:   MatDestroy(&Na->D);

    410:   Na->A  = A00;
    411:   Na->Ap = Ap00;
    412:   Na->B  = A01;
    413:   Na->C  = A10;
    414:   Na->D  = A11;

    416:   KSPSetOperators(Na->ksp,A00,Ap00);
    417:   return(0);
    418: }

    421: /*@C
    422:   MatSchurComplementGetSubMatrices - Get the individual submatrices in the Schur complement

    424:   Collective on S

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

    429:   Output Parameters:
    430: + A00,A01,A10,A11  - the four parts of the original matrix A = [A00 A01; A10 A11] (A11 is optional)
    431: - Ap00             - preconditioning matrix for use in ksp(A00,Ap00) to approximate the action of A^{-1}.

    433:   Note: A11 is optional, and thus can be NULL.  The submatrices are not increfed before they are returned and should not be modified or destroyed.

    435:   Level: intermediate

    437: .seealso: MatCreateNormal(), MatMult(), MatCreate(), MatSchurComplementGetKSP(), MatCreateSchurComplement(), MatSchurComplementUpdateSubMatrices()
    438: @*/
    439: PetscErrorCode  MatSchurComplementGetSubMatrices(Mat S,Mat *A00,Mat *Ap00,Mat *A01,Mat *A10,Mat *A11)
    440: {
    441:   Mat_SchurComplement *Na = (Mat_SchurComplement*) S->data;
    442:   PetscErrorCode      ierr;
    443:   PetscBool           flg;

    447:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&flg);
    448:   if (!flg) SETERRQ1(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONG,"Not for type %s",((PetscObject)S)->type_name);
    449:   if (A00) *A00 = Na->A;
    450:   if (Ap00) *Ap00 = Na->Ap;
    451:   if (A01) *A01 = Na->B;
    452:   if (A10) *A10 = Na->C;
    453:   if (A11) *A11 = Na->D;
    454:   return(0);
    455: }

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

    459: /*@
    460:   MatSchurComplementComputeExplicitOperator - Compute the Schur complement matrix explicitly

    462:   Collective on M

    464:   Input Parameter:
    465: . M - the matrix obtained with MatCreateSchurComplement()

    467:   Output Parameter:
    468: . S - the Schur complement matrix

    470:   Note: This can be expensive, so it is mainly for testing

    472:   Level: advanced

    474: .seealso: MatCreateSchurComplement(), MatSchurComplementUpdate()
    475: @*/
    476: PetscErrorCode MatSchurComplementComputeExplicitOperator(Mat M, Mat *S)
    477: {
    478:   Mat            B, C, D;
    479:   KSP            ksp;
    480:   PC             pc;
    481:   PetscBool      isLU, isILU;
    482:   PetscReal      fill = 2.0;

    486:   MatSchurComplementGetSubMatrices(M, NULL, NULL, &B, &C, &D);
    487:   MatSchurComplementGetKSP(M, &ksp);
    488:   KSPGetPC(ksp, &pc);
    489:   PetscObjectTypeCompare((PetscObject) pc, PCLU, &isLU);
    490:   PetscObjectTypeCompare((PetscObject) pc, PCILU, &isILU);
    491:   if (isLU || isILU) {
    492:     Mat       fact, Bd, AinvB, AinvBd;
    493:     PetscReal eps = 1.0e-10;

    495:     /* This can be sped up for banded LU */
    496:     KSPSetUp(ksp);
    497:     PCFactorGetMatrix(pc, &fact);
    498:     MatConvert(B, MATDENSE, MAT_INITIAL_MATRIX, &Bd);
    499:     MatDuplicate(Bd, MAT_DO_NOT_COPY_VALUES, &AinvBd);
    500:     MatMatSolve(fact, Bd, AinvBd);
    501:     MatDestroy(&Bd);
    502:     MatChop(AinvBd, eps);
    503:     MatConvert(AinvBd, MATAIJ, MAT_INITIAL_MATRIX, &AinvB);
    504:     MatDestroy(&AinvBd);
    505:     MatMatMult(C, AinvB, MAT_INITIAL_MATRIX, fill, S);
    506:     MatDestroy(&AinvB);
    507:   } else {
    508:     Mat Ainv;

    510:     PCComputeOperator(pc, MATAIJ, &Ainv);
    511: #if 0
    512:     /* Symmetric version */
    513:     MatPtAP(Ainv, B, MAT_INITIAL_MATRIX, fill, S);
    514: #else
    515:     /* Nonsymmetric version */
    516:     MatMatMatMult(C, Ainv, B, MAT_INITIAL_MATRIX, fill, S);
    517: #endif
    518:     MatDestroy(&Ainv);
    519:   }
    520:   if (D) {
    521:     MatAXPY(*S, -1.0, D, DIFFERENT_NONZERO_PATTERN);
    522:   }
    523:   MatScale(*S,-1.0);
    524:   return(0);
    525: }

    527: /* Developer Notes:
    528:     This should be implemented with a MatCreate_SchurComplement() as that is the standard design for new Mat classes. */
    529: PetscErrorCode MatGetSchurComplement_Basic(Mat mat,IS isrow0,IS iscol0,IS isrow1,IS iscol1,MatReuse mreuse,Mat *newmat,MatSchurComplementAinvType ainvtype, MatReuse preuse,Mat *newpmat)
    530: {
    532:   Mat            A=NULL,Ap=NULL,B=NULL,C=NULL,D=NULL;
    533:   MatReuse       reuse;

    545:   if (mreuse == MAT_IGNORE_MATRIX && preuse == MAT_IGNORE_MATRIX) return(0);

    549:   if (mat->factortype) SETERRQ(PetscObjectComm((PetscObject)mat),PETSC_ERR_ARG_WRONGSTATE,"Not for factored matrix");

    551:   reuse = MAT_INITIAL_MATRIX;
    552:   if (mreuse == MAT_REUSE_MATRIX) {
    553:     MatSchurComplementGetSubMatrices(*newmat,&A,&Ap,&B,&C,&D);
    554:     if (!A || !Ap || !B || !C) SETERRQ(PetscObjectComm((PetscObject)mat),PETSC_ERR_ARG_WRONGSTATE,"Attempting to reuse matrix but Schur complement matrices unset");
    555:     if (A != Ap) SETERRQ(PetscObjectComm((PetscObject)mat),PETSC_ERR_ARG_WRONGSTATE,"Preconditioning matrix does not match operator");
    556:     MatDestroy(&Ap); /* get rid of extra reference */
    557:     reuse = MAT_REUSE_MATRIX;
    558:   }
    559:   MatCreateSubMatrix(mat,isrow0,iscol0,reuse,&A);
    560:   MatCreateSubMatrix(mat,isrow0,iscol1,reuse,&B);
    561:   MatCreateSubMatrix(mat,isrow1,iscol0,reuse,&C);
    562:   MatCreateSubMatrix(mat,isrow1,iscol1,reuse,&D);
    563:   switch (mreuse) {
    564:   case MAT_INITIAL_MATRIX:
    565:     MatCreateSchurComplement(A,A,B,C,D,newmat);
    566:     break;
    567:   case MAT_REUSE_MATRIX:
    568:     MatSchurComplementUpdateSubMatrices(*newmat,A,A,B,C,D);
    569:     break;
    570:   default:
    571:     if (mreuse != MAT_IGNORE_MATRIX) SETERRQ1(PetscObjectComm((PetscObject)mat),PETSC_ERR_SUP,"Unrecognized value of mreuse %d",(int)mreuse);
    572:   }
    573:   if (preuse != MAT_IGNORE_MATRIX) {
    574:     MatCreateSchurComplementPmat(A,B,C,D,ainvtype,preuse,newpmat);
    575:   }
    576:   MatDestroy(&A);
    577:   MatDestroy(&B);
    578:   MatDestroy(&C);
    579:   MatDestroy(&D);
    580:   return(0);
    581: }

    583: /*@
    584:     MatGetSchurComplement - Obtain the Schur complement from eliminating part of the matrix in another part.

    586:     Collective on A

    588:     Input Parameters:
    589: +   A      - matrix in which the complement is to be taken
    590: .   isrow0 - rows to eliminate
    591: .   iscol0 - columns to eliminate, (isrow0,iscol0) should be square and nonsingular
    592: .   isrow1 - rows in which the Schur complement is formed
    593: .   iscol1 - columns in which the Schur complement is formed
    594: .   mreuse - MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX, use MAT_IGNORE_MATRIX to put nothing in S
    595: .   ainvtype - the type of approximation used for the inverse of the (0,0) block used in forming Sp:
    596:                        MAT_SCHUR_COMPLEMENT_AINV_DIAG, MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG, or MAT_SCHUR_COMPLEMENT_AINV_LUMP
    597: -   preuse - MAT_INITIAL_MATRIX or MAT_REUSE_MATRIX, use MAT_IGNORE_MATRIX to put nothing in Sp

    599:     Output Parameters:
    600: +   S      - exact Schur complement, often of type MATSCHURCOMPLEMENT which is difficult to use for preconditioning
    601: -   Sp     - approximate Schur complement from which a preconditioner can be built

    603:     Note:
    604:     Since the real Schur complement is usually dense, providing a good approximation to newpmat usually requires
    605:     application-specific information.  The default for assembled matrices is to use the inverse of the diagonal of
    606:     the (0,0) block A00 in place of A00^{-1}. This rarely produce a scalable algorithm. Optionally, A00 can be lumped
    607:     before forming inv(diag(A00)).

    609:     Sometimes users would like to provide problem-specific data in the Schur complement, usually only for special row
    610:     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
    611:     "MatGetSchurComplement_C" to their function.  If their function needs to fall back to the default implementation, it
    612:     should call MatGetSchurComplement_Basic().

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

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

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

    621:     Developer Notes:
    622:     The API that includes MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementGetPmat() should be refactored to
    623:     remove redundancy and be clearer and simplier.

    625:     Level: advanced

    627: .seealso: MatCreateSubMatrix(), PCFIELDSPLIT, MatCreateSchurComplement(), MatSchurComplementAinvType
    628: @*/
    629: PetscErrorCode  MatGetSchurComplement(Mat A,IS isrow0,IS iscol0,IS isrow1,IS iscol1,MatReuse mreuse,Mat *S,MatSchurComplementAinvType ainvtype,MatReuse preuse,Mat *Sp)
    630: {
    631:   PetscErrorCode ierr,(*f)(Mat,IS,IS,IS,IS,MatReuse,Mat*,MatReuse,Mat*) = NULL;

    645:   if (A->factortype) SETERRQ(PetscObjectComm((PetscObject)A),PETSC_ERR_ARG_WRONGSTATE,"Not for factored matrix");
    646:   f = NULL;
    647:   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. */
    648:     PetscObjectQueryFunction((PetscObject)*S,"MatGetSchurComplement_C",&f);
    649:   }
    650:   if (f) {
    651:     (*f)(A,isrow0,iscol0,isrow1,iscol1,mreuse,S,preuse,Sp);
    652:   } else {
    653:     MatGetSchurComplement_Basic(A,isrow0,iscol0,isrow1,iscol1,mreuse,S,ainvtype,preuse,Sp);
    654:   }
    655:   return(0);
    656: }

    658: /*@
    659:     MatSchurComplementSetAinvType - set the type of approximation used for the inverse of the (0,0) block used in forming Sp in MatSchurComplementGetPmat()

    661:     Not collective.

    663:     Input Parameters:
    664: +   S        - matrix obtained with MatCreateSchurComplement() (or equivalent) and implementing the action of A11 - A10 ksp(A00,Ap00) A01
    665: -   ainvtype - type of approximation used to form A00inv from A00 when assembling Sp = A11 - A10 A00inv A01:
    666:                       MAT_SCHUR_COMPLEMENT_AINV_DIAG, MAT_SCHUR_COMPLEMENT_AINV_LUMP, or MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG

    668:     Options database:
    669:     -mat_schur_complement_ainv_type diag | lump | blockdiag

    671:     Note:
    672:     Since the real Schur complement is usually dense, providing a good approximation to newpmat usually requires
    673:     application-specific information.  The default for assembled matrices is to use the inverse of the diagonal of
    674:     the (0,0) block A00 in place of A00^{-1}. This rarely produces a scalable algorithm. Optionally, A00 can be lumped
    675:     before forming inv(diag(A00)).

    677:     Level: advanced

    679: .seealso: MatSchurComplementAinvType, MatCreateSchurComplement(), MatGetSchurComplement(), MatSchurComplementGetPmat(), MatSchurComplementGetAinvType()
    680: @*/
    681: PetscErrorCode  MatSchurComplementSetAinvType(Mat S,MatSchurComplementAinvType ainvtype)
    682: {
    683:   PetscErrorCode      ierr;
    684:   PetscBool           isschur;
    685:   Mat_SchurComplement *schur;

    689:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    690:   if (!isschur) return(0);
    692:   schur = (Mat_SchurComplement*)S->data;
    693:   if (ainvtype != MAT_SCHUR_COMPLEMENT_AINV_DIAG && ainvtype != MAT_SCHUR_COMPLEMENT_AINV_LUMP && ainvtype != MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Unknown MatSchurComplementAinvType: %d",(int)ainvtype);
    694:   schur->ainvtype = ainvtype;
    695:   return(0);
    696: }

    698: /*@
    699:     MatSchurComplementGetAinvType - get the type of approximation for the inverse of the (0,0) block used in forming Sp in MatSchurComplementGetPmat()

    701:     Not collective.

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

    706:     Output Parameter:
    707: .   ainvtype - type of approximation used to form A00inv from A00 when assembling Sp = A11 - A10 A00inv A01:
    708:                       MAT_SCHUR_COMPLEMENT_AINV_DIAG, MAT_SCHUR_COMPLEMENT_AINV_LUMP, or MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG

    710:     Note:
    711:     Since the real Schur complement is usually dense, providing a good approximation to newpmat usually requires
    712:     application-specific information.  The default for assembled matrices is to use the inverse of the diagonal of
    713:     the (0,0) block A00 in place of A00^{-1}. This rarely produce a scalable algorithm. Optionally, A00 can be lumped
    714:     before forming inv(diag(A00)).

    716:     Level: advanced

    718: .seealso: MatSchurComplementAinvType, MatCreateSchurComplement(), MatGetSchurComplement(), MatSchurComplementGetPmat(), MatSchurComplementSetAinvType()
    719: @*/
    720: PetscErrorCode  MatSchurComplementGetAinvType(Mat S,MatSchurComplementAinvType *ainvtype)
    721: {
    722:   PetscErrorCode      ierr;
    723:   PetscBool           isschur;
    724:   Mat_SchurComplement *schur;

    728:   PetscObjectTypeCompare((PetscObject)S,MATSCHURCOMPLEMENT,&isschur);
    729:   if (!isschur) SETERRQ1(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONG,"Not for type %s",((PetscObject)S)->type_name);
    730:   schur = (Mat_SchurComplement*)S->data;
    731:   if (ainvtype) *ainvtype = schur->ainvtype;
    732:   return(0);
    733: }

    735: /*@
    736:     MatCreateSchurComplementPmat - create a preconditioning matrix for the Schur complement by assembling Sp = A11 - A10 inv(diag(A00)) A01

    738:     Collective on A00

    740:     Input Parameters:
    741: +   A00,A01,A10,A11      - the four parts of the original matrix A = [A00 A01; A10 A11] (A01,A10, and A11 are optional, implying zero matrices)
    742: .   ainvtype             - type of approximation for inv(A00) used when forming Sp = A11 - A10 inv(A00) A01
    743: -   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

    745:     Output Parameter:
    746: -   Spmat                - approximate Schur complement suitable for preconditioning S = A11 - A10 inv(diag(A00)) A01

    748:     Note:
    749:     Since the real Schur complement is usually dense, providing a good approximation to newpmat usually requires
    750:     application-specific information.  The default for assembled matrices is to use the inverse of the diagonal of
    751:     the (0,0) block A00 in place of A00^{-1}. This rarely produce a scalable algorithm. Optionally, A00 can be lumped
    752:     before forming inv(diag(A00)).

    754:     Level: advanced

    756: .seealso: MatCreateSchurComplement(), MatGetSchurComplement(), MatSchurComplementGetPmat(), MatSchurComplementAinvType
    757: @*/
    758: PetscErrorCode  MatCreateSchurComplementPmat(Mat A00,Mat A01,Mat A10,Mat A11,MatSchurComplementAinvType ainvtype,MatReuse preuse,Mat *Spmat)
    759: {
    761:   PetscInt       N00;

    764:   /* Use an appropriate approximate inverse of A00 to form A11 - A10 inv(diag(A00)) A01; a NULL A01, A10 or A11 indicates a zero matrix. */
    765:   /* TODO: Perhaps should create an appropriately-sized zero matrix of the same type as A00? */
    766:   if ((!A01 || !A10) & !A11) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONGSTATE,"Cannot assemble Spmat: A01, A10 and A11 are all NULL.");
    768:   if (preuse == MAT_IGNORE_MATRIX) return(0);

    770:   /* A zero size A00 or empty A01 or A10 imply S = A11. */
    771:   MatGetSize(A00,&N00,NULL);
    772:   if (!A01 || !A10 || !N00) {
    773:     if (preuse == MAT_INITIAL_MATRIX) {
    774:       MatDuplicate(A11,MAT_COPY_VALUES,Spmat);
    775:     } else { /* MAT_REUSE_MATRIX */
    776:       /* TODO: when can we pass SAME_NONZERO_PATTERN? */
    777:       MatCopy(A11,*Spmat,DIFFERENT_NONZERO_PATTERN);
    778:     }
    779:   } else {
    780:     Mat AdB;
    781:     Vec diag;

    783:     if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_LUMP || ainvtype == MAT_SCHUR_COMPLEMENT_AINV_DIAG) {
    784:       MatDuplicate(A01,MAT_COPY_VALUES,&AdB);
    785:       MatCreateVecs(A00,&diag,NULL);
    786:       if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_LUMP) {
    787:         MatGetRowSum(A00,diag);
    788:       } else {
    789:         MatGetDiagonal(A00,diag);
    790:       }
    791:       VecReciprocal(diag);
    792:       MatDiagonalScale(AdB,diag,NULL);
    793:       VecDestroy(&diag);
    794:     } else if (ainvtype == MAT_SCHUR_COMPLEMENT_AINV_BLOCK_DIAG) {
    795:       Mat      A00_inv;
    796:       MatType  type;
    797:       MPI_Comm comm;

    799:       PetscObjectGetComm((PetscObject)A00,&comm);
    800:       MatGetType(A00,&type);
    801:       MatCreate(comm,&A00_inv);
    802:       MatSetType(A00_inv,type);
    803:       MatInvertBlockDiagonalMat(A00,A00_inv);
    804:       MatMatMult(A00_inv,A01,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&AdB);
    805:       MatDestroy(&A00_inv);
    806:     } else SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Unknown MatSchurComplementAinvType: %D", ainvtype);
    807:     /* Cannot really reuse Spmat in MatMatMult() because of MatAYPX() -->
    808:          MatAXPY() --> MatHeaderReplace() --> MatDestroy_XXX_MatMatMult()  */
    809:     MatDestroy(Spmat);
    810:     MatMatMult(A10,AdB,MAT_INITIAL_MATRIX,PETSC_DEFAULT,Spmat);
    811:     if (!A11) {
    812:       MatScale(*Spmat,-1.0);
    813:     } else {
    814:       /* TODO: when can we pass SAME_NONZERO_PATTERN? */
    815:       MatAYPX(*Spmat,-1,A11,DIFFERENT_NONZERO_PATTERN);
    816:     }
    817:     MatDestroy(&AdB);
    818:   }
    819:   return(0);
    820: }

    822: PetscErrorCode  MatSchurComplementGetPmat_Basic(Mat S,MatReuse preuse,Mat *Spmat)
    823: {
    824:   Mat A,B,C,D;
    825:   Mat_SchurComplement *schur = (Mat_SchurComplement *)S->data;
    826:   PetscErrorCode      ierr;

    829:   if (preuse == MAT_IGNORE_MATRIX) return(0);
    830:   MatSchurComplementGetSubMatrices(S,&A,NULL,&B,&C,&D);
    831:   if (!A) SETERRQ(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONGSTATE,"Schur complement component matrices unset");
    832:   MatCreateSchurComplementPmat(A,B,C,D,schur->ainvtype,preuse,Spmat);
    833:   return(0);
    834: }

    836: /*@
    837:     MatSchurComplementGetPmat - Obtain a preconditioning matrix for the Schur complement by assembling Sp = A11 - A10 inv(diag(A00)) A01

    839:     Collective on S

    841:     Input Parameters:
    842: +   S      - matrix obtained with MatCreateSchurComplement() (or equivalent) and implementing the action of A11 - A10 ksp(A00,Ap00) A01
    843: -   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

    845:     Output Parameter:
    846: -   Sp     - approximate Schur complement suitable for preconditioning S = A11 - A10 inv(diag(A00)) A01

    848:     Note:
    849:     Since the real Schur complement is usually dense, providing a good approximation to newpmat usually requires
    850:     application-specific information.  The default for assembled matrices is to use the inverse of the diagonal of
    851:     the (0,0) block A00 in place of A00^{-1}. This rarely produce a scalable algorithm. Optionally, A00 can be lumped
    852:     before forming inv(diag(A00)).

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

    859:     Developer Notes:
    860:     The API that includes MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementGetPmat() should be refactored to
    861:     remove redundancy and be clearer and simplier.

    863:     Level: advanced

    865: .seealso: MatCreateSubMatrix(), PCFIELDSPLIT, MatGetSchurComplement(), MatCreateSchurComplement(), MatSchurComplementSetAinvType()
    866: @*/
    867: PetscErrorCode  MatSchurComplementGetPmat(Mat S,MatReuse preuse,Mat *Sp)
    868: {
    869:   PetscErrorCode ierr,(*f)(Mat,MatReuse,Mat*);

    877:   if (S->factortype) SETERRQ(PetscObjectComm((PetscObject)S),PETSC_ERR_ARG_WRONGSTATE,"Not for factored matrix");

    879:   PetscObjectQueryFunction((PetscObject)S,"MatSchurComplementGetPmat_C",&f);
    880:   if (f) {
    881:     (*f)(S,preuse,Sp);
    882:   } else {
    883:     MatSchurComplementGetPmat_Basic(S,preuse,Sp);
    884:   }
    885:   return(0);
    886: }

    888: PETSC_EXTERN PetscErrorCode MatCreate_SchurComplement(Mat N)
    889: {
    890:   PetscErrorCode      ierr;
    891:   Mat_SchurComplement *Na;

    894:   PetscNewLog(N,&Na);
    895:   N->data = (void*) Na;

    897:   N->ops->destroy        = MatDestroy_SchurComplement;
    898:   N->ops->getvecs        = MatCreateVecs_SchurComplement;
    899:   N->ops->view           = MatView_SchurComplement;
    900:   N->ops->mult           = MatMult_SchurComplement;
    901:   N->ops->multtranspose  = MatMultTranspose_SchurComplement;
    902:   N->ops->multadd        = MatMultAdd_SchurComplement;
    903:   N->ops->setfromoptions = MatSetFromOptions_SchurComplement;
    904:   N->assembled           = PETSC_FALSE;
    905:   N->preallocated        = PETSC_FALSE;

    907:   KSPCreate(PetscObjectComm((PetscObject)N),&Na->ksp);
    908:   PetscObjectChangeTypeName((PetscObject)N,MATSCHURCOMPLEMENT);
    909:   return(0);
    910: }