Actual source code: tomographyADMM.c

      1: #include <petsctao.h>
      2: /*
      3: Description:   ADMM tomography reconstruction example .
      4:                0.5*||Ax-b||^2 + lambda*g(x)
      5: Reference:     BRGN Tomography Example
      6: */

      8: static char help[] = "Finds the ADMM solution to the under constraint linear model Ax = b, with regularizer. \n\
      9:                       A is a M*N real matrix (M<N), x is sparse. A good regularizer is an L1 regularizer. \n\
     10:                       We first split the operator into 0.5*||Ax-b||^2, f(x), and lambda*||x||_1, g(z), where lambda is user specified weight. \n\
     11:                       g(z) could be either ||z||_1, or ||z||_2^2. Default closed form solution for NORM1 would be soft-threshold, which is \n\
     12:                       natively supported in admm.c with -tao_admm_regularizer_type soft-threshold. Or user can use regular TAO solver for  \n\
     13:                       either NORM1 or NORM2 or TAOSHELL, with -reg {1,2,3} \n\
     14:                       Then, we augment both f and g, and solve it via ADMM. \n\
     15:                       D is the M*N transform matrix so that D*x is sparse. \n";

     17: typedef struct {
     18:   PetscInt  M,N,K,reg;
     19:   PetscReal lambda,eps,mumin;
     20:   Mat       A,ATA,H,Hx,D,Hz,DTD,HF;
     21:   Vec       c,xlb,xub,x,b,workM,workN,workN2,workN3,xGT;    /* observation b, ground truth xGT, the lower bound and upper bound of x*/
     22: } AppCtx;

     24: /*------------------------------------------------------------*/

     26: PetscErrorCode NullJacobian(Tao tao,Vec X,Mat J,Mat Jpre,void *ptr)
     27: {
     29:   return(0);
     30: }

     32: /*------------------------------------------------------------*/

     34: static PetscErrorCode TaoShellSolve_SoftThreshold(Tao tao)
     35: {
     37:   PetscReal      lambda, mu;
     38:   AppCtx         *user;
     39:   Vec            out,work,y,x;
     40:   Tao            admm_tao,misfit;

     43:   user = NULL;
     44:   mu   = 0;
     45:   TaoGetADMMParentTao(tao,&admm_tao);
     46:   TaoADMMGetMisfitSubsolver(admm_tao, &misfit);
     47:   TaoADMMGetSpectralPenalty(admm_tao,&mu);
     48:   TaoShellGetContext(tao, (void**) &user);

     50:   lambda = user->lambda;
     51:   work   = user->workN;
     52:   TaoGetSolutionVector(tao, &out);
     53:   TaoGetSolutionVector(misfit, &x);
     54:   TaoADMMGetDualVector(admm_tao, &y);

     56:   /* Dx + y/mu */
     57:   MatMult(user->D,x,work);
     58:   VecAXPY(work,1/mu,y);

     60:   /* soft thresholding */
     61:   TaoSoftThreshold(work, -lambda/mu, lambda/mu, out);
     62:   return(0);
     63: }

     65: /*------------------------------------------------------------*/

     67: PetscErrorCode MisfitObjectiveAndGradient(Tao tao,Vec X,PetscReal *f,Vec g,void *ptr)
     68: {
     69:   AppCtx         *user = (AppCtx*)ptr;

     73:   /* Objective  0.5*||Ax-b||_2^2 */
     74:   MatMult(user->A,X,user->workM);
     75:   VecAXPY(user->workM,-1,user->b);
     76:   VecDot(user->workM,user->workM,f);
     77:   *f  *= 0.5;
     78:   /* Gradient. ATAx-ATb */
     79:   MatMult(user->ATA,X,user->workN);
     80:   MatMultTranspose(user->A,user->b,user->workN2);
     81:   VecWAXPY(g,-1.,user->workN2,user->workN);
     82:   return(0);
     83: }

     85: /*------------------------------------------------------------*/

     87: PetscErrorCode RegularizerObjectiveAndGradient1(Tao tao,Vec X,PetscReal *f_reg,Vec G_reg,void *ptr)
     88: {
     89:   AppCtx         *user = (AppCtx*)ptr;

     93:   /* compute regularizer objective
     94:    * f = f + lambda*sum(sqrt(y.^2+epsilon^2) - epsilon), where y = D*x */
     95:   VecCopy(X,user->workN2);
     96:   VecPow(user->workN2,2.);
     97:   VecShift(user->workN2,user->eps*user->eps);
     98:   VecSqrtAbs(user->workN2);
     99:   VecCopy(user->workN2, user->workN3);
    100:   VecShift(user->workN2,-user->eps);
    101:   VecSum(user->workN2,f_reg);
    102:   *f_reg *= user->lambda;
    103:   /* compute regularizer gradient = lambda*x */
    104:   VecPointwiseDivide(G_reg,X,user->workN3);
    105:   VecScale(G_reg,user->lambda);
    106:   return(0);
    107: }

    109: /*------------------------------------------------------------*/

    111: PetscErrorCode RegularizerObjectiveAndGradient2(Tao tao,Vec X,PetscReal *f_reg,Vec G_reg,void *ptr)
    112: {
    113:   AppCtx         *user = (AppCtx*)ptr;
    115:   PetscReal      temp;

    118:   /* compute regularizer objective = lambda*|z|_2^2 */
    119:   VecDot(X,X,&temp);
    120:   *f_reg = 0.5*user->lambda*temp;
    121:   /* compute regularizer gradient = lambda*z */
    122:   VecCopy(X,G_reg);
    123:   VecScale(G_reg,user->lambda);
    124:   return(0);
    125: }

    127: /*------------------------------------------------------------*/

    129: static PetscErrorCode HessianMisfit(Tao tao, Vec x, Mat H, Mat Hpre, void *ptr)
    130: {
    132:   return(0);
    133: }

    135: /*------------------------------------------------------------*/

    137: static PetscErrorCode HessianReg(Tao tao, Vec x, Mat H, Mat Hpre, void *ptr)
    138: {
    139:   AppCtx         *user = (AppCtx*)ptr;

    143:   MatMult(user->D,x,user->workN);
    144:   VecPow(user->workN2,2.);
    145:   VecShift(user->workN2,user->eps*user->eps);
    146:   VecSqrtAbs(user->workN2);
    147:   VecShift(user->workN2,-user->eps);
    148:   VecReciprocal(user->workN2);
    149:   VecScale(user->workN2,user->eps*user->eps);
    150:   MatDiagonalSet(H,user->workN2,INSERT_VALUES);
    151:   return(0);
    152: }

    154: /*------------------------------------------------------------*/

    156: PetscErrorCode FullObjGrad(Tao tao,Vec X,PetscReal *f,Vec g,void *ptr)
    157: {
    158:   AppCtx         *user = (AppCtx*)ptr;
    160:   PetscReal      f_reg;

    163:   /* Objective  0.5*||Ax-b||_2^2 + lambda*||x||_2^2*/
    164:   MatMult(user->A,X,user->workM);
    165:   VecAXPY(user->workM,-1,user->b);
    166:   VecDot(user->workM,user->workM,f);
    167:   VecNorm(X,NORM_2,&f_reg);
    168:   *f  *= 0.5;
    169:   *f  += user->lambda*f_reg*f_reg;
    170:   /* Gradient. ATAx-ATb + 2*lambda*x */
    171:   MatMult(user->ATA,X,user->workN);
    172:   MatMultTranspose(user->A,user->b,user->workN2);
    173:   VecWAXPY(g,-1.,user->workN2,user->workN);
    174:   VecAXPY(g,2*user->lambda,X);
    175:   return(0);
    176: }
    177: /*------------------------------------------------------------*/

    179: static PetscErrorCode HessianFull(Tao tao, Vec x, Mat H, Mat Hpre, void *ptr)
    180: {
    182:   return(0);
    183: }
    184: /*------------------------------------------------------------*/

    187: PetscErrorCode InitializeUserData(AppCtx *user)
    188: {
    189:   char           dataFile[] = "tomographyData_A_b_xGT";   /* Matrix A and vectors b, xGT(ground truth) binary files generated by Matlab. Debug: change from "tomographyData_A_b_xGT" to "cs1Data_A_b_xGT". */
    190:   PetscViewer    fd;   /* used to load data from file */
    192:   PetscInt       k,n;
    193:   PetscScalar    v;

    196:   /* Load the A matrix, b vector, and xGT vector from a binary file. */
    197:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,dataFile,FILE_MODE_READ,&fd);
    198:   MatCreate(PETSC_COMM_WORLD,&user->A);
    199:   MatSetType(user->A,MATAIJ);
    200:   MatLoad(user->A,fd);
    201:   VecCreate(PETSC_COMM_WORLD,&user->b);
    202:   VecLoad(user->b,fd);
    203:   VecCreate(PETSC_COMM_WORLD,&user->xGT);
    204:   VecLoad(user->xGT,fd);
    205:   PetscViewerDestroy(&fd);

    207:   MatGetSize(user->A,&user->M,&user->N);

    209:   MatCreate(PETSC_COMM_WORLD,&user->D);
    210:   MatSetSizes(user->D,PETSC_DECIDE,PETSC_DECIDE,user->N,user->N);
    211:   MatSetFromOptions(user->D);
    212:   MatSetUp(user->D);
    213:   for (k=0; k<user->N; k++) {
    214:     v = 1.0;
    215:     n = k+1;
    216:     if (k< user->N -1) {
    217:       MatSetValues(user->D,1,&k,1,&n,&v,INSERT_VALUES);
    218:     }
    219:     v    = -1.0;
    220:     MatSetValues(user->D,1,&k,1,&k,&v,INSERT_VALUES);
    221:   }
    222:   MatAssemblyBegin(user->D,MAT_FINAL_ASSEMBLY);
    223:   MatAssemblyEnd(user->D,MAT_FINAL_ASSEMBLY);

    225:   MatTransposeMatMult(user->D,user->D,MAT_INITIAL_MATRIX,PETSC_DEFAULT,&user->DTD);

    227:   MatCreate(PETSC_COMM_WORLD,&user->Hz);
    228:   MatSetSizes(user->Hz,PETSC_DECIDE,PETSC_DECIDE,user->N,user->N);
    229:   MatSetFromOptions(user->Hz);
    230:   MatSetUp(user->Hz);
    231:   MatAssemblyBegin(user->Hz,MAT_FINAL_ASSEMBLY);
    232:   MatAssemblyEnd(user->Hz,MAT_FINAL_ASSEMBLY);

    234:   VecCreate(PETSC_COMM_WORLD,&(user->x));
    235:   VecCreate(PETSC_COMM_WORLD,&(user->workM));
    236:   VecCreate(PETSC_COMM_WORLD,&(user->workN));
    237:   VecCreate(PETSC_COMM_WORLD,&(user->workN2));
    238:   VecSetSizes(user->x,PETSC_DECIDE,user->N);
    239:   VecSetSizes(user->workM,PETSC_DECIDE,user->M);
    240:   VecSetSizes(user->workN,PETSC_DECIDE,user->N);
    241:   VecSetSizes(user->workN2,PETSC_DECIDE,user->N);
    242:   VecSetFromOptions(user->x);
    243:   VecSetFromOptions(user->workM);
    244:   VecSetFromOptions(user->workN);
    245:   VecSetFromOptions(user->workN2);

    247:   VecDuplicate(user->workN,&(user->workN3));
    248:   VecDuplicate(user->x,&(user->xlb));
    249:   VecDuplicate(user->x,&(user->xub));
    250:   VecDuplicate(user->x,&(user->c));
    251:   VecSet(user->xlb,0.0);
    252:   VecSet(user->c,0.0);
    253:   VecSet(user->xub,PETSC_INFINITY);

    255:   MatTransposeMatMult(user->A,user->A, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &(user->ATA));
    256:   MatTransposeMatMult(user->A,user->A, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &(user->Hx));
    257:   MatTransposeMatMult(user->A,user->A, MAT_INITIAL_MATRIX, PETSC_DEFAULT, &(user->HF));

    259:   MatAssemblyBegin(user->ATA,MAT_FINAL_ASSEMBLY);
    260:   MatAssemblyEnd(user->ATA,MAT_FINAL_ASSEMBLY);
    261:   MatAssemblyBegin(user->Hx,MAT_FINAL_ASSEMBLY);
    262:   MatAssemblyEnd(user->Hx,MAT_FINAL_ASSEMBLY);
    263:   MatAssemblyBegin(user->HF,MAT_FINAL_ASSEMBLY);
    264:   MatAssemblyEnd(user->HF,MAT_FINAL_ASSEMBLY);

    266:   user->lambda = 1.e-8;
    267:   user->eps    = 1.e-3;
    268:   user->reg    = 2;
    269:   user->mumin  = 5.e-6;

    271:   PetscOptionsBegin(PETSC_COMM_WORLD, NULL, "Configure separable objection example", "tomographyADMM.c");
    272:   PetscOptionsInt("-reg","Regularization scheme for z solver (1,2)", "tomographyADMM.c", user->reg, &(user->reg), NULL);
    273:   PetscOptionsReal("-lambda", "The regularization multiplier. 1 default", "tomographyADMM.c", user->lambda, &(user->lambda), NULL);
    274:   PetscOptionsReal("-eps", "L1 norm epsilon padding", "tomographyADMM.c", user->eps, &(user->eps), NULL);
    275:   PetscOptionsReal("-mumin", "Minimum value for ADMM spectral penalty", "tomographyADMM.c", user->mumin, &(user->mumin), NULL);
    276:   PetscOptionsEnd();
    277:   return(0);
    278: }

    280: /*------------------------------------------------------------*/

    282: PetscErrorCode DestroyContext(AppCtx *user)
    283: {

    287:   MatDestroy(&user->A);
    288:   MatDestroy(&user->ATA);
    289:   MatDestroy(&user->Hx);
    290:   MatDestroy(&user->Hz);
    291:   MatDestroy(&user->HF);
    292:   MatDestroy(&user->D);
    293:   MatDestroy(&user->DTD);
    294:   VecDestroy(&user->xGT);
    295:   VecDestroy(&user->xlb);
    296:   VecDestroy(&user->xub);
    297:   VecDestroy(&user->b);
    298:   VecDestroy(&user->x);
    299:   VecDestroy(&user->c);
    300:   VecDestroy(&user->workN3);
    301:   VecDestroy(&user->workN2);
    302:   VecDestroy(&user->workN);
    303:   VecDestroy(&user->workM);
    304:   return(0);
    305: }

    307: /*------------------------------------------------------------*/

    309: int main(int argc,char **argv)
    310: {
    312:   Tao            tao,misfit,reg;
    313:   PetscReal      v1,v2;
    314:   AppCtx*        user;
    315:   PetscViewer    fd;
    316:   char           resultFile[] = "tomographyResult_x";

    318:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
    319:   PetscNew(&user);
    320:   InitializeUserData(user);

    322:   TaoCreate(PETSC_COMM_WORLD, &tao);
    323:   TaoSetType(tao, TAOADMM);
    324:   TaoSetInitialVector(tao, user->x);
    325:   /* f(x) + g(x) for parent tao */
    326:   TaoADMMSetSpectralPenalty(tao,1.);
    327:   TaoSetObjectiveAndGradientRoutine(tao, FullObjGrad, (void*)user);
    328:   MatShift(user->HF,user->lambda);
    329:   TaoSetHessianRoutine(tao, user->HF, user->HF, HessianFull, (void*)user);

    331:   /* f(x) for misfit tao */
    332:   TaoADMMSetMisfitObjectiveAndGradientRoutine(tao, MisfitObjectiveAndGradient, (void*)user);
    333:   TaoADMMSetMisfitHessianRoutine(tao, user->Hx, user->Hx, HessianMisfit, (void*)user);
    334:   TaoADMMSetMisfitHessianChangeStatus(tao,PETSC_FALSE);
    335:   TaoADMMSetMisfitConstraintJacobian(tao,user->D,user->D,NullJacobian,(void*)user);

    337:   /* g(x) for regularizer tao */
    338:   if (user->reg == 1) {
    339:     TaoADMMSetRegularizerObjectiveAndGradientRoutine(tao, RegularizerObjectiveAndGradient1, (void*)user);
    340:     TaoADMMSetRegularizerHessianRoutine(tao, user->Hz, user->Hz, HessianReg, (void*)user);
    341:     TaoADMMSetRegHessianChangeStatus(tao,PETSC_TRUE);
    342:   } else if (user->reg == 2) {
    343:     TaoADMMSetRegularizerObjectiveAndGradientRoutine(tao, RegularizerObjectiveAndGradient2, (void*)user);
    344:     MatShift(user->Hz,1);
    345:     MatScale(user->Hz,user->lambda);
    346:     TaoADMMSetRegularizerHessianRoutine(tao, user->Hz, user->Hz, HessianMisfit, (void*)user);
    347:     TaoADMMSetRegHessianChangeStatus(tao,PETSC_TRUE);
    348:   } else if (user->reg != 3) SETERRQ(PETSC_COMM_WORLD, 1, "Incorrect Reg type"); /* TaoShell case */

    350:   /* Set type for the misfit solver */
    351:   TaoADMMGetMisfitSubsolver(tao, &misfit);
    352:   TaoADMMGetRegularizationSubsolver(tao, &reg);
    353:   TaoSetType(misfit,TAONLS);
    354:   if (user->reg == 3) {
    355:     TaoSetType(reg,TAOSHELL);
    356:     TaoShellSetContext(reg, (void*) user);
    357:     TaoShellSetSolve(reg, TaoShellSolve_SoftThreshold);
    358:   } else {
    359:     TaoSetType(reg,TAONLS);
    360:   }
    361:   TaoSetVariableBounds(misfit,user->xlb,user->xub);

    363:   /* Soft Thresholding solves the ADMM problem with the L1 regularizer lambda*||z||_1 and the x-z=0 constraint */
    364:   TaoADMMSetRegularizerCoefficient(tao, user->lambda);
    365:   TaoADMMSetRegularizerConstraintJacobian(tao,NULL,NULL,NullJacobian,(void*)user);
    366:   TaoADMMSetMinimumSpectralPenalty(tao,user->mumin);

    368:   TaoADMMSetConstraintVectorRHS(tao,user->c);
    369:   TaoSetFromOptions(tao);
    370:   TaoSolve(tao);

    372:   /* Save x (reconstruction of object) vector to a binary file, which maybe read from Matlab and convert to a 2D image for comparison. */
    373:   PetscViewerBinaryOpen(PETSC_COMM_WORLD,resultFile,FILE_MODE_WRITE,&fd);
    374:   VecView(user->x,fd);
    375:   PetscViewerDestroy(&fd);

    377:   /* compute the error */
    378:   VecAXPY(user->x,-1,user->xGT);
    379:   VecNorm(user->x,NORM_2,&v1);
    380:   VecNorm(user->xGT,NORM_2,&v2);
    381:   PetscPrintf(PETSC_COMM_WORLD, "relative reconstruction error: ||x-xGT||/||xGT|| = %6.4e.\n", (double)(v1/v2));

    383:   /* Free TAO data structures */
    384:   TaoDestroy(&tao);
    385:   DestroyContext(user);
    386:   PetscFree(user);
    387:   PetscFinalize();
    388:   return ierr;
    389: }

    391: /*TEST

    393:    build:
    394:       requires: !complex !single !__float128 !define(PETSC_USE_64BIT_INDICES)

    396:    test:
    397:       suffix: 1
    398:       localrunfiles: tomographyData_A_b_xGT
    399:       args:  -lambda 1.e-8 -tao_monitor -tao_type nls -tao_nls_pc_type icc

    401:    test:
    402:       suffix: 2
    403:       localrunfiles: tomographyData_A_b_xGT
    404:       args:  -reg 2 -lambda 1.e-8 -tao_admm_dual_update update_basic -tao_admm_regularizer_type regularizer_user -tao_max_it 20 -tao_monitor -tao_admm_tolerance_update_factor 1.e-8  -misfit_tao_nls_pc_type icc -misfit_tao_monitor -reg_tao_monitor

    406:    test:
    407:       suffix: 3
    408:       localrunfiles: tomographyData_A_b_xGT
    409:       args:  -lambda 1.e-8 -tao_admm_dual_update update_basic -tao_admm_regularizer_type regularizer_soft_thresh -tao_max_it 20 -tao_monitor -tao_admm_tolerance_update_factor 1.e-8 -misfit_tao_nls_pc_type icc -misfit_tao_monitor

    411:    test:
    412:       suffix: 4
    413:       localrunfiles: tomographyData_A_b_xGT
    414:       args:  -lambda 1.e-8 -tao_admm_dual_update update_adaptive -tao_admm_regularizer_type regularizer_soft_thresh -tao_max_it 20 -tao_monitor -misfit_tao_monitor -misfit_tao_nls_pc_type icc

    416:    test:
    417:       suffix: 5
    418:       localrunfiles: tomographyData_A_b_xGT
    419:       args:  -reg 2 -lambda 1.e-8 -tao_admm_dual_update update_adaptive -tao_admm_regularizer_type regularizer_user -tao_max_it 20 -tao_monitor -tao_admm_tolerance_update_factor 1.e-8 -misfit_tao_monitor -reg_tao_monitor -misfit_tao_nls_pc_type icc

    421:    test:
    422:       suffix: 6
    423:       localrunfiles: tomographyData_A_b_xGT
    424:       args:  -reg 3 -lambda 1.e-8 -tao_admm_dual_update update_adaptive -tao_admm_regularizer_type regularizer_user -tao_max_it 20 -tao_monitor -tao_admm_tolerance_update_factor 1.e-8 -misfit_tao_monitor -reg_tao_monitor -misfit_tao_nls_pc_type icc

    426: TEST*/