xref: /petsc/src/tao/leastsquares/impls/brgn/brgn.c (revision 20fe612c600813636c9f603c3d964832e0d810cf)

#include <../src/tao/leastsquares/impls/brgn/brgn.h>

/* Old code
static PetscErrorCode GNHessianProd(Mat H, Vec in, Vec out)
{
  TAO_BRGN              *gn;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = MatShellGetContext(H, &gn);CHKERRQ(ierr);
  ierr = MatMult(gn->subsolver->ls_jac, in, gn->r_work);CHKERRQ(ierr);
  ierr = MatMultTranspose(gn->subsolver->ls_jac, gn->r_work, out);CHKERRQ(ierr);
  ierr = VecAXPY(out, gn->lambda, in);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode GNObjectiveGradientEval(Tao tao, Vec X, PetscReal *fcn, Vec G, void *ptr)
{
  TAO_BRGN              *gn = (TAO_BRGN *)ptr;
  PetscScalar           xnorm2;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = TaoComputeResidual(tao, X, tao->ls_res);CHKERRQ(ierr);
  ierr = VecDotBegin(tao->ls_res, tao->ls_res, fcn);CHKERRQ(ierr);
  ierr = VecAXPBYPCZ(gn->x_work, 1.0, -1.0, 0.0, X, gn->x_old);CHKERRQ(ierr);
  ierr = VecDotBegin(gn->x_work, gn->x_work, &xnorm2);CHKERRQ(ierr);
  ierr = VecDotEnd(tao->ls_res, tao->ls_res, fcn);CHKERRQ(ierr);
  ierr = VecDotEnd(gn->x_work, gn->x_work, &xnorm2);CHKERRQ(ierr);
  *fcn = 0.5*(*fcn) + 0.5*gn->lambda*xnorm2;

  ierr = TaoComputeResidualJacobian(tao, X, tao->ls_jac, tao->ls_jac_pre);CHKERRQ(ierr);
  ierr = MatMultTranspose(tao->ls_jac, tao->ls_res, G);CHKERRQ(ierr);
  ierr = VecAXPBYPCZ(G, gn->lambda, -gn->lambda, 1.0, X, gn->x_old);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}
*/

static PetscErrorCode GNHessianProd(Mat H, Vec in, Vec out)
{
  TAO_BRGN              *gn;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = MatShellGetContext(H, &gn);CHKERRQ(ierr);
  ierr = MatMult(gn->subsolver->ls_jac, in, gn->r_work);CHKERRQ(ierr);
  ierr = MatMultTranspose(gn->subsolver->ls_jac, gn->r_work, out);CHKERRQ(ierr);
  /* out = out + lambda*D'*(diag.*(D*in)) */
  ierr = MatMult(gn->D, in, gn->y);CHKERRQ(ierr);  /* y = D*in */
  ierr = VecPointwiseMult(gn->y_work, gn->diag, gn->y);CHKERRQ(ierr);     /* y_work = diag.*(D*in), where diag = epsilon^2 ./ sqrt(x.^2+epsilon^2).^3 */
  ierr = MatMultTranspose(gn->D, gn->y_work, gn->x_work);CHKERRQ(ierr);   /* x_work = D'*(diag.*(D*in)) */
  ierr = VecAXPY(out, gn->lambda, gn->x_work);CHKERRQ(ierr);

  PetscFunctionReturn(0);
}

static PetscErrorCode GNObjectiveGradientEval(Tao tao, Vec X, PetscReal *fcn, Vec G, void *ptr)
{
  TAO_BRGN              *gn = (TAO_BRGN *)ptr;
  PetscInt              Ny;                    /* dimension of D*X */
  PetscScalar           yESum;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  /* compute objective */
  /* compute first term ||ls_res||^2 */
  ierr = TaoComputeResidual(tao, X, tao->ls_res);CHKERRQ(ierr);
  ierr = VecDotBegin(tao->ls_res, tao->ls_res, fcn);CHKERRQ(ierr);
  ierr = VecDotEnd(tao->ls_res, tao->ls_res, fcn);CHKERRQ(ierr);
  /* add the second term lambda*sum(sqrt(y.^2+epsilon^2) - epsilon), where y = D*x*/
  ierr = MatMult(gn->D, X, gn->y);CHKERRQ(ierr);  /* y = D*x */
  ierr = VecPointwiseMult(gn->y_work, gn->y, gn->y);CHKERRQ(ierr);
  ierr = VecShift(gn->y_work, gn->epsilon*gn->epsilon);CHKERRQ(ierr);
  ierr = VecSqrtAbs(gn->y_work);CHKERRQ(ierr);      /* gn->y_work = sqrt(y.^2+epsilon^2) */
  ierr = VecSum(gn->y_work, &yESum);CHKERRQ(ierr);CHKERRQ(ierr);
  ierr = VecGetSize(gn->y, &Ny);CHKERRQ(ierr);
  *fcn = 0.5*(*fcn) + gn->lambda*(yESum - Ny*gn->epsilon);

  /* compute gradient G */
  ierr = TaoComputeResidualJacobian(tao, X, tao->ls_jac, tao->ls_jac_pre);CHKERRQ(ierr);
  ierr = MatMultTranspose(tao->ls_jac, tao->ls_res, G);CHKERRQ(ierr);
  /* compute G = G + lambda*D'*(y./sqrt(y.^2+epsilon^2)), where y = D*x */
  ierr = VecPointwiseDivide(gn->y_work, gn->y, gn->y_work);CHKERRQ(ierr); /* reuse y_work = y./sqrt(y.^2+epsilon^2) */
  ierr = MatMultTranspose(gn->D, gn->y_work, gn->x_work);CHKERRQ(ierr);
  ierr = VecAXPY(G, gn->lambda, gn->x_work);CHKERRQ(ierr);

  PetscFunctionReturn(0);
}


static PetscErrorCode GNComputeHessian(Tao tao, Vec X, Mat H, Mat Hpre, void *ptr)
{
  TAO_BRGN              *gn = (TAO_BRGN *)ptr;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = TaoComputeResidualJacobian(tao, X, tao->ls_jac, tao->ls_jac_pre);CHKERRQ(ierr);

  /* calculate and store diagonal matrix as a vector: diag = epsilon^2 ./ sqrt(x.^2+epsilon^2).^3* --> diag = epsilon^2 ./ sqrt(y.^2+epsilon^2).^3, where y = D*x */
  ierr = MatMult(gn->D, X, gn->y);CHKERRQ(ierr);  /* y = D*x */
  ierr = VecPointwiseMult(gn->y_work, gn->y, gn->y);CHKERRQ(ierr);
  ierr = VecShift(gn->y_work, gn->epsilon*gn->epsilon);CHKERRQ(ierr);
  ierr = VecCopy(gn->y_work, gn->diag);CHKERRQ(ierr);                     /* gn->diag = y.^2+epsilon^2 */
  ierr = VecSqrtAbs(gn->y_work);CHKERRQ(ierr);                            /* gn->y_work = sqrt(y.^2+epsilon^2) */
  ierr = VecPointwiseMult(gn->diag, gn->y_work, gn->diag);CHKERRQ(ierr);  /* gn->diag = sqrt(y.^2+epsilon^2).^3 */
  ierr = VecReciprocal(gn->diag);CHKERRQ(ierr);
  ierr = VecScale(gn->diag, gn->epsilon*gn->epsilon);CHKERRQ(ierr);

  PetscFunctionReturn(0);
}

static PetscErrorCode GNHookFunction(Tao tao, PetscInt iter)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->user_update;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  /* Update basic tao information from the subsolver */
  gn->parent->nfuncs = tao->nfuncs;
  gn->parent->ngrads = tao->ngrads;
  gn->parent->nfuncgrads = tao->nfuncgrads;
  gn->parent->nhess = tao->nhess;
  gn->parent->niter = tao->niter;
  gn->parent->ksp_its = tao->ksp_its;
  gn->parent->ksp_tot_its = tao->ksp_tot_its;
  ierr = TaoGetConvergedReason(tao, &gn->parent->reason);CHKERRQ(ierr);
  /* Update the solution vectors */
  if (iter == 0) {
    ierr = VecSet(gn->x_old, 0.0);CHKERRQ(ierr);
  } else {
    ierr = VecCopy(tao->solution, gn->x_old);CHKERRQ(ierr);
    ierr = VecCopy(tao->solution, gn->parent->solution);CHKERRQ(ierr);
  }
  /* Update the gradient */
  ierr = VecCopy(tao->gradient, gn->parent->gradient);CHKERRQ(ierr);
  /* Call general purpose update function */
  if (gn->parent->ops->update) {
    ierr = (*gn->parent->ops->update)(gn->parent, gn->parent->niter);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TaoSolve_BRGN(Tao tao)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->data;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = TaoSolve(gn->subsolver);CHKERRQ(ierr);
  /* Update basic tao information from the subsolver */
  tao->nfuncs = gn->subsolver->nfuncs;
  tao->ngrads = gn->subsolver->ngrads;
  tao->nfuncgrads = gn->subsolver->nfuncgrads;
  tao->nhess = gn->subsolver->nhess;
  tao->niter = gn->subsolver->niter;
  tao->ksp_its = gn->subsolver->ksp_its;
  tao->ksp_tot_its = gn->subsolver->ksp_tot_its;
  ierr = TaoGetConvergedReason(gn->subsolver, &tao->reason);CHKERRQ(ierr);
  /* Update vectors */
  ierr = VecCopy(gn->subsolver->solution, tao->solution);CHKERRQ(ierr);
  ierr = VecCopy(gn->subsolver->gradient, tao->gradient);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TaoSetFromOptions_BRGN(PetscOptionItems *PetscOptionsObject,Tao tao)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->data;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  /* old Tikhonov regularization code
  ierr = PetscOptionsHead(PetscOptionsObject,"Gauss-Newton method for least-squares problems using Tikhonov regularization");CHKERRQ(ierr);
  ierr = PetscOptionsReal("-tao_brgn_lambda", "Tikhonov regularization factor", "", gn->lambda, &gn->lambda, NULL);CHKERRQ(ierr);
  */
  ierr = PetscOptionsHead(PetscOptionsObject,"least-squares problems with L1 regularizer: ||f(x)||^2 + lambda*||x||_1. Currently L1-norm is approximated with smooth form");CHKERRQ(ierr);
  ierr = PetscOptionsReal("-tao_brgn_lambda", "L1-norm regularizer weight", "", gn->lambda, &gn->lambda, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-tao_brgn_epsilon", "L1-norm smooth approximation parameter: ||x||_1 = sum(sqrt(x.^2+epsilon^2)-epsilon)", "", gn->epsilon, &gn->epsilon, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  ierr = TaoSetFromOptions(gn->subsolver);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TaoView_BRGN(Tao tao, PetscViewer viewer)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->data;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = PetscViewerASCIIPushTab(viewer);CHKERRQ(ierr);
  ierr = TaoView(gn->subsolver, viewer);CHKERRQ(ierr);
  ierr = PetscViewerASCIIPopTab(viewer);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode TaoSetUp_BRGN(Tao tao)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->data;
  PetscErrorCode        ierr;
  PetscBool             is_bnls, is_bntr, is_bntl;
  PetscInt              i, nx, Nx, Ny; /* XH: added Ny  for size of y=D*x*/
  PetscScalar           v = 1.0; /* XH: hack to set value of matrix */

  PetscFunctionBegin;
  if (!tao->ls_res) SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ORDER, "TaoSetResidualRoutine() must be called before setup!");
  ierr = PetscObjectTypeCompare((PetscObject)gn->subsolver, TAOBNLS, &is_bnls);CHKERRQ(ierr);
  ierr = PetscObjectTypeCompare((PetscObject)gn->subsolver, TAOBNTR, &is_bntr);CHKERRQ(ierr);
  ierr = PetscObjectTypeCompare((PetscObject)gn->subsolver, TAOBNTL, &is_bntl);CHKERRQ(ierr);
  if ((is_bnls || is_bntr || is_bntl) && !tao->ls_jac) SETERRQ(PetscObjectComm((PetscObject)tao), PETSC_ERR_ORDER, "TaoSetResidualJacobianRoutine() must be called before setup!");
  if (!tao->gradient){
    ierr = VecDuplicate(tao->solution, &tao->gradient);CHKERRQ(ierr);
  }
  if (!gn->x_work){
    ierr = VecDuplicate(tao->solution, &gn->x_work);CHKERRQ(ierr);
  }
  if (!gn->r_work){
    ierr = VecDuplicate(tao->ls_res, &gn->r_work);CHKERRQ(ierr);
  }
  if (!gn->x_old) {
    ierr = VecDuplicate(tao->solution, &gn->x_old);CHKERRQ(ierr);
    ierr = VecSet(gn->x_old, 0.0);CHKERRQ(ierr);
  }


  /* XH: hack: following works only when D is a squared matrix, to fix dimension of gn->diag/y/y_work, when D is not squared matrix */
  ierr = VecGetSize(tao->solution, &Nx);CHKERRQ(ierr);
  /* Ny = Nx; */ /* for identity matrix */
  Ny = Nx - 1;  /* for gradient matrix */
  if (!gn->y){
    ierr = VecCreate(PETSC_COMM_SELF,&gn->y);CHKERRQ(ierr);
    ierr = VecSetSizes(gn->y,PETSC_DECIDE,Ny);CHKERRQ(ierr);
    ierr = VecSetFromOptions(gn->y);CHKERRQ(ierr);
    ierr = VecSet(gn->y,0.0);CHKERRQ(ierr);

  }
  if (!gn->y_work){
    ierr = VecDuplicate(gn->y, &gn->y_work);CHKERRQ(ierr);
  }
  if (!gn->diag){
    ierr = VecDuplicate(gn->y, &gn->diag);CHKERRQ(ierr);
    ierr = VecSet(gn->diag, 0.0);CHKERRQ(ierr);
  }

  /* XH: hack: set gn->D as identity/gradient  matrix here for text , how to set D matrix from user data?*/
  if (!gn->D){
    ierr = MatCreate(PETSC_COMM_SELF,&gn->D);CHKERRQ(ierr);
    ierr = MatSetSizes(gn->D,PETSC_DECIDE,PETSC_DECIDE,Ny,Nx);CHKERRQ(ierr);
    ierr = MatSetFromOptions(gn->D);CHKERRQ(ierr);
    ierr = MatSetUp(gn->D);CHKERRQ(ierr);
    /* identity matrix */
    /*
    for (i=0; i<Ny; i++) {
        ierr = MatSetValues(gn->D,1,&i,1,&i,&v,INSERT_VALUES);CHKERRQ(ierr);
    }
    */
    /* gradient matrix */
    /* [-1, 1, 0,...; 0, -1, 1, 0, ...] */
    for (i=0; i<Ny; i++) {
        v = 1.0;
        nx = i+1;
        ierr = MatSetValues(gn->D,1,&i,1,&nx,&v,INSERT_VALUES);CHKERRQ(ierr);
        v = -1.0;
        ierr = MatSetValues(gn->D,1,&i,1,&i,&v,INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = MatAssemblyBegin(gn->D,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
    ierr = MatAssemblyEnd(gn->D,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  }

  /* XH: debug: check matrix */
  PetscPrintf(PETSC_COMM_SELF, "-------- Check D matrix. -------- \n");
  ierr = MatView(gn->D,PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);


  if (!tao->setupcalled) {
    /* Hessian setup */
    ierr = VecGetLocalSize(tao->solution, &nx);CHKERRQ(ierr);
    ierr = VecGetSize(tao->solution, &Nx);CHKERRQ(ierr);
    ierr = MatSetSizes(gn->H, nx, nx, Nx, Nx);CHKERRQ(ierr);
    ierr = MatSetType(gn->H, MATSHELL);CHKERRQ(ierr);
    ierr = MatSetUp(gn->H);CHKERRQ(ierr);
    ierr = MatShellSetOperation(gn->H, MATOP_MULT, (void (*)(void))GNHessianProd);CHKERRQ(ierr);
    ierr = MatShellSetContext(gn->H, (void*)gn);CHKERRQ(ierr);
    /* Subsolver setup */
    ierr = TaoSetUpdate(gn->subsolver, GNHookFunction, (void*)gn);CHKERRQ(ierr);
    ierr = TaoSetInitialVector(gn->subsolver, tao->solution);CHKERRQ(ierr);
    if (tao->bounded) {
      ierr = TaoSetVariableBounds(gn->subsolver, tao->XL, tao->XU);CHKERRQ(ierr);
    }
    ierr = TaoSetResidualRoutine(gn->subsolver, tao->ls_res, tao->ops->computeresidual, tao->user_lsresP);CHKERRQ(ierr);
    ierr = TaoSetJacobianResidualRoutine(gn->subsolver, tao->ls_jac, tao->ls_jac, tao->ops->computeresidualjacobian, tao->user_lsjacP);CHKERRQ(ierr);
    ierr = TaoSetObjectiveAndGradientRoutine(gn->subsolver, GNObjectiveGradientEval, (void*)gn);CHKERRQ(ierr);
    ierr = TaoSetHessianRoutine(gn->subsolver, gn->H, gn->H, GNComputeHessian, (void*)gn);CHKERRQ(ierr);
    /* Propagate some options down */
    ierr = TaoSetTolerances(gn->subsolver, tao->gatol, tao->grtol, tao->gttol);CHKERRQ(ierr);
    ierr = TaoSetMaximumIterations(gn->subsolver, tao->max_it);CHKERRQ(ierr);
    ierr = TaoSetMaximumFunctionEvaluations(gn->subsolver, tao->max_funcs);CHKERRQ(ierr);
    for (i=0; i<tao->numbermonitors; ++i) {
      ierr = TaoSetMonitor(gn->subsolver, tao->monitor[i], tao->monitorcontext[i], tao->monitordestroy[i]);CHKERRQ(ierr);
      ierr = PetscObjectReference((PetscObject)(tao->monitorcontext[i]));CHKERRQ(ierr);
    }
    ierr = TaoSetUp(gn->subsolver);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode TaoDestroy_BRGN(Tao tao)
{
  TAO_BRGN              *gn = (TAO_BRGN *)tao->data;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  if (tao->setupcalled) {
    ierr = VecDestroy(&tao->gradient);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->x_work);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->r_work);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->x_old);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->diag);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->y);CHKERRQ(ierr);
    ierr = VecDestroy(&gn->y_work);CHKERRQ(ierr);
  }
  ierr = MatDestroy(&gn->H);CHKERRQ(ierr);
  ierr = MatDestroy(&gn->D);CHKERRQ(ierr);
  ierr = TaoDestroy(&gn->subsolver);CHKERRQ(ierr);
  gn->parent = NULL;
  ierr = PetscFree(tao->data);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*MC
  TAOBRGN - Bounded Regularized Gauss-Newton method for solving nonlinear least-squares
            problems with bound constraints. This algorithm is a thin wrapper around TAOBNTL
            that constructs the Guass-Newton problem with the user-provided least-squares
            residual and Jacobian. The problem is regularized with an L2-norm proximal point
            term.

  Options Database Keys:
  + -tao_bqnk_max_cg_its - maximum number of bounded conjugate-gradient iterations taken in each Newton loop
  . -tao_bqnk_init_type - trust radius initialization method ("constant", "direction", "interpolation")
  . -tao_bqnk_update_type - trust radius update method ("step", "direction", "interpolation")
  - -tao_bqnk_as_type - active-set estimation method ("none", "bertsekas")

  Level: beginner
M*/
PETSC_EXTERN PetscErrorCode TaoCreate_BRGN(Tao tao)
{
  TAO_BRGN       *gn;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscNewLog(tao,&gn);CHKERRQ(ierr);

  tao->ops->destroy = TaoDestroy_BRGN;
  tao->ops->setup = TaoSetUp_BRGN;
  tao->ops->setfromoptions = TaoSetFromOptions_BRGN;
  tao->ops->view = TaoView_BRGN;
  tao->ops->solve = TaoSolve_BRGN;

  tao->data = (void*)gn;
  gn->lambda = 1e-4;
  gn->epsilon = 1e-6;
  gn->parent = tao;

  ierr = MatCreate(PetscObjectComm((PetscObject)tao), &gn->H);CHKERRQ(ierr);
  ierr = MatSetOptionsPrefix(gn->H, "tao_brgn_hessian_");CHKERRQ(ierr);

  ierr = TaoCreate(PetscObjectComm((PetscObject)tao), &gn->subsolver);CHKERRQ(ierr);
  ierr = TaoSetType(gn->subsolver, TAOBNLS);CHKERRQ(ierr);
  ierr = TaoSetOptionsPrefix(gn->subsolver, "tao_brgn_subsolver_");CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  TaoBRGNGetSubsolver - Get the pointer to the subsolver inside BRGN

  Collective on Tao

  Level: developer

  Input Parameters:
+  tao - the Tao solver context
-  subsolver - the Tao sub-solver context
@*/
PetscErrorCode TaoBRGNGetSubsolver(Tao tao, Tao *subsolver)
{
  TAO_BRGN       *gn = (TAO_BRGN *)tao->data;

  PetscFunctionBegin;
  *subsolver = gn->subsolver;
  PetscFunctionReturn(0);
}

/*@C
  TaoBRGNSetTikhonovLambda - Set the Tikhonov regularization factor for the Gauss-Newton least-squares algorithm

  Collective on Tao

  Level: developer

  Input Parameters:
+  tao - the Tao solver context
-  lambda - Tikhonov regularization factor
@*/
PetscErrorCode TaoBRGNSetTikhonovLambda(Tao tao, PetscReal lambda)
{
  TAO_BRGN       *gn = (TAO_BRGN *)tao->data;

  /* Initialize lambda here */

  PetscFunctionBegin;
  gn->lambda = lambda;
  PetscFunctionReturn(0);
}

/*@C
  TaoBRGNSetL1SmoothEpsilon - Set the L1-norm smooth approximation parameter for L1-regularized least-squares algorithm

  Collective on Tao

  Level: developer

  Input Parameters:
+  tao - the Tao solver context
-  epsilon - L1-norm smooth approximation parameter
@*/
PetscErrorCode TaoBRGNSetL1SmoothEpsilon(Tao tao, PetscReal epsilon)
{
  TAO_BRGN       *gn = (TAO_BRGN *)tao->data;

  /* Initialize epsilon here */

  PetscFunctionBegin;
  gn->epsilon = epsilon;
  PetscFunctionReturn(0);
}
/* XH: done TaoBRGNSetL1SmoothEpsilon by copy TaoBRGNSetTikhonovLambda in peststao.h, brgn.c and zbrgnf.c.
 maybe change the name of Tikhonov in TaoBRGNSetTikhonovLambda() etc, as lambda is no longer the Tikhonov regularizer weight but the L1 regularizer weight
 Maybe change D*x to D(x), and  A*x to A(x) as function handle
 Maybe need to also keep y = D*x, to avoid duplicate frequent computation of D*x
 */