impls/qn/qn.c

#include <petsc-private/snesimpl.h> /*I "petscsnes.h" I*/

#define H(i,j)  qn->dXdFmat[i*qn->m + j]

const char *const SNESQNScaleTypes[] =        {"NONE","SHANNO","LINESEARCH","JACOBIAN","SNESQNScaleType","SNES_QN_SCALING_",0};
const char *const SNESQNRestartTypes[] =      {"NONE","POWELL","PERIODIC","SNESQNRestartType","SNES_QN_RESTART_",0};
const char *const SNESQNTypes[] =             {"LBFGS","BROYDEN","BADBROYDEN","SNESQNType","SNES_QN_RESTART_",0};

typedef enum {SNES_QN_LBFGS      = 0,
              SNES_QN_BROYDEN    = 1,
              SNES_QN_BADBROYDEN = 2
             } SNESQNType;

typedef struct {
  Vec                   *U;               /* Stored past states (vary from method to method) */
  Vec                   *V;               /* Stored past states (vary from method to method) */
  PetscInt              m;                /* The number of kept previous steps */
  PetscScalar           *alpha, *beta;
  PetscScalar           *dXtdF, *dFtdX, *YtdX;
  PetscBool             singlereduction;  /* Aggregated reduction implementation */
  PetscScalar           *dXdFmat;         /* A matrix of values for dX_i dot dF_j */
  PetscViewer           monitor;
  PetscReal             powell_gamma;     /* Powell angle restart condition */
  PetscReal             powell_downhill;  /* Powell descent restart condition */
  PetscReal             scaling;          /* scaling of H0 */
  SNESQNType            type;             /* the type of quasi-newton method used */
  SNESQNScaleType       scale_type;       /* the type of scaling used */
  SNESQNRestartType     restart_type;     /* determine the frequency and type of restart conditions */
} SNES_QN;

#undef __FUNCT__
#define __FUNCT__ "SNESQNApply_Broyden"
PetscErrorCode SNESQNApply_Broyden(SNES snes,PetscInt it,Vec Y,Vec X,Vec Xold, Vec D, Vec Dold)
{
  PetscErrorCode     ierr;
  SNES_QN            *qn = (SNES_QN*)snes->data;
  Vec                W   = snes->work[3];
  Vec                *U  = qn->U;
  Vec                *V  = qn->V;
  KSPConvergedReason kspreason;
  MatStructure       flg = DIFFERENT_NONZERO_PATTERN;
  PetscInt           k,i,lits;
  PetscInt           m = qn->m;
  PetscScalar        gdot;
  PetscInt           l = m;
  Mat                jac,jac_pre;

  PetscFunctionBegin;
  if (it < m) l = it;
  if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
    ierr = SNESGetJacobian(snes,&jac,&jac_pre,PETSC_NULL,PETSC_NULL);CHKERRQ(ierr);
    ierr = KSPSetOperators(snes->ksp,jac,jac_pre,flg);CHKERRQ(ierr);
    ierr = SNES_KSPSolve(snes,snes->ksp,D,W);CHKERRQ(ierr);
    ierr = KSPGetConvergedReason(snes->ksp,&kspreason);CHKERRQ(ierr);
    if (kspreason < 0) {
      if (++snes->numLinearSolveFailures >= snes->maxLinearSolveFailures) {
        ierr = PetscInfo2(snes,"iter=%D, number linear solve failures %D greater than current SNES allowed, stopping solve\n",snes->iter,snes->numLinearSolveFailures);CHKERRQ(ierr);
        snes->reason = SNES_DIVERGED_LINEAR_SOLVE;
        PetscFunctionReturn(0);
      }
    }
    ierr              = KSPGetIterationNumber(snes->ksp,&lits);CHKERRQ(ierr);
    snes->linear_its += lits;
    ierr              = VecCopy(W,Y);CHKERRQ(ierr);
  } else {
    ierr = VecCopy(D,Y);CHKERRQ(ierr);
    ierr = VecScale(Y,qn->scaling);CHKERRQ(ierr);
  }

  /* inward recursion starting at the first update and working forward */
  if (it > 1) {
    for (i = 0; i < l-1; i++) {
      k    = (it+i-l)%l;
      ierr = VecDot(U[k],Y,&gdot);CHKERRQ(ierr);
      ierr = VecAXPY(Y,gdot,V[k]);CHKERRQ(ierr);
      if (qn->monitor) {
        ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(qn->monitor, "it: %d k: %d gdot: %14.12e\n", it, k, PetscRealPart(gdot));CHKERRQ(ierr);
        ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      }
    }
  }
  if (it < m) l = it;

  /* set W to be the last step, post-linesearch */
  ierr = VecCopy(Xold,W);CHKERRQ(ierr);
  ierr = VecAXPY(W,-1.0,X);CHKERRQ(ierr);

  if (l > 0) {
    k    = (it-1)%l;
    ierr = VecCopy(W,U[k]);CHKERRQ(ierr);
    ierr = VecAXPY(W,-1.0,Y);CHKERRQ(ierr);
    ierr = VecDot(U[k],W,&gdot);CHKERRQ(ierr);
    ierr = VecCopy(Y,V[k]);CHKERRQ(ierr);
    ierr = VecScale(V[k],1.0/gdot);CHKERRQ(ierr);
    ierr = VecDot(U[k],Y,&gdot);CHKERRQ(ierr);
    ierr = VecAXPY(Y,gdot,V[k]);CHKERRQ(ierr);
    if (qn->monitor) {
      ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(qn->monitor, "update: %d k: %d gdot: %14.12e\n", it, k, PetscRealPart(gdot));CHKERRQ(ierr);
      ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESQNApply_BadBroyden"
PetscErrorCode SNESQNApply_BadBroyden(SNES snes,PetscInt it,Vec Y,Vec X,Vec Xold,Vec D,Vec Dold)
{
  PetscErrorCode ierr;
  SNES_QN        *qn = (SNES_QN*)snes->data;
  Vec            W   = snes->work[3];
  Vec            *U  = qn->U;
  Vec            *T  = qn->V;

  /* ksp thing for jacobian scaling */
  KSPConvergedReason kspreason;
  MatStructure       flg = DIFFERENT_NONZERO_PATTERN;
  PetscInt           k, i, lits;
  PetscInt           m = qn->m;
  PetscScalar        gdot;
  PetscInt           l = m;
  Mat                jac, jac_pre;

  PetscFunctionBegin;
  if (it < m) l = it;
  ierr = VecCopy(D,Y);CHKERRQ(ierr);
  if (l > 0) {
    k    = (it-1)%l;
    ierr = VecCopy(Dold,U[k]);CHKERRQ(ierr);
    ierr = VecAXPY(U[k],-1.0,D);CHKERRQ(ierr);
    ierr = VecDot(U[k],U[k],&gdot);CHKERRQ(ierr);
    ierr = VecCopy(D,T[k]);CHKERRQ(ierr);
    ierr = VecScale(T[k],1.0/gdot);CHKERRQ(ierr);
    ierr = VecDot(U[k],Y,&gdot);CHKERRQ(ierr);
    ierr = VecAXPY(Y,gdot,T[k]);CHKERRQ(ierr);
    if (qn->monitor) {
      ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(qn->monitor, "update: %d k: %d gdot: %14.12e\n", it, k, PetscRealPart(gdot));CHKERRQ(ierr);
      ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
    }
  }

  /* inward recursion starting at the first update and working forward */
  if (it > 1) {
    for (i = 0; i < l-1; i++) {
      k    = (it+i-l)%l;
      ierr = VecDot(U[k],Y,&gdot);CHKERRQ(ierr);
      ierr = VecAXPY(Y,gdot,T[k]);CHKERRQ(ierr);
      if (qn->monitor) {
        ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(qn->monitor, "it: %d k: %d gdot: %14.12e\n", it, k, PetscRealPart(gdot));CHKERRQ(ierr);
        ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      }
    }
  }

  if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
    ierr = SNESGetJacobian(snes, &jac, &jac_pre, PETSC_NULL, PETSC_NULL);CHKERRQ(ierr);
    ierr = KSPSetOperators(snes->ksp,jac,jac_pre,flg);CHKERRQ(ierr);
    ierr = SNES_KSPSolve(snes,snes->ksp,Y,W);CHKERRQ(ierr);
    ierr = KSPGetConvergedReason(snes->ksp,&kspreason);CHKERRQ(ierr);
    if (kspreason < 0) {
      if (++snes->numLinearSolveFailures >= snes->maxLinearSolveFailures) {
        ierr         = PetscInfo2(snes,"iter=%D, number linear solve failures %D greater than current SNES allowed, stopping solve\n",snes->iter,snes->numLinearSolveFailures);CHKERRQ(ierr);
        snes->reason = SNES_DIVERGED_LINEAR_SOLVE;
        PetscFunctionReturn(0);
      }
    }
    ierr              = KSPGetIterationNumber(snes->ksp,&lits);CHKERRQ(ierr);
    snes->linear_its += lits;
    ierr              = VecCopy(W,Y);CHKERRQ(ierr);
  } else {
    ierr = VecScale(Y,qn->scaling);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESQNApply_LBFGS"
PetscErrorCode SNESQNApply_LBFGS(SNES snes,PetscInt it,Vec Y,Vec X,Vec Xold,Vec D,Vec Dold)
{
  PetscErrorCode ierr;
  SNES_QN        *qn    = (SNES_QN*)snes->data;
  Vec            W      = snes->work[3];
  Vec            *dX    = qn->U;
  Vec            *dF    = qn->V;
  PetscScalar    *alpha = qn->alpha;
  PetscScalar    *beta  = qn->beta;
  PetscScalar    *dXtdF = qn->dXtdF;
  PetscScalar    *dFtdX = qn->dFtdX;
  PetscScalar    *YtdX  = qn->YtdX;

  /* ksp thing for jacobian scaling */
  KSPConvergedReason kspreason;
  MatStructure       flg = DIFFERENT_NONZERO_PATTERN;
  PetscInt           k,i,j,g,lits;
  PetscInt           m = qn->m;
  PetscScalar        t;
  PetscInt           l = m;
  Mat                jac,jac_pre;

  PetscFunctionBegin;
  if (it < m) l = it;
  if (it > 0) {
    k    = (it - 1) % l;
    ierr = VecCopy(D, dF[k]);CHKERRQ(ierr);
    ierr = VecAXPY(dF[k], -1.0, Dold);CHKERRQ(ierr);
    ierr = VecCopy(X, dX[k]);CHKERRQ(ierr);
    ierr = VecAXPY(dX[k], -1.0, Xold);CHKERRQ(ierr);
    if (qn->singlereduction) {
      ierr = VecMDot(dF[k], l, dX, dXtdF);CHKERRQ(ierr);
      ierr = VecMDot(dX[k], l, dF, dFtdX);CHKERRQ(ierr);
      for (j = 0; j < l; j++) {
        H(k, j) = dFtdX[j];
        H(j, k) = dXtdF[j];
      }
      /* copy back over to make the computation of alpha and beta easier */
      for (j = 0; j < l; j++) dXtdF[j] = H(j, j);
    } else {
      ierr = VecDot(dX[k], dF[k], &dXtdF[k]);CHKERRQ(ierr);
    }
    if (qn->scale_type == SNES_QN_SCALE_SHANNO) {
      PetscReal dFtdF;
      ierr        = VecDotRealPart(dF[k],dF[k],&dFtdF);CHKERRQ(ierr);
      qn->scaling = PetscRealPart(dXtdF[k])/dFtdF;
    } else if (qn->scale_type == SNES_QN_SCALE_LINESEARCH) {
      ierr = SNESLineSearchGetLambda(snes->linesearch,&qn->scaling);CHKERRQ(ierr);
    }
  }

  ierr = VecCopy(D,Y);CHKERRQ(ierr);

  if (qn->singlereduction) {
    ierr = VecMDot(Y,l,dX,YtdX);CHKERRQ(ierr);
  }
  /* outward recursion starting at iteration k's update and working back */
  for (i=0; i<l; i++) {
    k = (it-i-1)%l;
    if (qn->singlereduction) {
      /* construct t = dX[k] dot Y as Y_0 dot dX[k] + sum(-alpha[j]dX[k]dF[j]) */
      t = YtdX[k];
      for (j=0; j<i; j++) {
        g  = (it-j-1)%l;
        t += -alpha[g]*H(g, k);
      }
      alpha[k] = t/H(k,k);
    } else {
      ierr     = VecDot(dX[k],Y,&t);CHKERRQ(ierr);
      alpha[k] = t/dXtdF[k];
    }
    if (qn->monitor) {
      ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(qn->monitor, "it: %d k: %d alpha:        %14.12e\n", it, k, PetscRealPart(alpha[k]));CHKERRQ(ierr);
      ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
    }
    ierr = VecAXPY(Y,-alpha[k],dF[k]);CHKERRQ(ierr);
  }

  if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
    ierr = SNESGetJacobian(snes, &jac, &jac_pre, PETSC_NULL, PETSC_NULL);CHKERRQ(ierr);
    ierr = KSPSetOperators(snes->ksp,jac,jac_pre,flg);CHKERRQ(ierr);
    ierr = SNES_KSPSolve(snes,snes->ksp,Y,W);CHKERRQ(ierr);
    ierr = KSPGetConvergedReason(snes->ksp,&kspreason);CHKERRQ(ierr);
    if (kspreason < 0) {
      if (++snes->numLinearSolveFailures >= snes->maxLinearSolveFailures) {
        ierr         = PetscInfo2(snes,"iter=%D, number linear solve failures %D greater than current SNES allowed, stopping solve\n",snes->iter,snes->numLinearSolveFailures);CHKERRQ(ierr);
        snes->reason = SNES_DIVERGED_LINEAR_SOLVE;
        PetscFunctionReturn(0);
      }
    }
    ierr              = KSPGetIterationNumber(snes->ksp,&lits);CHKERRQ(ierr);
    snes->linear_its += lits;
    ierr              = VecCopy(W, Y);CHKERRQ(ierr);
  } else {
    ierr = VecScale(Y, qn->scaling);CHKERRQ(ierr);
  }
  if (qn->singlereduction) {
    ierr = VecMDot(Y,l,dF,YtdX);CHKERRQ(ierr);
  }
  /* inward recursion starting at the first update and working forward */
  for (i = 0; i < l; i++) {
    k = (it + i - l) % l;
    if (qn->singlereduction) {
      t = YtdX[k];
      for (j = 0; j < i; j++) {
        g  = (it + j - l) % l;
        t += (alpha[g] - beta[g])*H(k, g);
      }
      beta[k] = t / H(k, k);
    } else {
      ierr    = VecDot(dF[k], Y, &t);CHKERRQ(ierr);
      beta[k] = t / dXtdF[k];
    }
    ierr = VecAXPY(Y, (alpha[k] - beta[k]), dX[k]);CHKERRQ(ierr);
    if (qn->monitor) {
      ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      ierr = PetscViewerASCIIPrintf(qn->monitor, "it: %d k: %d alpha - beta: %14.12e\n", it, k, PetscRealPart(alpha[k] - beta[k]));CHKERRQ(ierr);
      ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESSolve_QN"
static PetscErrorCode SNESSolve_QN(SNES snes)
{
  PetscErrorCode      ierr;
  SNES_QN             *qn = (SNES_QN*) snes->data;
  Vec                 X,Xold;
  Vec                 F,B;
  Vec                 Y,FPC,D,Dold;
  SNESConvergedReason reason;
  PetscInt            i, i_r;
  PetscReal           fnorm,xnorm,ynorm,gnorm;
  PetscBool           lssucceed,powell,periodic;
  PetscScalar         DolddotD,DolddotDold,DdotD,YdotD;
  MatStructure        flg = DIFFERENT_NONZERO_PATTERN;

  /* basically just a regular newton's method except for the application of the jacobian */

  PetscFunctionBegin;
  F = snes->vec_func;                   /* residual vector */
  Y = snes->vec_sol_update;             /* search direction generated by J^-1D*/
  B = snes->vec_rhs;

  X    = snes->vec_sol;                 /* solution vector */
  Xold = snes->work[0];

  /* directions generated by the preconditioned problem with F_pre = F or x - M(x, b) */
  D    = snes->work[1];
  Dold = snes->work[2];

  snes->reason = SNES_CONVERGED_ITERATING;

  ierr       = PetscObjectTakeAccess(snes);CHKERRQ(ierr);
  snes->iter = 0;
  snes->norm = 0.;
  ierr       = PetscObjectGrantAccess(snes);CHKERRQ(ierr);
  if (!snes->vec_func_init_set) {
    ierr = SNESComputeFunction(snes,X,F);CHKERRQ(ierr);
    if (snes->domainerror) {
      snes->reason = SNES_DIVERGED_FUNCTION_DOMAIN;
      PetscFunctionReturn(0);
    }
  } else snes->vec_func_init_set = PETSC_FALSE;

  if (!snes->norm_init_set) {
    ierr = VecNorm(F, NORM_2, &fnorm);CHKERRQ(ierr); /* fnorm <- ||F||  */
    if (PetscIsInfOrNanReal(fnorm)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_FP,"Infinite or not-a-number generated in norm");
  } else {
    fnorm               = snes->norm_init;
    snes->norm_init_set = PETSC_FALSE;
  }

  ierr       = PetscObjectTakeAccess(snes);CHKERRQ(ierr);
  snes->norm = fnorm;
  ierr       = PetscObjectGrantAccess(snes);CHKERRQ(ierr);
  SNESLogConvHistory(snes,fnorm,0);
  ierr = SNESMonitor(snes,0,fnorm);CHKERRQ(ierr);

  /* set parameter for default relative tolerance convergence test */
  snes->ttol = fnorm*snes->rtol;
  /* test convergence */
  ierr = (*snes->ops->converged)(snes,0,0.0,0.0,fnorm,&snes->reason,snes->cnvP);CHKERRQ(ierr);
  if (snes->reason) PetscFunctionReturn(0);

  /* composed solve */
  if (snes->pc && snes->pcside == PC_RIGHT) {
    ierr = SNESSetInitialFunction(snes->pc, F);CHKERRQ(ierr);
    ierr = SNESSetInitialFunctionNorm(snes->pc, fnorm);CHKERRQ(ierr);

    ierr = PetscLogEventBegin(SNES_NPCSolve,snes->pc,X,B,0);CHKERRQ(ierr);
    ierr = SNESSolve(snes->pc, B, X);CHKERRQ(ierr);
    ierr = PetscLogEventEnd(SNES_NPCSolve,snes->pc,X,B,0);CHKERRQ(ierr);

    ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
    if (reason < 0 && (reason != SNES_DIVERGED_MAX_IT)) {
      snes->reason = SNES_DIVERGED_INNER;
      PetscFunctionReturn(0);
    }
    ierr = SNESGetFunction(snes->pc, &FPC, PETSC_NULL, PETSC_NULL);CHKERRQ(ierr);
    ierr = VecCopy(FPC, F);CHKERRQ(ierr);
    ierr = SNESGetFunctionNorm(snes->pc, &fnorm);CHKERRQ(ierr);
    ierr = VecCopy(F, Y);CHKERRQ(ierr);
  } else {
    ierr = VecCopy(F, Y);CHKERRQ(ierr);
  }
  ierr = VecCopy(Y, D);CHKERRQ(ierr);

  /* scale the initial update */
  if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
    ierr = SNESComputeJacobian(snes,X,&snes->jacobian,&snes->jacobian_pre,&flg);CHKERRQ(ierr);
  }

  for (i = 0, i_r = 0; i < snes->max_its; i++, i_r++) {
    switch (qn->type) {
    case SNES_QN_BADBROYDEN:
      ierr = SNESQNApply_BadBroyden(snes,i_r,Y,X,Xold,D,Dold);CHKERRQ(ierr);
      break;
    case SNES_QN_BROYDEN:
      ierr = SNESQNApply_Broyden(snes,i_r,Y,X,Xold,D,Dold);CHKERRQ(ierr);
      break;
    case SNES_QN_LBFGS:
      SNESQNApply_LBFGS(snes,i_r,Y,X,Xold,D,Dold);CHKERRQ(ierr);
      break;
    }
    /* line search for lambda */
    ynorm = 1; gnorm = fnorm;
    ierr  = VecCopy(D, Dold);CHKERRQ(ierr);
    ierr  = VecCopy(X, Xold);CHKERRQ(ierr);
    ierr  = SNESLineSearchApply(snes->linesearch, X, F, &fnorm, Y);CHKERRQ(ierr);
    if (snes->reason == SNES_DIVERGED_FUNCTION_COUNT) break;
    if (snes->domainerror) {
      snes->reason = SNES_DIVERGED_FUNCTION_DOMAIN;
      PetscFunctionReturn(0);
    }
    ierr = SNESLineSearchGetSuccess(snes->linesearch, &lssucceed);CHKERRQ(ierr);
    if (!lssucceed) {
      if (++snes->numFailures >= snes->maxFailures) {
        snes->reason = SNES_DIVERGED_LINE_SEARCH;
        break;
      }
    }
    ierr = SNESLineSearchGetNorms(snes->linesearch, &xnorm, &fnorm, &ynorm);CHKERRQ(ierr);
    if (qn->scale_type == SNES_QN_SCALE_LINESEARCH) {
      ierr = SNESLineSearchGetLambda(snes->linesearch, &qn->scaling);CHKERRQ(ierr);
    }

    /* convergence monitoring */
    ierr = PetscInfo4(snes,"fnorm=%18.16e, gnorm=%18.16e, ynorm=%18.16e, lssucceed=%d\n",(double)fnorm,(double)gnorm,(double)ynorm,(int)lssucceed);CHKERRQ(ierr);

    ierr = SNESSetIterationNumber(snes, i+1);CHKERRQ(ierr);
    ierr = SNESSetFunctionNorm(snes, fnorm);CHKERRQ(ierr);

    SNESLogConvHistory(snes,snes->norm,snes->iter);
    ierr = SNESMonitor(snes,snes->iter,snes->norm);CHKERRQ(ierr);
    /* set parameter for default relative tolerance convergence test */
    ierr = (*snes->ops->converged)(snes,snes->iter,xnorm,ynorm,fnorm,&snes->reason,snes->cnvP);CHKERRQ(ierr);
    if (snes->reason) PetscFunctionReturn(0);

    if (snes->pc && snes->pcside == PC_RIGHT) {
      ierr = SNESSetInitialFunction(snes->pc, F);CHKERRQ(ierr);
      ierr = SNESSetInitialFunctionNorm(snes->pc, fnorm);CHKERRQ(ierr);
      ierr = SNESSolve(snes->pc, B, X);CHKERRQ(ierr);
      ierr = SNESGetConvergedReason(snes->pc,&reason);CHKERRQ(ierr);
      if (reason < 0 && (reason != SNES_DIVERGED_MAX_IT)) {
        snes->reason = SNES_DIVERGED_INNER;
        PetscFunctionReturn(0);
      }
      ierr = SNESGetFunction(snes->pc, &FPC, PETSC_NULL, PETSC_NULL);CHKERRQ(ierr);
      ierr = VecCopy(FPC, F);CHKERRQ(ierr);
      ierr = SNESGetFunctionNorm(snes->pc, &fnorm);CHKERRQ(ierr);
      ierr = VecCopy(F, D);CHKERRQ(ierr);
    } else {
      ierr = VecCopy(F, D);CHKERRQ(ierr);
    }

    powell = PETSC_FALSE;
    if (qn->restart_type == SNES_QN_RESTART_POWELL) {
      /* check restart by Powell's Criterion: |F^T H_0 Fold| > 0.2 * |Fold^T H_0 Fold| */
      ierr = VecDotBegin(Dold, Dold, &DolddotDold);CHKERRQ(ierr);
      ierr = VecDotBegin(Dold, D, &DolddotD);CHKERRQ(ierr);
      ierr = VecDotBegin(D, D, &DdotD);CHKERRQ(ierr);
      ierr = VecDotBegin(Y, D, &YdotD);CHKERRQ(ierr);
      ierr = VecDotEnd(Dold, Dold, &DolddotDold);CHKERRQ(ierr);
      ierr = VecDotEnd(Dold, D, &DolddotD);CHKERRQ(ierr);
      ierr = VecDotEnd(D, D, &DdotD);CHKERRQ(ierr);
      ierr = VecDotEnd(Y, D, &YdotD);CHKERRQ(ierr);
      if (PetscAbs(PetscRealPart(DolddotD)) > qn->powell_gamma*PetscAbs(PetscRealPart(DolddotDold))) powell = PETSC_TRUE;
    }
    periodic = PETSC_FALSE;
    if (qn->restart_type == SNES_QN_RESTART_PERIODIC) {
      if (i_r>qn->m-1) periodic = PETSC_TRUE;
    }
    /* restart if either powell or periodic restart is satisfied. */
    if (powell || periodic) {
      if (qn->monitor) {
        ierr = PetscViewerASCIIAddTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
        ierr = PetscViewerASCIIPrintf(qn->monitor, "restart! |%14.12e| > %4.2f*|%14.12e| or i_r = %d\n", PetscRealPart(DolddotD), qn->powell_gamma, PetscRealPart(DolddotDold), i_r);CHKERRQ(ierr);
        ierr = PetscViewerASCIISubtractTab(qn->monitor,((PetscObject)snes)->tablevel+2);CHKERRQ(ierr);
      }
      i_r = -1;
      /* general purpose update */
      if (snes->ops->update) {
        ierr = (*snes->ops->update)(snes, snes->iter);CHKERRQ(ierr);
      }
      if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
        ierr = SNESComputeJacobian(snes,X,&snes->jacobian,&snes->jacobian_pre,&flg);CHKERRQ(ierr);
      }
    }
    /* general purpose update */
    if (snes->ops->update) {
      ierr = (*snes->ops->update)(snes, snes->iter);CHKERRQ(ierr);
    }
  }
  if (i == snes->max_its) {
    ierr = PetscInfo1(snes, "Maximum number of iterations has been reached: %D\n", snes->max_its);CHKERRQ(ierr);
    if (!snes->reason) snes->reason = SNES_DIVERGED_MAX_IT;
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESSetUp_QN"
static PetscErrorCode SNESSetUp_QN(SNES snes)
{
  SNES_QN        *qn = (SNES_QN*)snes->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = VecDuplicateVecs(snes->vec_sol, qn->m, &qn->U);CHKERRQ(ierr);
  ierr = VecDuplicateVecs(snes->vec_sol, qn->m, &qn->V);CHKERRQ(ierr);
  ierr = PetscMalloc3(qn->m, PetscScalar, &qn->alpha,
                      qn->m, PetscScalar, &qn->beta,
                      qn->m, PetscScalar, &qn->dXtdF);CHKERRQ(ierr);

  if (qn->singlereduction) {
    ierr = PetscMalloc3(qn->m*qn->m, PetscScalar, &qn->dXdFmat,
                        qn->m, PetscScalar, &qn->dFtdX,
                        qn->m, PetscScalar, &qn->YtdX);CHKERRQ(ierr);
  }
  ierr = SNESDefaultGetWork(snes,4);CHKERRQ(ierr);

  /* set up the line search */
  if (qn->scale_type == SNES_QN_SCALE_JACOBIAN) {
    ierr = SNESSetUpMatrices(snes);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESReset_QN"
static PetscErrorCode SNESReset_QN(SNES snes)
{
  PetscErrorCode ierr;
  SNES_QN        *qn;

  PetscFunctionBegin;
  if (snes->data) {
    qn = (SNES_QN*)snes->data;
    if (qn->U) {
      ierr = VecDestroyVecs(qn->m, &qn->U);CHKERRQ(ierr);
    }
    if (qn->V) {
      ierr = VecDestroyVecs(qn->m, &qn->V);CHKERRQ(ierr);
    }
    if (qn->singlereduction) {
      ierr = PetscFree3(qn->dXdFmat, qn->dFtdX, qn->YtdX);CHKERRQ(ierr);
    }
    ierr = PetscFree3(qn->alpha, qn->beta, qn->dXtdF);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESDestroy_QN"
static PetscErrorCode SNESDestroy_QN(SNES snes)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = SNESReset_QN(snes);CHKERRQ(ierr);
  ierr = PetscFree(snes->data);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)snes,"","",PETSC_NULL);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESSetFromOptions_QN"
static PetscErrorCode SNESSetFromOptions_QN(SNES snes)
{

  PetscErrorCode    ierr;
  SNES_QN           *qn    = (SNES_QN*)snes->data;
  PetscBool         monflg = PETSC_FALSE,flg;
  SNESLineSearch    linesearch;
  SNESQNRestartType rtype = qn->restart_type;
  SNESQNScaleType   stype = qn->scale_type;

  PetscFunctionBegin;
  ierr = PetscOptionsHead("SNES QN options");CHKERRQ(ierr);
  ierr = PetscOptionsInt("-snes_qn_m","Number of past states saved for L-BFGS methods","SNESQN",qn->m,&qn->m,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-snes_qn_powell_gamma","Powell angle tolerance",          "SNESQN", qn->powell_gamma, &qn->powell_gamma, PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-snes_qn_powell_downhill","Powell descent tolerance",        "SNESQN", qn->powell_downhill, &qn->powell_downhill, PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-snes_qn_monitor",         "Monitor for the QN methods",      "SNESQN", monflg, &monflg, PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-snes_qn_single_reduction", "Aggregate reductions",           "SNESQN", qn->singlereduction, &qn->singlereduction, PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnum("-snes_qn_scale_type","Scaling type","SNESQNSetScaleType",SNESQNScaleTypes,(PetscEnum)stype,(PetscEnum*)&stype,&flg);CHKERRQ(ierr);
  if (flg) ierr = SNESQNSetScaleType(snes,stype);CHKERRQ(ierr);

  ierr = PetscOptionsEnum("-snes_qn_restart_type","Restart type","SNESQNSetRestartType",SNESQNRestartTypes,(PetscEnum)rtype,(PetscEnum*)&rtype,&flg);CHKERRQ(ierr);
  if (flg) ierr = SNESQNSetRestartType(snes,rtype);CHKERRQ(ierr);

  ierr = PetscOptionsEnum("-snes_qn_type","Quasi-Newton update type","",SNESQNTypes,
                          (PetscEnum)qn->type,(PetscEnum*)&qn->type,PETSC_NULL);CHKERRQ(ierr);
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  if (!snes->linesearch) {
    ierr = SNESGetSNESLineSearch(snes, &linesearch);CHKERRQ(ierr);
    ierr = SNESLineSearchSetType(linesearch, SNESLINESEARCHCP);CHKERRQ(ierr);
  }
  if (monflg) {
    qn->monitor = PETSC_VIEWER_STDOUT_(((PetscObject)snes)->comm);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}


#undef __FUNCT__
#define __FUNCT__ "SNESQNSetRestartType"
/*@
    SNESQNSetRestartType - Sets the restart type for SNESQN.

    Logically Collective on SNES

    Input Parameters:
+   snes - the iterative context
-   rtype - restart type

    Options Database:
+   -snes_qn_restart_type<powell,periodic,none> - set the restart type
-   -snes_qn_restart[10] - sets the number of iterations before restart for periodic

    Level: intermediate

    SNESQNRestartTypes:
+   SNES_QN_RESTART_NONE - never restart
.   SNES_QN_RESTART_POWELL - restart based upon descent criteria
-   SNES_QN_RESTART_PERIODIC - restart after a fixed number of iterations

    Notes:
    The default line search used is the L2 line search and it requires two additional function evaluations.

.keywords: SNES, SNESQN, restart, type, set SNESLineSearch
@*/
PetscErrorCode SNESQNSetRestartType(SNES snes, SNESQNRestartType rtype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes,SNES_CLASSID,1);
  ierr = PetscTryMethod(snes,"SNESQNSetRestartType_C",(SNES,SNESQNRestartType),(snes,rtype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESQNSetScaleType"
/*@
    SNESQNSetScaleType - Sets the scaling type for the inner inverse jacobian in SNESQN.

    Logically Collective on SNES

    Input Parameters:
+   snes - the iterative context
-   stype - scale type

    Options Database:
.   -snes_qn_scale_type<shanno,none,linesearch,jacobian>

    Level: intermediate

    SNESQNSelectTypes:
+   SNES_QN_SCALE_NONE - don't scale the problem
.   SNES_QN_SCALE_SHANNO - use shanno scaling
.   SNES_QN_SCALE_LINESEARCH - scale based upon line search lambda
-   SNES_QN_SCALE_JACOBIAN - scale by inverting a previously computed Jacobian.

.keywords: SNES, SNESQN, scaling, type, set SNESLineSearch
@*/

PetscErrorCode SNESQNSetScaleType(SNES snes, SNESQNScaleType stype)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes,SNES_CLASSID,1);
  ierr = PetscTryMethod(snes,"SNESQNSetScaleType_C",(SNES,SNESQNScaleType),(snes,stype));CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "SNESQNSetScaleType_QN"
PetscErrorCode SNESQNSetScaleType_QN(SNES snes, SNESQNScaleType stype)
{
  SNES_QN *qn = (SNES_QN*)snes->data;

  PetscFunctionBegin;
  qn->scale_type = stype;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "SNESQNSetRestartType_QN"
PetscErrorCode SNESQNSetRestartType_QN(SNES snes, SNESQNRestartType rtype)
{
  SNES_QN *qn = (SNES_QN*)snes->data;

  PetscFunctionBegin;
  qn->restart_type = rtype;
  PetscFunctionReturn(0);
}
EXTERN_C_END

/* -------------------------------------------------------------------------- */
/*MC
      SNESQN - Limited-Memory Quasi-Newton methods for the solution of nonlinear systems.

      Options Database:

+     -snes_qn_m - Number of past states saved for the L-Broyden methods.
.     -snes_qn_powell_angle - Angle condition for restart.
.     -snes_qn_powell_descent - Descent condition for restart.
.     -snes_linesearch_type <cp, l2, basic> - Type of line search.
-     -snes_qn_monitor - Monitors the quasi-newton jacobian.

      Notes: This implements the L-BFGS, Broyden, and "Bad" Broyden algorithms for the solution of F(x) = b using
      previous change in F(x) and x to form the approximate inverse Jacobian using a series of multiplicative rank-one
      updates.

      When using a nonlinear preconditioner, one has two options as to how the preconditioner is applied.  The first of
      these options, sequential, uses the preconditioner to generate a new solution and function and uses those at this
      iteration as the current iteration's values when constructing the approximate jacobian.  The second, composed,
      perturbs the problem the jacobian represents to be P(x, b) - x = 0, where P(x, b) is the preconditioner.

      References:

      L-Broyden Methods: a generalization of the L-BFGS method to the limited memory Broyden family, M. B. Reed,
      International Journal of Computer Mathematics, vol. 86, 2009.

      Experiments with Quasi-Newton Methods in Solving Stiff ODE Systems, Peter N. Brown, Alan C. Hindmarsh, Homer F. Walker
      SIAM J. Sci. Stat. Comput. Vol 6(2), April 1985.


      Level: beginner

.seealso:  SNESCreate(), SNES, SNESSetType(), SNESNEWTONLS, SNESNEWTONTR

M*/
EXTERN_C_BEGIN
#undef __FUNCT__
#define __FUNCT__ "SNESCreate_QN"
PetscErrorCode  SNESCreate_QN(SNES snes)
{
  PetscErrorCode ierr;
  SNES_QN        *qn;

  PetscFunctionBegin;
  snes->ops->setup          = SNESSetUp_QN;
  snes->ops->solve          = SNESSolve_QN;
  snes->ops->destroy        = SNESDestroy_QN;
  snes->ops->setfromoptions = SNESSetFromOptions_QN;
  snes->ops->view           = 0;
  snes->ops->reset          = SNESReset_QN;

  snes->usespc  = PETSC_TRUE;
  snes->usesksp = PETSC_FALSE;

  if (!snes->tolerancesset) {
    snes->max_funcs = 30000;
    snes->max_its   = 10000;
  }

  ierr                = PetscNewLog(snes,SNES_QN,&qn);CHKERRQ(ierr);
  snes->data          = (void*) qn;
  qn->m               = 10;
  qn->scaling         = 1.0;
  qn->U               = PETSC_NULL;
  qn->V               = PETSC_NULL;
  qn->dXtdF           = PETSC_NULL;
  qn->dFtdX           = PETSC_NULL;
  qn->dXdFmat         = PETSC_NULL;
  qn->monitor         = PETSC_NULL;
  qn->singlereduction = PETSC_FALSE;
  qn->powell_gamma    = 0.9999;
  qn->scale_type      = SNES_QN_SCALE_SHANNO;
  qn->restart_type    = SNES_QN_RESTART_POWELL;
  qn->type            = SNES_QN_LBFGS;

  ierr = PetscObjectComposeFunctionDynamic((PetscObject)snes,"SNESQNSetScaleType_C","SNESQNSetScaleType_QN",SNESQNSetScaleType_QN);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunctionDynamic((PetscObject)snes,"SNESQNSetRestartType_C","SNESQNSetRestartType_QN",SNESQNSetRestartType_QN);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

EXTERN_C_END