ordinal_type
Stokhos::MonomialGramSchmidtPCEBasis<ordinal_type, value_type>::
buildReducedBasis(
ordinal_type max_p,
value_type threshold,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& A,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& F,
const Teuchos::Array<value_type>& weights,
Teuchos::Array< Stokhos::MultiIndex<ordinal_type> >& terms_,
Teuchos::Array<ordinal_type>& num_terms_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Qp_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Q_)
{
// Compute basis terms -- 2-D array giving powers for each linear index
ordinal_type max_sz;
CPBUtils::compute_terms(max_p, this->d, max_sz, terms_, num_terms_);
// Compute B matrix -- monomials in F
// for i=0,...,nqp-1
// for j=0,...,sz-1
// B(i,j) = F(i,1)^terms_[j][1] * ... * F(i,d)^terms_[j][d]
// where sz is the total size of a basis up to order p and terms_[j]
// is an array of powers for each term in the total-order basis
ordinal_type nqp = weights.size();
SDM B(nqp, max_sz);
for (ordinal_type i=0; i<nqp; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
B(i,j) = 1.0;
for (ordinal_type k=0; k<this->d; k++)
B(i,j) *= std::pow(F(i,k), terms_[j][k]);
}
}
// Rescale columns of B to have unit norm
for (ordinal_type j=0; j<max_sz; j++) {
value_type nrm = 0.0;
for (ordinal_type i=0; i<nqp; i++)
nrm += B(i,j)*B(i,j)*weights[i];
nrm = std::sqrt(nrm);
for (ordinal_type i=0; i<nqp; i++)
B(i,j) /= nrm;
}
// Compute our new basis -- each column of Q is the new basis evaluated
// at the original quadrature points. Constraint pivoting so first d+1
// columns and included in Q.
SDM R;
Teuchos::Array<ordinal_type> piv(max_sz);
for (int i=0; i<this->d+1; i++)
piv[i] = 1;
typedef Stokhos::OrthogonalizationFactory<ordinal_type,value_type> SOF;
ordinal_type sz_ = SOF::createOrthogonalBasis(
this->orthogonalization_method, threshold, this->verbose, B, weights,
Q_, R, piv);
// Compute Qp = A^T*W*Q
SDM tmp(nqp, sz_);
Qp_.reshape(this->pce_sz, sz_);
for (ordinal_type i=0; i<nqp; i++)
for (ordinal_type j=0; j<sz_; j++)
tmp(i,j) = Q_(i,j)*weights[i];
ordinal_type ret =
Qp_.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, A, tmp, 0.0);
TEUCHOS_ASSERT(ret == 0);
// It isn't clear that Qp is orthogonal, but if you derive the projection
// matrix from the original space to the reduced, you end up with
// Q^T*W*A = Qp^T
return sz_;
}
ordinal_type
Stokhos::MonomialProjGramSchmidtPCEBasis<ordinal_type, value_type>::
buildReducedBasis(
ordinal_type max_p,
value_type threshold,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& A,
const Teuchos::SerialDenseMatrix<ordinal_type,value_type>& F,
const Teuchos::Array<value_type>& weights,
Teuchos::Array< Stokhos::MultiIndex<ordinal_type> >& terms_,
Teuchos::Array<ordinal_type>& num_terms_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Qp_,
Teuchos::SerialDenseMatrix<ordinal_type,value_type>& Q_)
{
// Compute basis terms -- 2-D array giving powers for each linear index
ordinal_type max_sz;
CPBUtils::compute_terms(max_p, this->d, max_sz, terms_, num_terms_);
// Compute B matrix -- monomials in F
// for i=0,...,nqp-1
// for j=0,...,sz-1
// B(i,j) = F(i,1)^terms_[j][1] * ... * F(i,d)^terms_[j][d]
// where sz is the total size of a basis up to order p and terms_[j]
// is an array of powers for each term in the total-order basis
ordinal_type nqp = weights.size();
SDM B(nqp, max_sz);
for (ordinal_type i=0; i<nqp; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
B(i,j) = 1.0;
for (ordinal_type k=0; k<this->d; k++)
B(i,j) *= std::pow(F(i,k), terms_[j][k]);
}
}
// Project B into original basis -- should use SPAM for this
SDM Bp(this->pce_sz, max_sz);
const Teuchos::Array<value_type>& basis_norms =
this->pce_basis->norm_squared();
for (ordinal_type i=0; i<this->pce_sz; i++) {
for (ordinal_type j=0; j<max_sz; j++) {
Bp(i,j) = 0.0;
for (ordinal_type k=0; k<nqp; k++)
Bp(i,j) += weights[k]*B(k,j)*A(k,i);
Bp(i,j) /= basis_norms[i];
}
}
// Rescale columns of Bp to have unit norm
for (ordinal_type j=0; j<max_sz; j++) {
value_type nrm = 0.0;
for (ordinal_type i=0; i<this->pce_sz; i++)
nrm += Bp(i,j)*Bp(i,j)*basis_norms[i];
nrm = std::sqrt(nrm);
for (ordinal_type i=0; i<this->pce_sz; i++)
Bp(i,j) /= nrm;
}
// Compute our new basis -- each column of Qp is the coefficients of the
// new basis in the original basis. Constraint pivoting so first d+1
// columns and included in Qp.
Teuchos::Array<value_type> w(this->pce_sz, 1.0);
SDM R;
Teuchos::Array<ordinal_type> piv(max_sz);
for (int i=0; i<this->d+1; i++)
piv[i] = 1;
typedef Stokhos::OrthogonalizationFactory<ordinal_type,value_type> SOF;
ordinal_type sz_ = SOF::createOrthogonalBasis(
this->orthogonalization_method, threshold, this->verbose, Bp, w,
Qp_, R, piv);
// Evaluate new basis at original quadrature points
Q_.reshape(nqp, sz_);
ordinal_type ret =
Q_.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, A, Qp_, 0.0);
TEUCHOS_ASSERT(ret == 0);
return sz_;
}
//GMRES
int gmres(const Teuchos::SerialDenseMatrix<int, double> & A, Teuchos::SerialDenseMatrix<int,double> X,const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance)
{
int n;
int k;
double resid;
k=1;
n=A.numRows();
std::cout << "A= " << A << std::endl;
std::cout << "B= " << B << std::endl;
//Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
//Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
//r0-=Ax;
resid=r0.normFrobenius();
std::cout << "resid= " << resid << std::endl;
//define vector v=r/norm(r) where r=b-Ax
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++){
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter){
std::cout << "k = " << k << std::endl;
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, A, vk, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++){
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/w.normFrobenius());
//add column vk+1=w to V
for (int i=0; i<n; i++){
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++){
// double hi=h(i,k-1);
// double hi1=h(i+1,k-1);
// h(i,k-1)=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
// h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
// h(i,k-1)=c(i,0)*hi+s(i,0)*hi1;
// h(i+1,k-1)=-1*s(i,0)*hi+c(i,0)*hi1;
double q=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
h(i,k-1)=q;
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
std::cout << "c " << c(k-1,0)<<std::endl;
std::cout << "s " << s(k-1,0)<<std::endl;
// Givens rotation on h and bb
// h(k-1,k-1)=l;
// h(k,k-1)=0;
double hk=h(k,k-1);
double hk1=h(k-1,k-1);
h(k-1,k-1)=c(k-1,0)*hk1+s(k-1,0)*hk;
h(k,k-1)=-1*s(k-1,0)*hk1+c(k-1,0)*hk;
std::cout << "l = " << l <<std::endl;
std::cout << "h(k-1,k-1) = should be l " << h(k-1,k-1) <<std::endl;
std::cout << "h(k,k-1) = should be 0 " << h(k,k-1) <<std::endl;
bb(k,0)=-1*s(k-1,0)*bb(k-1,0);
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
//Determine residual
resid =fabs(bb(k,0));
std::cout << "resid = " << resid <<std::endl;
k++;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
std::cout << "bb pre solve = " << bb << std::endl;
std::cout << "h= " << h << std::endl;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
V.reshape(n,k-1);
std::cout << "V= " << V << std::endl;
std::cout << "y= " << bb << std::endl;
X.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 1.0);
std::cout << "X= " << X << std::endl;
//Check V is orthogoanl
// Teuchos::SerialDenseMatrix<int, double> vtv(V);
// vtv.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, V, V, 0.0);
// std::cout << "Vtv" << vtv << std::endl;
return 0;
}
int
Stokhos::GMRESDivisionExpansionStrategy<ordinal_type,value_type,node_type>::
GMRES(const Teuchos::SerialDenseMatrix<int, double> & A, Teuchos::SerialDenseMatrix<int,double> & X, const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance, int prec_iter, int order, int dim, int PrecNum, const Teuchos::SerialDenseMatrix<int, double> & M, int diag)
{
int n = A.numRows();
int k = 1;
double resid;
Teuchos::SerialDenseMatrix<int, double> P(n,n);
Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
r0-=Ax;
resid=r0.normFrobenius();
//define vector v=r/norm(r) where r=b-Ax
Teuchos::SerialDenseMatrix<int, double> v(n,1);
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++) {
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter) {
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
//Preconditioning step: solve Mz=vk
Teuchos::SerialDenseMatrix<int, double> z(vk);
if (PrecNum == 1) {
Stokhos::DiagPreconditioner precond(M);
precond.ApplyInverse(vk,z,prec_iter);
}
else if (PrecNum == 2) {
Stokhos::JacobiPreconditioner precond(M);
precond.ApplyInverse(vk,z,2);
}
else if (PrecNum == 3) {
Stokhos::GSPreconditioner precond(M,1);
precond.ApplyInverse(vk,z,1);
}
else if (PrecNum == 4) {
Stokhos::SchurPreconditioner precond(M, order, dim, diag);
precond.ApplyInverse(vk,z,prec_iter);
}
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1, A, z, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++) {
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/h(k,k-1));
//add column vk+1=w to V
for (int i=0; i<n; i++) {
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++) {
double q=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-1*s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
h(i,k-1)=q;
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
// Givens rotation on h and bb
h(k-1,k-1)=l;
h(k,k-1)=0;
bb(k,0)=-s(k-1,0)*bb(k-1,0);
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
//Determine residual
resid = fabs(bb(k,0));
k++;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
Teuchos::SerialDenseMatrix<int, double> ans(X);
V.reshape(n,k-1);
ans.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 0.0);
if (PrecNum == 1) {
Stokhos::DiagPreconditioner precond(M);
precond.ApplyInverse(ans,ans,prec_iter);
}
else if (PrecNum == 2) {
Stokhos::JacobiPreconditioner precond(M);
precond.ApplyInverse(ans,ans,2);
}
else if (PrecNum == 3) {
Stokhos::GSPreconditioner precond(M,1);
precond.ApplyInverse(ans,ans,1);
}
else if (PrecNum == 4) {
Stokhos::SchurPreconditioner precond(M, order, dim, diag);
precond.ApplyInverse(ans,ans,prec_iter);
}
X+=ans;
std::cout << "iteration count= " << k-1 << std::endl;
return 0;
}
//Mean-Based Preconditioned GMRES
int pregmres(const Teuchos::SerialDenseMatrix<int, double> & A, const Teuchos::SerialDenseMatrix<int,double> & X,const Teuchos::SerialDenseMatrix<int,double> & B, int max_iter, double tolerance)
{
int n;
int k;
double resid;
k=1;
n=A.numRows();
std::cout << A << std::endl;
Teuchos::SerialDenseMatrix<int, double> D(n,1);
//Get diagonal entries of A
for (int i=0; i<n; i++){
D(i,0)=A(i,i);
}
Teuchos::SerialDenseMatrix<int, double> Ax(n,1);
Ax.multiply(Teuchos::NO_TRANS,Teuchos::NO_TRANS,1.0, A, X, 0.0);
Teuchos::SerialDenseMatrix<int, double> r0(B);
r0-=Ax;
resid=r0.normFrobenius();
//define vector v=r/norm(r) where r=b-Ax
Teuchos::SerialDenseMatrix<int, double> v(n,1);
r0.scale(1/resid);
Teuchos::SerialDenseMatrix<int, double> h(1,1);
//Matrix of orthog basis vectors V
Teuchos::SerialDenseMatrix<int, double> V(n,1);
//Set v=r0/norm(r0) to be 1st col of V
for (int i=0; i<n; i++){
V(i,0)=r0(i,0);
}
//right hand side
Teuchos::SerialDenseMatrix<int, double> bb(1,1);
bb(0,0)=resid;
Teuchos::SerialDenseMatrix<int, double> w(n,1);
Teuchos::SerialDenseMatrix<int, double> c;
Teuchos::SerialDenseMatrix<int, double> s;
while (resid > tolerance && k < max_iter){
std::cout << "k = " << k << std::endl;
h.reshape(k+1,k);
//Arnoldi iteration(Gram-Schmidt )
V.reshape(n,k+1);
//set vk to be kth col of V
Teuchos::SerialDenseMatrix<int, double> vk(Teuchos::Copy, V, n,1,0,k-1);
//Preconditioning step w=AMj(-1)vj
w.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1/D(k-1,0), A, vk, 0.0);
Teuchos::SerialDenseMatrix<int, double> vi(n,1);
Teuchos::SerialDenseMatrix<int, double> ip(1,1);
for (int i=0; i<k; i++){
//set vi to be ith col of V
Teuchos::SerialDenseMatrix<int, double> vi(Teuchos::Copy, V, n,1,0,i);
//Calculate inner product
ip.multiply(Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, vi, w, 0.0);
h(i,k-1)= ip(0,0);
//scale vi by h(i,k-1)
vi.scale(ip(0,0));
w-=vi;
}
h(k,k-1)=w.normFrobenius();
w.scale(1.0/w.normFrobenius());
//add column vk+1=w to V
for (int i=0; i<n; i++){
V(i,k)=w(i,0);
}
//Solve upper hessenberg least squares problem via Givens rotations
//Compute previous Givens rotations
for (int i=0; i<k-1; i++){
h(i,k-1)=c(i,0)*h(i,k-1)+s(i,0)*h(i+1,k-1);
h(i+1,k-1)=-s(i,0)*h(i,k-1)+c(i,0)*h(i+1,k-1);
}
//Compute next Givens rotations
c.reshape(k,1);
s.reshape(k,1);
bb.reshape(k+1,1);
double l = sqrt(h(k-1,k-1)*h(k-1,k-1)+h(k,k-1)*h(k,k-1));
c(k-1,0)=h(k-1,k-1)/l;
s(k-1,0)=h(k,k-1)/l;
std::cout <<" h(k,k-1)= " << h(k,k-1) << std::endl;
// Givens rotation on h and bb
h(k-1,k-1)=l;
h(k,k-1)=0;
bb(k-1,0)=c(k-1,0)*bb(k-1,0);
bb(k,0)=-s(k-1,0)*bb(k-1,0);
//Determine residual
resid = fabs(bb(k,0));
std::cout << "resid = " << resid << std::endl;
k=k+1;
}
//Extract upper triangular square matrix
bb.reshape(h.numRows()-1 ,1);
//Solve linear system
int info;
Teuchos::LAPACK<int, double> lapack;
lapack.TRTRS('U', 'N', 'N', h.numRows()-1, 1, h.values(), h.stride(), bb.values(), bb.stride(),&info);
//Found y=Mx
for (int i=0; i<k-1; i++){
bb(i,0)=bb(i,0)/D(i,0);
}
V.reshape(n,k-1);
Teuchos::SerialDenseMatrix<int, double> ans(X);
ans.multiply(Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, V, bb, 1.0);
std::cout << "ans= " << ans << std::endl;
std::cout << "h= " << h << std::endl;
return 0;
}