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Matrix data structures, linear solvers, least squares methods, eigenvalue,
and singular value decompositions. For larger random dense matrices (above ~ 350 x 350)
matrix-matrix multiplication C = A.B is about 50% faster than MTJ.
/*
* Copyright (C) 2003-2006 Bjørn-Ove Heimsund
*
* This file is part of MTJ.
*
* This library is free software; you can redistribute it and/or modify it
* under the terms of the GNU Lesser General Public License as published by the
* Free Software Foundation; either version 2.1 of the License, or (at your
* option) any later version.
*
* This library is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License
* for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this library; if not, write to the Free Software Foundation,
* Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
*/
package no.uib.cipr.matrix.sparse;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashMap;
import java.util.HashSet;
import java.util.LinkedList;
import java.util.List;
import java.util.Map;
import java.util.Set;
import no.uib.cipr.matrix.DenseLU;
import no.uib.cipr.matrix.DenseMatrix;
import no.uib.cipr.matrix.DenseVector;
import no.uib.cipr.matrix.Matrix;
import no.uib.cipr.matrix.Vector;
/**
* Algebraic multigrid preconditioner. Uses the smoothed aggregation method
* described by Vanek, Mandel, and Brezina (1996).
*/
public class AMG implements Preconditioner {
/**
* Relaxations at each level
*/
private SSOR[] preM, postM;
/**
* The number of levels
*/
private int m;
/**
* System matrix at each level, except at the coarsest
*/
private CompRowMatrix[] A;
/**
* LU factorization at the coarsest level
*/
private DenseLU lu;
/**
* Solution, right-hand side, and residual vectors at each level
*/
private DenseVector[] u, f, r;
/**
* Interpolation operators going to a finer mesh
*/
private CompColMatrix[] I;
/**
* Smallest matrix size before terminating the AMG setup phase. Matrices
* smaller than this will be solved by a direct solver
*/
private final int min;
/**
* Number of times to perform the pre- and post-smoothings
*/
private final int nu1, nu2;
/**
* Determines cycle type. gamma=1 is V, gamma=2 is W
*/
private final int gamma;
/**
* Overrelaxation parameters in the pre- and post-smoothings, and with the
* possibility of distinct values in the forward and reverse sweeps
*/
private final double omegaPreF, omegaPreR, omegaPostF, omegaPostR;
/**
* Perform a reverse (backwards) smoothing sweep
*/
private final boolean reverse;
/**
* Jacobi damping parameter, between zero and one. If it equals zero, the
* method reduces to the standard aggregate multigrid method
*/
private final double omega;
/**
* Operating in transpose mode?
*/
private boolean transpose;
/**
* Sets up the algebraic multigrid preconditioner.
*
* @param omegaPreF
* Overrelaxation parameter in the forward sweep of the
* pre-smoothing
* @param omegaPreR
* Overrelaxation parameter in the backwards sweep of the
* pre-smoothing
* @param omegaPostF
* Overrelaxation parameter in the forward sweep of the
* post-smoothing
* @param omegaPostR
* Overrelaxation parameter in the backwards sweep of the
* post-smoothing
* @param nu1
* Number of pre-relaxations to perform
* @param nu2
* Number of post-relaxations to perform
* @param gamma
* Number of times to go to a coarser level
* @param min
* Smallest matrix size before using a direct solver
* @param omega
* Jacobi damping parameter, between zero and one. If it equals
* zero, the method reduces to the standard aggregate multigrid
* method
*/
public AMG(double omegaPreF, double omegaPreR, double omegaPostF,
double omegaPostR, int nu1, int nu2, int gamma, int min,
double omega) {
this.omegaPreF = omegaPreF;
this.omegaPreR = omegaPreR;
this.omegaPostF = omegaPostF;
this.omegaPostR = omegaPostR;
reverse = true;
this.nu1 = nu1;
this.nu2 = nu2;
this.gamma = gamma;
this.min = min;
this.omega = omega;
}
/**
* Sets up the algebraic multigrid preconditioner. Uses an SOR method,
* without the backward sweep in SSOR.
*
* @param omegaPre
* Overrelaxation parameter in the pre-smoothing
* @param omegaPost
* Overrelaxation parameter in the post-smoothing
* @param nu1
* Number of pre-relaxations to perform
* @param nu2
* Number of post-relaxations to perform
* @param gamma
* Number of times to go to a coarser level
* @param min
* Smallest matrix size before using a direct solver
* @param omega
* Jacobi damping parameter, between zero and one. If it equals
* zero, the method reduces to the standard aggregate multigrid
* method
*/
public AMG(double omegaPre, double omegaPost, int nu1, int nu2, int gamma,
int min, double omega) {
this.omegaPreF = omegaPre;
this.omegaPreR = omegaPre;
this.omegaPostF = omegaPost;
this.omegaPostR = omegaPost;
reverse = false;
this.nu1 = nu1;
this.nu2 = nu2;
this.gamma = gamma;
this.min = min;
this.omega = omega;
}
/**
* Sets up the algebraic multigrid preconditioner using some default
* parameters. In the presmoothing, omegaF=1 and
* omegaR=1.85, while in the postsmoothing,
* omegaF=1.85 and omegaR=1. Sets
* nu1=nu2=gamma=1, has a smallest matrix size of 40, and sets
* omega=2/3.
*/
public AMG() {
this(1, 1.85, 1.85, 1, 1, 1, 1, 40, 2. / 3);
}
public Vector apply(Vector b, Vector x) {
u[0].set(x);
f[0].set(b);
transpose = false;
cycle(0);
return x.set(u[0]);
}
public Vector transApply(Vector b, Vector x) {
u[0].set(x);
f[0].set(b);
transpose = true;
cycle(0);
return x.set(u[0]);
}
public void setMatrix(Matrix A) {
List Al = new LinkedList();
List Il = new LinkedList();
Al.add(new CompRowMatrix(A));
for (int k = 0; Al.get(k).numRows() > min; ++k) {
CompRowMatrix Af = Al.get(k);
double eps = 0.08 * Math.pow(0.5, k);
// Create the aggregates
Aggregator aggregator = new Aggregator(Af, eps);
// If no aggregates were created, no interpolation operator will be
// created, and the setup phase stops
if (aggregator.getAggregates().isEmpty())
break;
// Create an interpolation operator using smoothing. This also
// creates the Galerkin operator
Interpolator sa = new Interpolator(aggregator, Af, omega);
Al.add(sa.getGalerkinOperator());
Il.add(sa.getInterpolationOperator());
}
// Copy to array storage
m = Al.size();
if (m == 0)
throw new RuntimeException("Matrix too small for AMG");
I = new CompColMatrix[m - 1];
this.A = new CompRowMatrix[m - 1];
Il.toArray(I);
for (int i = 0; i < Al.size() - 1; ++i)
this.A[i] = Al.get(i);
// Create a LU decomposition of the smallest Galerkin matrix
DenseMatrix Ac = new DenseMatrix(Al.get(Al.size() - 1));
lu = new DenseLU(Ac.numRows(), Ac.numColumns());
lu.factor(Ac);
// Allocate vectors at each level
u = new DenseVector[m];
f = new DenseVector[m];
r = new DenseVector[m];
for (int k = 0; k < m; ++k) {
int n = Al.get(k).numRows();
u[k] = new DenseVector(n);
f[k] = new DenseVector(n);
r[k] = new DenseVector(n);
}
// Set up the SSOR relaxation schemes
preM = new SSOR[m - 1];
postM = new SSOR[m - 1];
for (int k = 0; k < m - 1; ++k) {
CompRowMatrix Ak = this.A[k];
preM[k] = new SSOR(Ak, reverse, omegaPreF, omegaPreR);
postM[k] = new SSOR(Ak, reverse, omegaPostF, omegaPostR);
preM[k].setMatrix(Ak);
postM[k].setMatrix(Ak);
}
}
/**
* Performs a multigrid cycle
*
* @param k
* Level to cycle at. Start by calling cycle(0)
*/
private void cycle(int k) {
if (k == m - 1)
directSolve();
else {
// Presmoothings
preRelax(k);
u[k + 1].zero();
// Compute the residual
A[k].multAdd(-1, u[k], r[k].set(f[k]));
// Restrict to the next coarser level
I[k].transMult(r[k], f[k + 1]);
// Recurse to next level
for (int i = 0; i < gamma; ++i)
cycle(k + 1);
// Add residual correction by prolongation
I[k].multAdd(u[k + 1], u[k]);
// Postsmoothings
postRelax(k);
}
}
/**
* Solves directly at the coarsest level
*/
private void directSolve() {
int k = m - 1;
u[k].set(f[k]);
DenseMatrix U = new DenseMatrix(u[k], false);
if (transpose)
lu.transSolve(U);
else
lu.solve(U);
}
/**
* Applies the relaxation scheme at the given level
*
* @param k
* Multigrid level
*/
private void preRelax(int k) {
for (int i = 0; i < nu1; ++i)
if (transpose)
preM[k].transApply(f[k], u[k]);
else
preM[k].apply(f[k], u[k]);
}
/**
* Applies the relaxation scheme at the given level
*
* @param k
* Multigrid level
*/
private void postRelax(int k) {
for (int i = 0; i < nu2; ++i)
if (transpose)
postM[k].transApply(f[k], u[k]);
else
postM[k].apply(f[k], u[k]);
}
/**
* Creates aggregates. These are disjoint sets, each of which represents one
* node at a coarser mesh by aggregating together a set of fine nodes
*/
private static class Aggregator {
/**
* The aggregates
*/
private List> C;
/**
* Diagonal indices into the sparse matrix
*/
private int[] diagind;
/**
* The strongly coupled node neighborhood of a given node
*/
private List> N;
/**
* Creates the aggregates
*
* @param A
* Sparse matrix
* @param eps
* Tolerance for selecting the strongly coupled node
* neighborhoods. Between zero and one.
*/
public Aggregator(CompRowMatrix A, double eps) {
diagind = findDiagonalIndices(A);
N = findNodeNeighborhood(A, diagind, eps);
/*
* Initialization. Remove isolated nodes from the aggregates
*/
boolean[] R = createInitialR(A);
/*
* Startup aggregation. Use disjoint strongly coupled neighborhoods
* as the initial aggregate approximation
*/
C = createInitialAggregates(N, R);
/*
* Enlargment of the aggregates. Add nodes to each aggregate based
* on how strongly connected the nodes are to a given aggregate
*/
C = enlargeAggregates(C, N, R);
/*
* Handling of the remenants. Put all remaining unallocated nodes
* into new aggregates defined by the intersection of N and R
*/
C = createFinalAggregates(C, N, R);
}
/**
* Gets the aggregates.
*/
public List> getAggregates() {
return C;
}
/**
* Returns the matrix diagonal indices. This is a by-product of the
* aggregation.
*/
public int[] getDiagonalIndices() {
return diagind;
}
/**
* Returns the strongly coupled node neighborhoods of a given node. This
* is a by-product of the aggregation.
*/
public List> getNodeNeighborhoods() {
return N;
}
/**
* Finds the diagonal indices of the matrix.
*/
private static int[] findDiagonalIndices(CompRowMatrix A) {
int[] rowptr = A.getRowPointers();
int[] colind = A.getColumnIndices();
int[] diagind = new int[A.numRows()];
for (int i = 0; i < A.numRows(); ++i) {
diagind[i] = no.uib.cipr.matrix.sparse.Arrays.binarySearch(
colind, i, rowptr[i], rowptr[i + 1]);
if (diagind[i] < 0)
throw new RuntimeException(
"Matrix is missing a diagonal entry on row "
+ (i + 1));
}
return diagind;
}
/**
* Finds the strongly coupled node neighborhoods.
*/
private List> findNodeNeighborhood(CompRowMatrix A,
int[] diagind, double eps) {
N = new ArrayList>(A.numRows());
int[] rowptr = A.getRowPointers();
int[] colind = A.getColumnIndices();
double[] data = A.getData();
for (int i = 0; i < A.numRows(); ++i) {
Set Ni = new HashSet();
double aii = data[diagind[i]];
for (int j = rowptr[i]; j < rowptr[i + 1]; ++j) {
double aij = data[j];
double ajj = data[diagind[colind[j]]];
if (Math.abs(aij) >= eps * Math.sqrt(aii * ajj))
Ni.add(colind[j]);
}
N.add(Ni);
}
return N;
}
/**
* Creates the initial R-set by including only the connected nodes.
*/
private static boolean[] createInitialR(CompRowMatrix A) {
boolean[] R = new boolean[A.numRows()];
int[] rowptr = A.getRowPointers();
int[] colind = A.getColumnIndices();
double[] data = A.getData();
for (int i = 0; i < A.numRows(); ++i) {
boolean hasOffDiagonal = false;
for (int j = rowptr[i]; j < rowptr[i + 1]; ++j)
if (colind[j] != i && data[j] != 0) {
hasOffDiagonal = true;
break;
}
R[i] = hasOffDiagonal;
}
return R;
}
/**
* Creates the initial aggregates
*/
private List> createInitialAggregates(
List> N, boolean[] R) {
C = new ArrayList>();
for (int i = 0; i < R.length; ++i) {
// Skip non-free nodes
if (!R[i])
continue;
// See if all nodes in the current N-set are free
boolean free = true;
for (int j : N.get(i))
free &= R[j];
// Create an aggregate out of N[i]
if (free) {
C.add(new HashSet(N.get(i)));
for (int j : N.get(i))
R[j] = false;
}
}
return C;
}
/**
* Enlarges the aggregates
*/
private static List> enlargeAggregates(List> C,
List> N, boolean[] R) {
// Contains the aggregates each node is coupled to
List> belong = new ArrayList>(R.length);
for (int i = 0; i < R.length; ++i)
belong.add(new ArrayList());
// Find which aggregate each node is coupled to. This is used for
// the intersection between Ni and Ck
for (int k = 0; k < C.size(); ++k)
for (int j : C.get(k))
belong.get(j).add(k);
// Number of nodes in the intersection between each C and Ni
int[] intersect = new int[C.size()];
for (int i = 0; i < R.length; ++i) {
// Skip non-free nodes
if (!R[i])
continue;
// Find the number of nodes intersecting Ni and every C, and
// keep a track on the largest overlap
Arrays.fill(intersect, 0);
int largest = 0, maxValue = 0;
for (int j : N.get(i))
// The k-index is to an aggregate coupled to node j
for (int k : belong.get(j)) {
intersect[k]++;
if (intersect[k] > maxValue) {
largest = k;
maxValue = intersect[largest];
}
}
// Add the node to the proper C-set, and mark it as used
// Also, check if the node actually does couple to a set
if (maxValue > 0) {
R[i] = false;
C.get(largest).add(i);
}
}
return C;
}
/**
* Creates final aggregates from the remaining unallocated nodes
*/
private static List> createFinalAggregates(List> C,
List> N, boolean[] R) {
for (int i = 0; i < R.length; ++i) {
// Skip non-free nodes
if (!R[i])
continue;
// Create new aggregate from the nodes in N[i] which are free
Set Cn = new HashSet();
for (int j : N.get(i))
if (R[j]) {
R[j] = false;
Cn.add(j);
}
if (!Cn.isEmpty())
C.add(Cn);
}
return C;
}
}
/**
* Creates interpolation (prolongation) operators using based on the
* aggregates. Can optionally smooth the aggregates.
*/
private static final class Interpolator {
/**
* The Galerkin coarse-space operator
*/
private CompRowMatrix Ac;
/**
* The interpolation (prolongation) matrix
*/
private CompColMatrix I;
/**
* Creates the interpolation (prolongation) and Galerkin operators.
*
* @param aggregator
* Aggregates
* @param A
* Matrix
* @param omega
* Jacobi damping parameter between zero and one. If zero, no
* smoothing is performed, and a faster algorithm for forming
* the Galerkin operator will be used.
*/
Interpolator(Aggregator aggregator, CompRowMatrix A, double omega) {
List> C = aggregator.getAggregates();
List> N = aggregator.getNodeNeighborhoods();
int[] diagind = aggregator.getDiagonalIndices();
// Create the tentative prolongation, in compressed form
int[] pt = createTentativeProlongation(C, A.numRows());
/*
* Apply Jacobi smoothing to the prolongator
*/
if (omega != 0) {
// Smooth the operator by a damped Jacobi method
List> P = createSmoothedProlongation(C, N,
A, diagind, omega, pt);
// Form a compressed column storage for the operator
I = createInterpolationMatrix(P, A.numRows());
// Create the Galerkin operator using a slow method
Ac = createGalerkinSlow(I, A);
}
/*
* Use the aggregates as-is
*/
else {
// Create the Galerkin operator using a fast method
Ac = createGalerkinFast(A, pt, C.size());
// Form an explicit interpolation operator
I = createInterpolationMatrix(pt, C.size());
}
}
/**
* Creates the tentative prolongation operator. Since the columns are
* all disjoint, and its entries are binary, it is possible to store it
* in a single array. Its length equals the number of fine nodes, and
* the entries are the indices to the corresponding aggregate (C-set).
*/
private static int[] createTentativeProlongation(List> C, int n) {
int[] pt = new int[n];
Arrays.fill(pt, -1);
for (int i = 0; i < C.size(); ++i)
for (int j : C.get(i))
pt[j] = i;
return pt;
}
/**
* Creates the Galerkin operator using the assumption of disjoint
* (non-smoothed) aggregates
*/
private static CompRowMatrix createGalerkinFast(CompRowMatrix A, int[] pt,
int c) {
int n = pt.length;
FlexCompRowMatrix Ac = new FlexCompRowMatrix(c, c);
int[] rowptr = A.getRowPointers();
int[] colind = A.getColumnIndices();
double[] data = A.getData();
for (int i = 0; i < n; ++i)
if (pt[i] != -1)
for (int j = rowptr[i]; j < rowptr[i + 1]; ++j)
if (pt[colind[j]] != -1)
Ac.add(pt[i], pt[colind[j]], data[j]);
return new CompRowMatrix(Ac);
}
/**
* Creates the interpolation (prolongation) matrix based on the smoothed
* aggregates
*/
private CompColMatrix createInterpolationMatrix(
List> P, int n) {
// Determine the sparsity pattern of I
int c = P.size();
int[][] nz = new int[c][];
for (int j = 0; j < c; ++j) {
Map Pj = P.get(j);
nz[j] = new int[Pj.size()];
int l = 0;
for (int k : Pj.keySet())
nz[j][l++] = k;
}
I = new CompColMatrix(n, c, nz);
// Populate it with numerical entries
for (int j = 0; j < c; ++j) {
Map Pj = P.get(j);
for (Map.Entry e : Pj.entrySet())
I.set(e.getKey(), j, e.getValue());
}
return I;
}
/**
* Creates the interpolation (prolongation) matrix based on the
* non-smoothed aggregates
*/
private static CompColMatrix createInterpolationMatrix(int[] pt, int c) {
FlexCompColMatrix If = new FlexCompColMatrix(pt.length, c);
for (int i = 0; i < pt.length; ++i)
if (pt[i] != -1)
If.set(i, pt[i], 1);
return new CompColMatrix(If);
}
/**
* Gets the interpolation (prolongation) operator
*/
public CompColMatrix getInterpolationOperator() {
return I;
}
/**
* Creates the smoothes interpolation (prolongation) operator by a
* single sweep of the damped Jacobi method
*/
private static List> createSmoothedProlongation(
List> C, List> N, CompRowMatrix A,
int[] diagind, double omega, int[] pt) {
int n = A.numRows(), c = C.size();
// Allocate the interpolation (prolongation) operator
// It is stored by columns, so the maps take row-indices as keys
List> P = new ArrayList>(
c);
for (int i = 0; i < c; ++i)
P.add(new HashMap());
int[] rowptr = A.getRowPointers();
int[] colind = A.getColumnIndices();
double[] data = A.getData();
double[] dot = new double[c];
// Apply the damped Jacobi smoother
for (int i = 0; i < n; ++i) {
if (pt[i] == -1)
continue;
Arrays.fill(dot, 0);
Set Ni = N.get(i);
// Calculate A*Pt, except for the diagonal
double weakAij = 0;
for (int j = rowptr[i]; j < rowptr[i + 1]; ++j) {
if (pt[colind[j]] == -1)
continue;
double aij = data[j];
// Off-diagonal, include only strong couplings, and add the
// weak couplings to the diagonal
if (aij != 0 && !Ni.contains(colind[j])) {
weakAij += aij;
continue;
}
dot[pt[colind[j]]] += aij;
}
// Subtract the weak couplings from the diagonal part of A*Pt
dot[pt[i]] -= weakAij;
// Scale by omega and the inverse of the diagonal (damping)
double scale = -omega / data[diagind[i]];
for (int j = 0; j < dot.length; ++j)
dot[j] *= scale;
// Set to (I-omega*D^{-1}*A)*Pt
dot[pt[i]]++;
// This has formed a whole row of P=(I-omega*D^{-1}*A)*Pt
// Store the non-zeros into the sparse structure
for (int j = 0; j < dot.length; ++j)
if (dot[j] != 0)
P.get(j).put(i, dot[j]);
}
return P;
}
/**
* Creates the entries of the Galerkin operator
* Ac = IT A I. This is a very time-consuming
* operation
*/
private static CompRowMatrix createGalerkinSlow(CompColMatrix I,
CompRowMatrix A) {
int n = I.numRows(), c = I.numColumns();
FlexCompRowMatrix Ac = new FlexCompRowMatrix(c, c);
double[] aiCol = new double[n];
double[] iCol = new double[n];
DenseVector aiV = new DenseVector(aiCol, false);
DenseVector iV = new DenseVector(iCol, false);
double[] itaiCol = new double[c];
DenseVector itaiV = new DenseVector(itaiCol, false);
int[] colptr = I.getColumnPointers();
int[] rowind = I.getRowIndices();
double[] Idata = I.getData();
for (int k = 0; k < c; ++k) {
// Expand column 'k' of I to dense storage
iV.zero();
for (int i = colptr[k]; i < colptr[k + 1]; ++i)
iCol[rowind[i]] = Idata[i];
// Form column 'k' of A*I
A.mult(iV, aiV);
// Form column 'k' of I'*A*I
I.transMult(aiV, itaiV);
// Store non-zeros into Ac
for (int i = 0; i < c; ++i)
if (itaiCol[i] != 0)
Ac.set(i, k, itaiCol[i]);
}
return new CompRowMatrix(Ac);
}
/**
* Gets the Galerkin operator
*/
CompRowMatrix getGalerkinOperator() {
return Ac;
}
}
}
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