Idrsstabl

unsupported/Eigen/IterativeSolvers

@@ -25,6 +25,7 @@
   *  - a BiCGSTAB(L) implementation
   *  - a DGMRES implementation
   *  - a MINRES implementation
+  *  - a IDRSTABL implementation
   *
   * Choosing the best solver for solving \c A \c x = \c b depends a lot on the preconditioner chosen as well as the properties of \c A. The following flowchart might help you.
   * \dot width=50%
@@ -89,6 +90,7 @@
 #include "src/IterativeSolvers/MINRES.h"
 #include "src/IterativeSolvers/IDRS.h"
 #include "src/IterativeSolvers/BiCGSTABL.h"
+#include "src/IterativeSolvers/IDRSTABL.h"
 
 #include "../../Eigen/src/Core/util/ReenableStupidWarnings.h"
 

unsupported/Eigen/src/IterativeSolvers/IDRSTABL.h

@@ -0,0 +1,476 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2020 Chris Schoutrop <c.e.m.schoutrop@tue.nl>
+// Copyright (C) 2020 Mischa Senders <m.j.senders@student.tue.nl>
+// Copyright (C) 2020 Lex Kuijpers <l.kuijpers@student.tue.nl>
+// Copyright (C) 2020 Jens Wehner <j.wehner@esciencecenter.nl>
+// Copyright (C) 2020 Jan van Dijk <j.v.dijk@tue.nl>
+// Copyright (C) 2020 Adithya Vijaykumar
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+/*
+
+The IDR(S)Stab(L) method is a combination of IDR(S) and BiCGStab(L)
+
+This implementation of IDRSTABL is based on
+1. Aihara, K., Abe, K., & Ishiwata, E. (2014). A variant of IDRstab with
+reliable update strategies for solving sparse linear systems. Journal of
+Computational and Applied Mathematics, 259, 244-258.
+   doi:10.1016/j.cam.2013.08.028
+                2. Aihara, K., Abe, K., & Ishiwata, E. (2015). Preconditioned
+IDRSTABL Algorithms for Solving Nonsymmetric Linear Systems. International
+Journal of Applied Mathematics, 45(3).
+                3. Saad, Y. (2003). Iterative Methods for Sparse Linear Systems:
+Second Edition. Philadelphia, PA: SIAM.
+                4. Sonneveld, P., & Van Gijzen, M. B. (2009). IDR(s): A Family
+of Simple and Fast Algorithms for Solving Large Nonsymmetric Systems of Linear
+Equations. SIAM Journal on Scientific Computing, 31(2), 1035-1062.
+   doi:10.1137/070685804
+                5. Sonneveld, P. (2012). On the convergence behavior of IDR (s)
+and related methods. SIAM Journal on Scientific Computing, 34(5), A2576-A2598.
+
+    Right-preconditioning based on Ref. 3 is implemented here.
+*/
+
+#ifndef EIGEN_IDRSTABL_H
+#define EIGEN_IDRSTABL_H
+
+namespace Eigen {
+
+namespace internal {
+
+template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
+bool idrstabl(const MatrixType &mat, const Rhs &rhs, Dest &x, const Preconditioner &precond, Index &iters,
+              typename Dest::RealScalar &tol_error, Index L, Index S) {
+  /*
+    Setup and type definitions.
+  */
+  using numext::abs;
+  using numext::sqrt;
+  typedef typename Dest::Scalar Scalar;
+  typedef typename Dest::RealScalar RealScalar;
+  typedef Matrix<Scalar, Dynamic, 1> VectorType;
+  typedef Matrix<Scalar, Dynamic, Dynamic, ColMajor> DenseMatrixType;
+
+  const Index N = x.rows();
+
+  Index k = 0;  // Iteration counter
+  const Index maxIters = iters;
+
+  const RealScalar rhs_norm = rhs.stableNorm();
+  const RealScalar tol = tol_error * rhs_norm;
+
+  if (rhs_norm == 0) {
+    /*
+      If b==0, then the exact solution is x=0.
+      rhs_norm is needed for other calculations anyways, this exit is a freebie.
+    */
+    x.setZero();
+    tol_error = 0.0;
+    return true;
+  }
+  // Construct decomposition objects beforehand.
+  FullPivLU<DenseMatrixType> lu_solver;
+
+  if (S >= N || L >= N) {
+    /*
+      The matrix is very small, or the choice of L and S is very poor
+      in that case solving directly will be best.
+    */
+    lu_solver.compute(DenseMatrixType(mat));
+    x = lu_solver.solve(rhs);
+    tol_error = (rhs - mat * x).stableNorm() / rhs_norm;
+    return true;
+  }
+
+  // Define maximum sizes to prevent any reallocation later on.
+  DenseMatrixType u(N, L + 1);
+  DenseMatrixType r(N, L + 1);
+
+  DenseMatrixType V(N * (L + 1), S);
+
+  VectorType alpha(S);
+  VectorType gamma(L);
+  VectorType update(N);
+
+  /*
+    Main IDRSTABL algorithm
+  */
+  // Set up the initial residual
+  VectorType x0 = x;
+  r.col(0) = rhs - mat * x;
+  x.setZero();  // The final solution will be x0+x
+
+  tol_error = r.col(0).stableNorm();
+
+  // FOM = Full orthogonalisation method
+  DenseMatrixType h_FOM = DenseMatrixType::Zero(S, S - 1);
+
+  // Construct an initial U matrix of size N x S
+  DenseMatrixType U(N * (L + 1), S);
+  for (Index col_index = 0; col_index < S; ++col_index) {
+    // Arnoldi-like process to generate a set of orthogonal vectors spanning
+    // {u,A*u,A*A*u,...,A^(S-1)*u}. This construction can be combined with the
+    // Full Orthogonalization Method (FOM) from Ref.3 to provide a possible
+    // early exit with no additional MV.
+    if (col_index != 0) {
+      /*
+      Modified Gram-Schmidt strategy:
+      */
+      VectorType w = mat * precond.solve(u.col(0));
+      for (Index i = 0; i < col_index; ++i) {
+        auto v = U.col(i).head(N);
+        h_FOM(i, col_index - 1) = v.dot(w);
+        w -= h_FOM(i, col_index - 1) * v;
+      }
+      u.col(0) = w;
+      h_FOM(col_index, col_index - 1) = u.col(0).stableNorm();
+
+      if (abs(h_FOM(col_index, col_index - 1)) != RealScalar(0)) {
+        /*
+        This only happens if u is NOT exactly zero. In case it is exactly zero
+        it would imply that that this u has no component in the direction of the
+        current residual.
+
+        By then setting u to zero it will not contribute any further (as it
+        should). Whereas attempting to normalize results in division by zero.
+
+        Such cases occur if:
+        1. The basis of dimension <S is sufficient to exactly solve the linear
+        system. I.e. the current residual is in span{r,Ar,...A^{m-1}r}, where
+        (m-1)<=S.
+        2. Two vectors vectors generated from r, Ar,... are (numerically)
+        parallel.
+
+        In case 1, the exact solution to the system can be obtained from the
+        "Full Orthogonalization Method" (Algorithm 6.4 in the book of Saad),
+        without any additional MV.
+
+        Contrary to what one would suspect, the comparison with ==0.0 for
+        floating-point types is intended here. Any arbritary non-zero u is fine
+        to continue, however if u contains either NaN or Inf the algorithm will
+        break down.
+        */
+        u.col(0) /= h_FOM(col_index, col_index - 1);
+      }
+    } else {
+      u.col(0) = r.col(0);
+      u.col(0).normalize();
+    }
+
+    U.col(col_index).head(N) = u.col(0);
+  }
+
+  if (S > 1) {
+    // Check for early FOM exit.
+    Scalar beta = r.col(0).stableNorm();
+    VectorType e1 = VectorType::Zero(S - 1);
+    e1(0) = beta;
+    lu_solver.compute(h_FOM.topLeftCorner(S - 1, S - 1));
+    VectorType y = lu_solver.solve(e1);
+    VectorType x2 = x + U.topLeftCorner(N, S - 1) * y;
+
+    // Using proposition 6.7 in Saad, one MV can be saved to calculate the
+    // residual
+    RealScalar FOM_residual = (h_FOM(S - 1, S - 2) * y(S - 2) * U.col(S - 1).head(N)).stableNorm();
+
+    if (FOM_residual < tol) {
+      // Exit, the FOM algorithm was already accurate enough
+      iters = k;
+      // Convert back to the unpreconditioned solution
+      x = precond.solve(x2);
+      // x contains the updates to x0, add those back to obtain the solution
+      x += x0;
+      tol_error = FOM_residual / rhs_norm;
+      return true;
+    }
+  }
+
+  /*
+    Select an initial (N x S) matrix R0.
+    1. Generate random R0, orthonormalize the result.
+    2. This results in R0, however to save memory and compute we only need the
+    adjoint of R0. This is given by the matrix R_T.\ Additionally, the matrix
+    (mat.adjoint()*R_tilde).adjoint()=R_tilde.adjoint()*mat by the
+    anti-distributivity property of the adjoint. This results in AR_T, which is
+    constant if R_T does not have to be regenerated and can be precomputed.
+    Based on reference 4, this has zero probability in exact arithmetic.
+  */
+
+  // Original IDRSTABL and Kensuke choose S random vectors:
+  const HouseholderQR<DenseMatrixType> qr(DenseMatrixType::Random(N, S));
+  DenseMatrixType R_T = (qr.householderQ() * DenseMatrixType::Identity(N, S)).adjoint();
+  DenseMatrixType AR_T = DenseMatrixType(R_T * mat);
+
+  // Pre-allocate sigma.
+  DenseMatrixType sigma(S, S);
+
+  bool reset_while = false;  // Should the while loop be reset for some reason?
+
+  while (k < maxIters) {
+    for (Index j = 1; j <= L; ++j) {
+      /*
+        The IDR Step
+      */
+      // Construction of the sigma-matrix, and the decomposition of sigma.
+      for (Index i = 0; i < S; ++i) {
+        sigma.col(i).noalias() = AR_T * precond.solve(U.block(N * (j - 1), i, N, 1));
+      }
+
+      lu_solver.compute(sigma);
+      // Obtain the update coefficients alpha
+      if (j == 1) {
+        // alpha=inverse(sigma)*(R_T*r_0);
+        alpha.noalias() = lu_solver.solve(R_T * r.col(0));
+      } else {
+        // alpha=inverse(sigma)*(AR_T*r_{j-2})
+        alpha.noalias() = lu_solver.solve(AR_T * precond.solve(r.col(j - 2)));
+      }
+
+      // Obtain new solution and residual from this update
+      update.noalias() = U.topRows(N) * alpha;
+      r.col(0) -= mat * precond.solve(update);
+      x += update;
+
+      for (Index i = 1; i <= j - 2; ++i) {
+        // This only affects the case L>2
+        r.col(i) -= U.block(N * (i + 1), 0, N, S) * alpha;
+      }
+      if (j > 1) {
+        // r=[r;A*r_{j-2}]
+        r.col(j - 1).noalias() = mat * precond.solve(r.col(j - 2));
+      }
+      tol_error = r.col(0).stableNorm();
+
+      if (tol_error < tol) {
+        // If at this point the algorithm has converged, exit.
+        reset_while = true;
+        break;
+      }
+
+      bool break_normalization = false;
+      for (Index q = 1; q <= S; ++q) {
+        if (q == 1) {
+          // u = r;
+          u.leftCols(j + 1) = r.leftCols(j + 1);
+        } else {
+          // u=[u_1;u_2;...;u_j]
+          u.leftCols(j) = u.middleCols(1, j);
+        }
+
+        // Obtain the update coefficients beta implicitly
+        // beta=lu_sigma.solve(AR_T * u.block(N * (j - 1), 0, N, 1)
+        u.reshaped().head(u.rows() * j) -= U.topRows(N * j) * lu_solver.solve(AR_T * precond.solve(u.col(j - 1)));
+
+        // u=[u;Au_{j-1}]
+        u.col(j).noalias() = mat * precond.solve(u.col(j - 1));
+
+        // Orthonormalize u_j to the columns of V_j(:,1:q-1)
+        if (q > 1) {
+          /*
+          Modified Gram-Schmidt-like procedure to make u orthogonal to the
+          columns of V from Ref. 1.
+
+          The vector mu from Ref. 1 is obtained implicitly:
+          mu=V.block(N * j, 0, N, q - 1).adjoint() * u.block(N * j, 0, N, 1).
+          */
+          for (Index i = 0; i <= q - 2; ++i) {
+            auto v = V.col(i).segment(N * j, N);
+            Scalar h = v.squaredNorm();
+            h = v.dot(u.col(j)) / h;
+            u.reshaped().head(u.rows() * (j + 1)) -= h * V.block(0, i, N * (j + 1), 1);
+          }
+        }
+        // Normalize u and assign to a column of V
+        Scalar normalization_constant = u.col(j).stableNorm();
+        //  If u is exactly zero, this will lead to a NaN. Small, non-zero u is
+        //  fine.
+        if (normalization_constant == RealScalar(0.0)) {
+          break_normalization = true;
+          break;
+        } else {
+          u.leftCols(j + 1) /= normalization_constant;
+        }
+
+        V.block(0, q - 1, N * (j + 1), 1).noalias() = u.reshaped().head(u.rows() * (j + 1));
+      }
+
+      if (break_normalization == false) {
+        U = V;
+      }
+    }
+    if (reset_while) {
+      break;
+    }
+
+    // r=[r;mat*r_{L-1}]
+    r.col(L).noalias() = mat * precond.solve(r.col(L - 1));
+
+    /*
+            The polynomial step
+    */
+    ColPivHouseholderQR<DenseMatrixType> qr_solver(r.rightCols(L));
+    gamma.noalias() = qr_solver.solve(r.col(0));
+
+    // Update solution and residual using the "minimized residual coefficients"
+    update.noalias() = r.leftCols(L) * gamma;
+    x += update;
+    r.col(0) -= mat * precond.solve(update);
+
+    // Update iteration info
+    ++k;
+    tol_error = r.col(0).stableNorm();
+
+    if (tol_error < tol) {
+      // Slightly early exit by moving the criterion before the update of U,
+      // after the main while loop the result of that calculation would not be
+      // needed.
+      break;
+    }
+
+    /*
+    U=U0-sum(gamma_j*U_j)
+    Consider the first iteration. Then U only contains U0, so at the start of
+    the while-loop U should be U0. Therefore only the first N rows of U have to
+    be updated.
+    */
+    for (Index i = 1; i <= L; ++i) {
+      U.topRows(N) -= U.block(N * i, 0, N, S) * gamma(i - 1);
+    }
+  }
+
+  /*
+          Exit after the while loop terminated.
+  */
+  iters = k;
+  // Convert back to the unpreconditioned solution
+  x = precond.solve(x);
+  // x contains the updates to x0, add those back to obtain the solution
+  x += x0;
+  tol_error = tol_error / rhs_norm;
+  return true;
+}
+
+}  // namespace internal
+
+template <typename MatrixType_, typename Preconditioner_ = DiagonalPreconditioner<typename MatrixType_::Scalar>>
+class IDRSTABL;
+
+namespace internal {
+
+template <typename MatrixType_, typename Preconditioner_>
+struct traits<IDRSTABL<MatrixType_, Preconditioner_>> {
+  typedef MatrixType_ MatrixType;
+  typedef Preconditioner_ Preconditioner;
+};
+
+}  // namespace internal
+
+/** \ingroup IterativeLinearSolvers_Module
+ * \brief The IDR(s)STAB(l) is a combination of IDR(s) and BiCGSTAB(l). It is a
+ * short-recurrences Krylov method for sparse square problems. It can outperform
+ * both IDR(s) and BiCGSTAB(l). IDR(s)STAB(l) generally closely follows the
+ * optimal GMRES convergence in terms of the number of Matrix-Vector products.
+ * However, without the increasing cost per iteration of GMRES. IDR(s)STAB(l) is
+ * suitable for both indefinite systems and systems with complex eigenvalues.
+ *
+ * This class allows solving for A.x = b sparse linear problems. The vectors x
+ * and b can be either dense or sparse.
+ *
+ * \tparam MatrixType_ the type of the sparse matrix A, can be a dense or a
+ * sparse matrix. \tparam Preconditioner_ the type of the preconditioner.
+ * Default is DiagonalPreconditioner
+ *
+ * \implsparsesolverconcept
+ *
+ * The maximum number of iterations and tolerance value can be controlled via
+ * the setMaxIterations() and setTolerance() methods. The defaults are the size
+ * of the problem for the maximum number of iterations and
+ * NumTraits<Scalar>::epsilon() for the tolerance.
+ *
+ * The tolerance is the maximum relative residual error: |Ax-b|/|b| for which
+ * the linear system is considered solved.
+ *
+ * \b Performance: When using sparse matrices, best performance is achieved for
+ * a row-major sparse matrix format. Moreover, in this case multi-threading can
+ * be exploited if the user code is compiled with OpenMP enabled. See \ref
+ * TopicMultiThreading for details.
+ *
+ * By default the iterations start with x=0 as an initial guess of the solution.
+ * One can control the start using the solveWithGuess() method.
+ *
+ * IDR(s)STAB(l) can also be used in a matrix-free context, see the following
+ * \link MatrixfreeSolverExample example \endlink.
+ *
+ * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
+ */
+
+template <typename MatrixType_, typename Preconditioner_>
+class IDRSTABL : public IterativeSolverBase<IDRSTABL<MatrixType_, Preconditioner_>> {
+  typedef IterativeSolverBase<IDRSTABL> Base;
+  using Base::m_error;
+  using Base::m_info;
+  using Base::m_isInitialized;
+  using Base::m_iterations;
+  using Base::matrix;
+  Index m_L;
+  Index m_S;
+
+ public:
+  typedef MatrixType_ MatrixType;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef Preconditioner_ Preconditioner;
+
+ public:
+  /** Default constructor. */
+  IDRSTABL() : m_L(2), m_S(4) {}
+
+  /**   Initialize the solver with matrix \a A for further \c Ax=b solving.
+
+  This constructor is a shortcut for the default constructor followed
+  by a call to compute().
+
+  \warning this class stores a reference to the matrix A as well as some
+  precomputed values that depend on it. Therefore, if \a A is changed
+  this class becomes invalid. Call compute() to update it with the new
+  matrix A, or modify a copy of A.
+          */
+  template <typename MatrixDerived>
+  explicit IDRSTABL(const EigenBase<MatrixDerived> &A) : Base(A.derived()), m_L(2), m_S(4) {}
+
+  /** \internal */
+  /**     Loops over the number of columns of b and does the following:
+                                  1. sets the tolerance and maxIterations
+                                  2. Calls the function that has the core solver
+     routine
+  */
+  template <typename Rhs, typename Dest>
+  void _solve_vector_with_guess_impl(const Rhs &b, Dest &x) const {
+    m_iterations = Base::maxIterations();
+    m_error = Base::m_tolerance;
+    bool ret = internal::idrstabl(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error, m_L, m_S);
+
+    m_info = (!ret) ? NumericalIssue : m_error <= 10 * Base::m_tolerance ? Success : NoConvergence;
+  }
+
+  /** Sets the parameter L, indicating the amount of minimize residual steps are
+   * used. */
+  void setL(Index L) {
+    eigen_assert(L >= 1 && "L needs to be positive");
+    m_L = L;
+  }
+  /** Sets the parameter S, indicating the dimension of the shadow residual
+   * space.. */
+  void setS(Index S) {
+    eigen_assert(S >= 1 && "S needs to be positive");
+    m_S = S;
+  }
+};
+
+}  // namespace Eigen
+
+#endif /* EIGEN_IDRSTABL_H */

unsupported/test/CMakeLists.txt

@@ -100,6 +100,7 @@ ei_add_test(dgmres)
 ei_add_test(minres)
 ei_add_test(idrs)
 ei_add_test(bicgstabl)
+ei_add_test(idrstabl)
 ei_add_test(levenberg_marquardt)
 ei_add_test(kronecker_product)
 ei_add_test(bessel_functions)

unsupported/test/idrstabl.cpp

@@ -0,0 +1,28 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2011 Gael Guennebaud <g.gael@free.fr>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#include "../../test/sparse_solver.h"
+#include <unsupported/Eigen/IterativeSolvers>
+
+template <typename T>
+void test_idrstabl_T() {
+  IDRSTABL<SparseMatrix<T>, DiagonalPreconditioner<T> > idrstabl_colmajor_diag;
+  IDRSTABL<SparseMatrix<T>, IncompleteLUT<T> > idrstabl_colmajor_ilut;
+
+  idrstabl_colmajor_diag.setTolerance(NumTraits<T>::epsilon() * 4);
+  idrstabl_colmajor_ilut.setTolerance(NumTraits<T>::epsilon() * 4);
+
+  CALL_SUBTEST(check_sparse_square_solving(idrstabl_colmajor_diag));
+  CALL_SUBTEST(check_sparse_square_solving(idrstabl_colmajor_ilut));
+}
+
+EIGEN_DECLARE_TEST(idrstabl) {
+  CALL_SUBTEST_1((test_idrstabl_T<double>()));
+  CALL_SUBTEST_2((test_idrstabl_T<std::complex<double> >()));
+}