2,041 research outputs found
A Lanczos Method for Approximating Composite Functions
We seek to approximate a composite function h(x) = g(f(x)) with a global
polynomial. The standard approach chooses points x in the domain of f and
computes h(x) at each point, which requires an evaluation of f and an
evaluation of g. We present a Lanczos-based procedure that implicitly
approximates g with a polynomial of f. By constructing a quadrature rule for
the density function of f, we can approximate h(x) using many fewer evaluations
of g. The savings is particularly dramatic when g is much more expensive than f
or the dimension of x is large. We demonstrate this procedure with two
numerical examples: (i) an exponential function composed with a rational
function and (ii) a Navier-Stokes model of fluid flow with a scalar input
parameter that depends on multiple physical quantities
Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method
Pipelined Krylov subspace methods (also referred to as communication-hiding
methods) have been proposed in the literature as a scalable alternative to
classic Krylov subspace algorithms for iteratively computing the solution to a
large linear system in parallel. For symmetric and positive definite system
matrices the pipelined Conjugate Gradient method outperforms its classic
Conjugate Gradient counterpart on large scale distributed memory hardware by
overlapping global communication with essential computations like the
matrix-vector product, thus hiding global communication. A well-known drawback
of the pipelining technique is the (possibly significant) loss of numerical
stability. In this work a numerically stable variant of the pipelined Conjugate
Gradient algorithm is presented that avoids the propagation of local rounding
errors in the finite precision recurrence relations that construct the Krylov
subspace basis. The multi-term recurrence relation for the basis vector is
replaced by two-term recurrences, improving stability without increasing the
overall computational cost of the algorithm. The proposed modification ensures
that the pipelined Conjugate Gradient method is able to attain a highly
accurate solution independently of the pipeline length. Numerical experiments
demonstrate a combination of excellent parallel performance and improved
maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm.
This work thus resolves one of the major practical restrictions for the
useability of pipelined Krylov subspace methods.Comment: 15 pages, 5 figures, 1 table, 2 algorithm
Lanczos algorithm with Matrix Product States for dynamical correlation functions
The density-matrix renormalization group (DMRG) algorithm can be adapted to
the calculation of dynamical correlation functions in various ways which all
represent compromises between computational efficiency and physical accuracy.
In this paper we reconsider the oldest approach based on a suitable
Lanczos-generated approximate basis and implement it using matrix product
states (MPS) for the representation of the basis states. The direct use of
matrix product states combined with an ex-post reorthogonalization method
allows to avoid several shortcomings of the original approach, namely the
multi-targeting and the approximate representation of the Hamiltonian inherent
in earlier Lanczos-method implementations in the DMRG framework, and to deal
with the ghost problem of Lanczos methods, leading to a much better convergence
of the spectral weights and poles. We present results for the dynamic spin
structure factor of the spin-1/2 antiferromagnetic Heisenberg chain. A
comparison to Bethe ansatz results in the thermodynamic limit reveals that the
MPS-based Lanczos approach is much more accurate than earlier approaches at
minor additional numerical cost.Comment: final version 11 pages, 11 figure
Recycling BiCGSTAB with an Application to Parametric Model Order Reduction
Krylov subspace recycling is a process for accelerating the convergence of
sequences of linear systems. Based on this technique, the recycling BiCG
algorithm has been developed recently. Here, we now generalize and extend this
recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving
sequences of dual linear systems, while the focus here is on efficiently
solving sequences of single linear systems (assuming non-symmetric matrices for
both recycling BiCG and recycling BiCGSTAB).
As compared with other methods for solving sequences of single linear systems
with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based
recycling algorithms, like recycling BiCGSTAB, have the advantage that they
involve a short-term recurrence, and hence, do not suffer from storage issues
and are also cheaper with respect to the orthogonalizations.
We modify the BiCGSTAB algorithm to use a recycle space, which is built from
left and right approximate invariant subspaces. Using our algorithm for a
parametric model order reduction example gives good results. We show about 40%
savings in the number of matrix-vector products and about 35% savings in
runtime.Comment: 18 pages, 5 figures, Extended version of Max Planck Institute report
(MPIMD/13-21
New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms
Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all
Solving large sparse eigenvalue problems on supercomputers
An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed
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