144 research outputs found
MONOTONE CONVERGENCE OF THE LANCZOS APPROXIMATIONS TO MATRIX FUNCTIONS OF HERMITIAN MATRICES
When A is a Hermitian matrix, the action f(A)b of a matrix function f(A) on a vector b can efficiently be approximated via the Lanczos method. In this note we use M-matrix theory to establish that the 2-norm of the error of the sequence of approximations is monotonically decreasing if f is a Stieltjes transform and A is positive definite. We discuss the relation of our approach to a recent, more general monotonicity result of Druskin for Laplace transforms. We also extend the class of functions to certain product type functions. This yields, for example, monotonicity when approximating sign(A)b with A indefinite if the Lanczos method is performed for A² rather than A
Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator
In lattice QCD computations a substantial amount of work is spent in solving
discretized versions of the Dirac equation. Conventional Krylov solvers show
critical slowing down for large system sizes and physically interesting
parameter regions. We present a domain decomposition adaptive algebraic
multigrid method used as a precondtioner to solve the "clover improved" Wilson
discretization of the Dirac equation. This approach combines and improves two
approaches, namely domain decomposition and adaptive algebraic multigrid, that
have been used seperately in lattice QCD before. We show in extensive numerical
test conducted with a parallel production code implementation that considerable
speed-up over conventional Krylov subspace methods, domain decomposition
methods and other hierarchical approaches for realistic system sizes can be
achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to
mixed-precision odd-even preconditioned BiCGStab. Results of numerical
experiments changed slightly due to more systematic use of odd-even
preconditionin
Many Masses on One Stroke: Economic Computation of Quark Propagators
The computational effort in the calculation of Wilson fermion quark
propagators in Lattice Quantum Chromodynamics can be considerably reduced by
exploiting the Wilson fermion matrix structure in inversion algorithms based on
the non-symmetric Lanczos process. We consider two such methods: QMR (quasi
minimal residual) and BCG (biconjugate gradients). Based on the decomposition
of the Wilson mass matrix, using QMR, one can carry
out inversions on a {\em whole} trajectory of masses simultaneously, merely at
the computational expense of a single propagator computation. In other words,
one has to compute the propagator corresponding to the lightest mass only,
while all the heavier masses are given for free, at the price of extra storage.
Moreover, the symmetry can be used to cut
the computational effort in QMR and BCG by a factor of two. We show that both
methods then become---in the critical regime of small quark
masses---competitive to BiCGStab and significantly better than the standard MR
method, with optimal relaxation factor, and CG as applied to the normal
equations.Comment: 17 pages, uuencoded compressed postscrip
Aggregation-based Multilevel Methods for Lattice QCD
In Lattice QCD computations a substantial amount of work is spent in solving
the Dirac equation. In the recent past it has been observed that conventional
Krylov solvers tend to critically slow down for large lattices and small quark
masses. We present a Schwarz alternating procedure (SAP) multilevel method as a
solver for the Clover improved Wilson discretization of the Dirac equation.
This approach combines two components (SAP and algebraic multigrid) that have
separately been used in lattice QCD before. In combination with a bootstrap
setup procedure we show that considerable speed-up over conventional Krylov
subspace methods for realistic configurations can be achieved.Comment: Talk presented at the XXIX International Symposium on Lattice Field
Theory, July 10-16, 2011, Lake Tahoe, Californi
Deflated Multigrid Multilevel Monte Carlo
In lattice QCD, the trace of the inverse of the discretized Dirac operator
appears in the disconnected fermion loop contribution to an observable. As
simulation methods get more and more precise, these contributions become
increasingly important. Hence, we consider here the problem of computing the
trace , with the Dirac operator. The Hutchinson
method, which is very frequently used to stochastically estimate the trace of a
function of a matrix, approximates the trace as the average over estimates of
the form , with the entries of the vector following a
certain probability distribution. For samples, the accuracy is
. In recent work, we have introduced multigrid
multilevel Monte Carlo: having a multigrid hierarchy with operators ,
and , for level , we can rewrite the trace
via a telescopic sum with difference-levels, written in
terms of the aforementioned operators and with a reduced variance. We have seen
significant reductions in the variance and the total work with respect to
exactly deflated Hutchinson. In this work, we explore the use of exact
deflation in combination with the multigrid multilevel Monte Carlo method, and
demonstrate how this leads to both algorithmic and computational gains
Operator splitting for semi-explicit differential-algebraic equations and port-Hamiltonian DAEs
Operator splitting methods allow to split the operator describing a complex
dynamical system into a sequence of simpler subsystems and treat each part
independently. In the modeling of dynamical problems, systems of (possibly
coupled) differential-algebraic equations (DAEs) arise. This motivates the
application of operator splittings which are aware of the various structural
forms of DAEs. Here, we present an approach for the splitting of coupled
index-1 DAE as well as for the splitting of port-Hamiltonian DAEs, taking
advantage of the energy-conservative and energy-dissipative parts. We provide
numerical examples illustrating our second-order convergence results
Analysis of stochastic probing methods for estimating the trace of functions of sparse symmetric matrices
We consider the problem of estimating the trace of a matrix function .
In certain situations, in particular if cannot be well approximated by a
low-rank matrix, combining probing methods based on graph colorings with
stochastic trace estimation techniques can yield accurate approximations at
moderate cost. So far, such methods have not been thoroughly analyzed, though,
but were rather used as efficient heuristics by practitioners. In this
manuscript, we perform a detailed analysis of stochastic probing methods and,
in particular, expose conditions under which the expected approximation error
in the stochastic probing method scales more favorably with the dimension of
the matrix than the error in non-stochastic probing. Extending results from [E.
Aune, D. P. Simpson, J. Eidsvik, Parameter estimation in high dimensional
Gaussian distributions, Stat. Comput., 24, pp. 247--263, 2014], we also
characterize situations in which using just one stochastic vector is always --
not only in expectation -- better than the deterministic probing method.
Several numerical experiments illustrate our theory and compare with existing
methods
Numerical Methods for the QCD Overlap Operator:III. Nested Iterations
The numerical and computational aspects of chiral fermions in lattice quantum
chromodynamics are extremely demanding. In the overlap framework, the
computation of the fermion propagator leads to a nested iteration where the
matrix vector multiplications in each step of an outer iteration have to be
accomplished by an inner iteration; the latter approximates the product of the
sign function of the hermitian Wilson fermion matrix with a vector. In this
paper we investigate aspects of this nested paradigm. We examine several Krylov
subspace methods to be used as an outer iteration for both propagator
computations and the Hybrid Monte-Carlo scheme. We establish criteria on the
accuracy of the inner iteration which allow to preserve an a priori given
precision for the overall computation. It will turn out that the accuracy of
the sign function can be relaxed as the outer iteration proceeds. Furthermore,
we consider preconditioning strategies, where the preconditioner is built upon
an inaccurate approximation to the sign function. Relaxation combined with
preconditioning allows for considerable savings in computational efforts up to
a factor of 4 as our numerical experiments illustrate. We also discuss the
possibility of projecting the squared overlap operator into one chiral sector.Comment: 33 Pages; citations adde
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