216 research outputs found
Efficient approximation of functions of some large matrices by partial fraction expansions
Some important applicative problems require the evaluation of functions
of large and sparse and/or \emph{localized} matrices . Popular and
interesting techniques for computing and , where
is a vector, are based on partial fraction expansions. However,
some of these techniques require solving several linear systems whose matrices
differ from by a complex multiple of the identity matrix for computing
or require inverting sequences of matrices with the same
characteristics for computing . Here we study the use and the
convergence of a recent technique for generating sequences of incomplete
factorizations of matrices in order to face with both these issues. The
solution of the sequences of linear systems and approximate matrix inversions
above can be computed efficiently provided that shows certain decay
properties. These strategies have good parallel potentialities. Our claims are
confirmed by numerical tests
Guaranteed passive parameterized model order reduction of the partial element equivalent circuit (PEEC) method
The decrease of IC feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the system under study as a function of design parameters, such as geometrical and substrate features, in addition to frequency (or time). Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters. We propose an innovative PMOR technique applicable to PEEC analysis, which combines traditional passivity-preserving model order reduction methods and positive interpolation schemes. It is able to provide parametric reduced-order models, stable, and passive by construction over a user-defined range of design parameter values. Numerical examples validate the proposed approach
A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators
We develop an algorithm for computing the solution of a large system of
linear ordinary differential equations (ODEs) with polynomial inhomogeneity.
This is equivalent to computing the action of a certain matrix function on the
vector representing the initial condition. The matrix function is a linear
combination of the matrix exponential and other functions related to the
exponential (the so-called phi-functions). Such computations are the major
computational burden in the implementation of exponential integrators, which
can solve general ODEs. Our approach is to compute the action of the matrix
function by constructing a Krylov subspace using Arnoldi or Lanczos iteration
and projecting the function on this subspace. This is combined with
time-stepping to prevent the Krylov subspace from growing too large. The
algorithm is fully adaptive: it varies both the size of the time steps and the
dimension of the Krylov subspace to reach the required accuracy. We implement
this algorithm in the Matlab function phipm and we give instructions on how to
obtain and use this function. Various numerical experiments show that the phipm
function is often significantly more efficient than the state-of-the-art.Comment: 20 pages, 3 colour figures, code available from
http://www.maths.leeds.ac.uk/~jitse/software.html . v2: Various changes to
improve presentation as suggested by the refere
Reliable Fast Frequency Sweep for Microwave Devices via the Reduced Basis Method
International audienceIn this paper, a reduced basis approximation-based model order reduction for fast and reliable sweep in time-harmonic Maxwell's equations is detailed. Contrary to what one may expect by observing the frequency response of different microwave circuits, the electromagnetic field within these devices does not drastically vary as frequency changes in a band of interest. Thus, instead of using computationally inefficient, large dimension, numerical approximations such as finite element or boundary element methods for each frequency in the band, the point in here is to approximate the dynamics of the electromagnetic field itself as frequency changes. A much lower dimension, reduced basis approximation sorts this problem out. Not only rapid frequency evaluation of the reduced order model is carried out within this approach, but also special emphasis is placed on a fast determination of the error mesure for each frequency in the band of interest. This certifies the accurate response of the reduced order model. The same scheme allows us, in a offline stage, to adaptively select the basis functions in the reduced basis approximation and automatically select the model order reduction process whenever a preestablished accuracy is required throughout the band of interest. Finally, real-life applications will illustrate the capabilities of this approach
Low-frequency expansion for probability amplitudes: An alternative approach to certain intramolecular dynamics problems
We present an algorithm to determine the averaged time evolution of the probability amplitude for a nonstationary state in a quantum mechanical system. The algorithm is based on a low‐frequency expansion of the probability amplitude and is related to the generalized moment expansion method which has been applied successfully to the description of dynamic correlation functions in stochastic systems. It is shown that the proposed algorithm gives excellent results for the description of quantum beats in the time evolution of the occupation probability for a nonstationary state in model systems. The relation of the algorithm to other theoretical approaches and the relevance for the description of intramolecular energy transfer processes is discussed
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