1,440 research outputs found
Exponential Krylov time integration for modeling multi-frequency optical response with monochromatic sources
Light incident on a layer of scattering material such as a piece of sugar or
white paper forms a characteristic speckle pattern in transmission and
reflection. The information hidden in the correlations of the speckle pattern
with varying frequency, polarization and angle of the incident light can be
exploited for applications such as biomedical imaging and high-resolution
microscopy. Conventional computational models for multi-frequency optical
response involve multiple solution runs of Maxwell's equations with
monochromatic sources. Exponential Krylov subspace time solvers are promising
candidates for improving efficiency of such models, as single monochromatic
solution can be reused for the other frequencies without performing full
time-domain computations at each frequency. However, we show that the
straightforward implementation appears to have serious limitations. We further
propose alternative ways for efficient solution through Krylov subspace
methods. Our methods are based on two different splittings of the unknown
solution into different parts, each of which can be computed efficiently.
Experiments demonstrate a significant gain in computation time with respect to
the standard solvers.Comment: 22 pages, 4 figure
A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers
We consider the problem of estimating parameters in large-scale weakly
nonlinear inverse problems for which the underlying governing equations is a
linear, time-dependent, parabolic partial differential equation. A major
challenge in solving these inverse problems using Newton-type methods is the
computational cost associated with solving the forward problem and with
repeated construction of the Jacobian, which represents the sensitivity of the
measurements to the unknown parameters. Forming the Jacobian can be
prohibitively expensive because it requires repeated solutions of the forward
and adjoint time-dependent parabolic partial differential equations
corresponding to multiple sources and receivers. We propose an efficient method
based on a Laplace transform-based exponential time integrator combined with a
flexible Krylov subspace approach to solve the resulting shifted systems of
equations efficiently. Our proposed solver speeds up the computation of the
forward and adjoint problems, thus yielding significant speedup in total
inversion time. We consider an application from Transient Hydraulic Tomography
(THT), which is an imaging technique to estimate hydraulic parameters related
to the subsurface from pressure measurements obtained by a series of pumping
tests. The algorithms discussed are applied to a synthetic example taken from
THT to demonstrate the resulting computational gains of this proposed method
MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks
We proposed MATEX, a distributed framework for transient simulation of power
distribution networks (PDNs). MATEX utilizes matrix exponential kernel with
Krylov subspace approximations to solve differential equations of linear
circuit. First, the whole simulation task is divided into subtasks based on
decompositions of current sources, in order to reduce the computational
overheads. Then these subtasks are distributed to different computing nodes and
processed in parallel. Within each node, after the matrix factorization at the
beginning of simulation, the adaptive time stepping solver is performed without
extra matrix re-factorizations. MATEX overcomes the stiff-ness hinder of
previous matrix exponential-based circuit simulator by rational Krylov subspace
method, which leads to larger step sizes with smaller dimensions of Krylov
subspace bases and highly accelerates the whole computation. MATEX outperforms
both traditional fixed and adaptive time stepping methods, e.g., achieving
around 13X over the trapezoidal framework with fixed time step for the IBM
power grid benchmarks.Comment: ACM/IEEE DAC 2014. arXiv admin note: substantial text overlap with
arXiv:1505.0669
Explicit schemes for time propagating many-body wavefunctions
Accurate theoretical data on many time-dependent processes in atomic and
molecular physics and in chemistry require the direct numerical solution of the
time-dependent Schr\"odinger equation, thereby motivating the development of
very efficient time propagators. These usually involve the solution of very
large systems of first order differential equations that are characterized by a
high degree of stiffness. We analyze and compare the performance of the
explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have
exactly the same stability function, therefore sharing the same stability
properties that turn out to be optimum. Their respective accuracy however
differs significantly and depends on the physical situation involved. In order
to test this accuracy, we use a predictor-corrector scheme in which the
predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully
implicit four-stage Radau IIA method of order 7. We consider two physical
processes. The first one is the ionization of an atomic system by a short and
intense electromagnetic pulse; the atomic systems include a one-dimensional
Gaussian model potential as well as atomic hydrogen and helium, both in full
dimensionality. The second process is the decoherence of two-electron quantum
states when a time independent perturbation is applied to a planar two-electron
quantum dot where both electrons are confined in an anharmonic potential. Even
though the Hamiltonian of this system is time independent the corresponding
differential equation shows a striking stiffness. For the one-dimensional
Gaussian potential we discuss in detail the possibility of monitoring the time
step for both explicit algorithms. In the other physical situations that are
much more demanding in term of computations, we show that the accuracy of both
algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure
An Arnoldi-frontal approach for the stability analysis of flows in a collapsible channel
In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods
Wave packet propagation by the Faber polynomial approximation in electrodynamics of passive media
Maxwell's equations for propagation of electromagnetic waves in dispersive
and absorptive (passive) media are represented in the form of the Schr\"odinger
equation , where is a linear
differential operator (Hamiltonian) acting on a multi-dimensional vector
composed of the electromagnetic fields and auxiliary matter fields describing
the medium response. In this representation, the initial value problem is
solved by applying the fundamental solution to the initial field
configuration. The Faber polynomial approximation of the fundamental solution
is used to develop a numerical algorithm for propagation of broad band wave
packets in passive media. The action of the Hamiltonian on the wave function
is approximated by the Fourier grid pseudospectral method. The algorithm
is global in time, meaning that the entire propagation can be carried out in
just a few time steps. A typical time step is much larger than that in finite
differencing schemes, . The accuracy and stability
of the algorithm is analyzed. The Faber propagation method is compared with the
Lanczos-Arnoldi propagation method with an example of scattering of broad band
laser pulses on a periodic grating made of a dielectric whose dispersive
properties are described by the Rocard-Powels-Debye model. The Faber algorithm
is shown to be more efficient. The Courant limit for time stepping, , is exceeded at least in 3000 times in the Faber propagation
scheme.Comment: Latex, 17 pages, 4 figures (separate png files); to appear in J.
Comput. Phy
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