1,256 research outputs found
Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution
We present a new solver for nonlinear parabolic problems that is L-stable and
achieves high order accuracy in space and time. The solver is built by first
constructing a single-dimensional heat equation solver that uses fast O(N)
convolution. This fundamental solver has arbitrary order of accuracy in space,
and is based on the use of the Green's function to invert a modified Helmholtz
equation. Higher orders of accuracy in time are then constructed through a
novel technique known as successive convolution (or resolvent expansions).
These resolvent expansions facilitate our proofs of stability and convergence,
and permit us to construct schemes that have provable stiff decay. The
multi-dimensional solver is built by repeated application of dimensionally
split independent fundamental solvers. Finally, we solve nonlinear parabolic
problems by using the integrating factor method, where we apply the basic
scheme to invert linear terms (that look like a heat equation), and make use of
Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our
solver is applied to several linear and nonlinear equations including heat,
Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two
dimensions
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
Computation and Learning in High Dimensions (hybrid meeting)
The most challenging problems in science often involve the learning and
accurate computation of high dimensional functions.
High-dimensionality is a typical feature for a multitude of problems
in various areas of science.
The so-called curse of dimensionality typically negates the use of
traditional numerical techniques for the solution of
high-dimensional problems. Instead, novel theoretical and
computational approaches need to be developed to make them tractable
and to capture fine resolutions and relevant features. Paradoxically,
increasing computational power may even serve to heighten this demand,
since the wealth of new computational data itself becomes a major
obstruction. Extracting essential information from complex
problem-inherent structures and developing rigorous models to quantify
the quality of information in a high-dimensional setting pose
challenging tasks from both theoretical and numerical perspective.
This has led to the emergence of several new computational methodologies,
accounting for the fact that by now well understood methods drawing on
spatial localization and mesh-refinement are in their original form no longer viable.
Common to these approaches is the nonlinearity of the solution method.
For certain problem classes, these methods have
drastically advanced the frontiers of computability.
The most visible of these new methods is deep learning. Although the use of deep neural
networks has been extremely successful in certain
application areas, their mathematical understanding is far from complete.
This workshop proposed to deepen the understanding of
the underlying mathematical concepts that drive this new evolution of
computational methods and to promote the exchange of ideas emerging in various
disciplines about how to treat multiscale and high-dimensional problems
Semi-Implicit Direct Kinetics Methodology for Deterministic, Time-Dependent, Three-Dimensional, and Fine-Energy Neutron Transport Solutions
Using a semi-implicit direct kinetics (SIDK) method that is developed in this dissertation, a finer neutron energy discretization and improved fidelity for transient radiation transport calculations are facilitated to reduce uncertainties and conservatisms in transient power and temperature predictions. These capabilities are implemented within a parallel computational solver framework, which is able to represent an arbitrary number of neutron energy groups, angles, and spatial discretizations, while internally coupled to an unstructured finite element multi-physics code for temperature and displacement calculations. This capability is demonstrated on a three-dimensional control rod ejection simulation run in parallel utilizing forty-four neutron energy groups.
An improved transient nuclear reactor simulation capability is developed by adapting the steady-state radiation transport code Denovo to solve the time-dependent Boltzmann transport equation for transient power distributions. The developed SIDK method is compared to fully-implicit direct kinetics, higher order time integration methods, as well as various computational benchmarks. Errors resulting from time integration, spatial discretization, angular treatment, multi-group treatment, homogenization of temperature, and power over the time step representation are explored.
For verification, the SIDK method is developed and tested externally and independently employing a few-group time-dependent neutron diffusion code which is compared to one and two-dimensional benchmarks with and without temperature feedbacks. The results of the semi-implicit direct kinetics method (SIDK) are shown to be accurate to within ~0.2% of direct kinetics and to execute roughly an order of magnitude faster, using a consistent space and time discretization. For sufficiently severe transients, the direct method is shown to produce lower errors with medium time steps than the SIDK method with fine steps, but proves to be subject to more severe oscillations at very coarse time steps than the SIDK method, in addition to producing similar errors (within 0.2 %) at medium spatial discretization with consistent time steps.
The objective of this dissertation is to provide developers of next generation high-performance computing neutron kinetics methods a guide to the benefits and costs of the dominant discretization strategies of time, space, neutron energy, and angle for the solution of the time-dependent Boltzmann transport equation
Application of Operator Splitting Methods in Finance
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps
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