334,842 research outputs found
A unified error analysis for the numerical solution of nonlinear wave-type equations with application to kinetic boundary conditions
In this thesis, a unified error analysis for discretizations of nonlinear first- and second-order wave-type equations is provided. For this, the wave equations as well as their space discretizations are considered as nonlinear evolution equations in Hilbert spaces. The space discretizations are supplemented with Runge-Kutta time discretizations. By employing stability properties of monotone operators, abstract error bounds for the space, time, and full discretizations are derived.
Further, for semilinear second-order wave-type equations, an implicit-explicit time integration scheme is presented. This scheme only requires the solution of a linear system of equations in each time step and it is stable under a step size restriction only depending on the nonlinearity. It is proven that the scheme converges with second order in time and in combination with the abstract space discretization of the unified error analysis, corresponding full discretization error bounds are derived.
The abstract results are used to derive convergence rates for an isoparametric finite element space discretization of a wave equation with kinetic boundary conditions and nonlinear forcing and damping terms.
For the combination of the finite element discretization with Runge-Kutta methods or the implicit-explicit scheme, respectively, error bounds of the resulting fully discrete schemes are proven. The theoretical results are illustrated by numerical experiments
hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems
We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order elliptic reaction-diffusion equations with mixed Dirichlet and Neumann boundary conditions. For simplicity of the presentation, we only consider boundary-value problems defined on an axiparallel polygonal domain whose solutions are approximated on subdivisions consisting of axiparallel elements. Our main concern is the generalisation of the error analysis of the hp-DGFEM for the case when shape-irregular (anisotropic) meshes and anisotropic polynomial degrees for the element basis functions are used. We shall present a general framework for deriving error bounds for the approximation error and we shall consider two important special cases. In the first of these we derive an error bound that is simultaneously optimal in h and p, for shape-regular elements and isotropic polynomial degrees, provided that the solution belongs to a certain anisotropic Sobolev space. The second result deals with the case where we have a uniform polynomial degree in every space direction and a shape-irregular mesh. Again we derive an error bound that is optimal both in h and in p. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement, in both cases considered. Finally, numerical experiments using shape-regular and shape-irregular elements are presented
Uniformly-convergent numerical methods for a system of coupled singularly perturbed convection–diffusion equations with mixed type boundary conditions
In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection - diffusion second order ordinary differential equations subject to the mixed type boundary conditions. We prove that the method has almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and also the numerical derivative are established. Numerical results are provided to illustrate the theoretical results.
First published online: 24 Oct 201
Complexity Analysis of Accelerated MCMC Methods for Bayesian Inversion
We study Bayesian inversion for a model elliptic PDE with unknown diffusion
coefficient. We provide complexity analyses of several Markov Chain-Monte Carlo
(MCMC) methods for the efficient numerical evaluation of expectations under the
Bayesian posterior distribution, given data . Particular attention is
given to bounds on the overall work required to achieve a prescribed error
level . Specifically, we first bound the computational complexity
of "plain" MCMC, based on combining MCMC sampling with linear complexity
multilevel solvers for elliptic PDE. Our (new) work versus accuracy bounds show
that the complexity of this approach can be quite prohibitive. Two strategies
for reducing the computational complexity are then proposed and analyzed:
first, a sparse, parametric and deterministic generalized polynomial chaos
(gpc) "surrogate" representation of the forward response map of the PDE over
the entire parameter space, and, second, a novel Multi-Level Markov Chain Monte
Carlo (MLMCMC) strategy which utilizes sampling from a multilevel
discretization of the posterior and of the forward PDE.
For both of these strategies we derive asymptotic bounds on work versus
accuracy, and hence asymptotic bounds on the computational complexity of the
algorithms. In particular we provide sufficient conditions on the regularity of
the unknown coefficients of the PDE, and on the approximation methods used, in
order for the accelerations of MCMC resulting from these strategies to lead to
complexity reductions over "plain" MCMC algorithms for Bayesian inversion of
PDEs.
The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods
The aim of the paper is to introduce a new systematic method that can produce
lower bounds for eigenvalues. The main idea is to use nonconforming finite
element methods. The general conclusion herein is that if local approximation
properties of nonconforming finite element spaces are better than global
continuity properties of , corresponding methods will produce lower bounds
for eigenvalues. More precisely, under three conditions on continuity and
approximation properties of nonconforming finite element spaces we first show
abstract error estimates of approximate eigenvalues and eigenfunctions.
Subsequently, we propose one more condition and prove that it is sufficient to
guarantee nonconforming finite element methods to produce lower bounds for
eigenvalues of symmetric elliptic operators. As one application, we show that
this condition hold for most nonconforming elements in literature. As another
important application, this condition provides a guidance to modify known
nonconforming elements in literature and to propose new nonconforming elements.
In fact, we enrich locally the Crouzeix-Raviart element such that the new
element satisfies the condition; we propose a new nonconforming element for
second order elliptic operators and prove that it will yield lower bounds for
eigenvalues. Finally, we prove the saturation condition for most nonconforming
elements.Comment: 24 page
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Finite state machine representation of digital signal processing systems
A new method for implementing digital filters is discussed. The met11od maximises the output signal to noise ratio of a filter by assigning at each of the filter variables an optimal quantization law. A filter optimised for a gaussian process is considered in detail. An error model is developed and applied to first and second order canonic form filter sections. Comparisons are drawn between the gaussian optimised filter and the equivalent fixed point arithmetic filter. The performance of gaussian optimised filters under sinusoidal input signal conditions is considered ; it is found that the gaussian optimised filter exhibits a lower approximation error than the equivalent fixed point arithmetic filter. It is shown that when high order filters are implemented as a cascade of second order sections - with if necessary one first order section - the section ordering has a very small effect on the overall signal to noise r atio performance. A similar result for the pairing of poles and zeroes is found. Bounds on the maximum limit cycle amplitude for first and second order all-pole sections are presented. It is shown that for a first order all-pole the maximum limit cycle amplitude is lower than would be expected in the equivalent fixed point arithmetic filter, whereas , for the second order all- pole the bound is twice as large. Examples of a low-pass , band-pass and wideband differentiating filter,designed using free quantization law techniques,are presented. This new design method leads to a filter whose arithmetic operations can not be performed using fixed point arithmetic hardware. Instead, the filter must be represented as a finite state machine and then implemented using sequential logic circuit synthesis techniques. The logic complexity is found to depend - amongst other considerations - on the so called state (code) assignment. Some preliminary results on this problem are presented for the case of a next state function computed using the AND/EXCLUSIVE- OR (ring-sum) logic expansion. A review of the state assignment techniques in the literature is included. A part of the state assignment problem - for the case of AND/EX'·/OR logic - requires the numerous and consequently rapid computation of the Reed-Muller Transformation. A hardware processor - designed as an add-on to a minicomputer - is described; speed comparisons are drawn with the equivalent software algorithm.Digitisation of this thesis was sponsored by Arcadia Fund, a charitable fund of Lisbet Rausing and Peter Baldwin
Scanning and Sequential Decision Making for Multi-Dimensional Data - Part I: the Noiseless Case
We investigate the problem of scanning and prediction ("scandiction", for
short) of multidimensional data arrays. This problem arises in several aspects
of image and video processing, such as predictive coding, for example, where an
image is compressed by coding the error sequence resulting from scandicting it.
Thus, it is natural to ask what is the optimal method to scan and predict a
given image, what is the resulting minimum prediction loss, and whether there
exist specific scandiction schemes which are universal in some sense.
Specifically, we investigate the following problems: First, modeling the data
array as a random field, we wish to examine whether there exists a scandiction
scheme which is independent of the field's distribution, yet asymptotically
achieves the same performance as if this distribution was known. This question
is answered in the affirmative for the set of all spatially stationary random
fields and under mild conditions on the loss function. We then discuss the
scenario where a non-optimal scanning order is used, yet accompanied by an
optimal predictor, and derive bounds on the excess loss compared to optimal
scanning and prediction.
This paper is the first part of a two-part paper on sequential decision
making for multi-dimensional data. It deals with clean, noiseless data arrays.
The second part deals with noisy data arrays, namely, with the case where the
decision maker observes only a noisy version of the data, yet it is judged with
respect to the original, clean data.Comment: 46 pages, 2 figures. Revised version: title changed, section 1
revised, section 3.1 added, a few minor/technical corrections mad
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