Weighted finite-state transducers in speech recognition : a compaction algorithm for non-determinizable transducers

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Finite difference calculations of permeability in large domains in a wide porosity range.

Determining effective hydraulic, thermal, mechanical and electrical properties of porous materials by means of classical physical experiments is often time-consuming and expensive. Thus, accurate numerical calculations of material properties are of increasing interest in geophysical, manufacturing, bio-mechanical and environmental applications, among other fields. Characteristic material properties (e.g. intrinsic permeability, thermal conductivity and elastic moduli) depend on morphological details on the porescale such as shape and size of pores and pore throats or cracks. To obtain reliable predictions of these properties it is necessary to perform numerical analyses of sufficiently large unit cells. Such representative volume elements require optimized numerical simulation techniques. Current state-of-the-art simulation tools to calculate effective permeabilities of porous materials are based on various methods, e.g. lattice Boltzmann, finite volumes or explicit jump Stokes methods. All approaches still have limitations in the maximum size of the simulation domain. In response to these deficits of the well-established methods we propose an efficient and reliable numerical method which allows to calculate intrinsic permeabilities directly from voxel-based data obtained from 3D imaging techniques like X-ray microtomography. We present a modelling framework based on a parallel finite differences solver, allowing the calculation of large domains with relative low computing requirements (i.e. desktop computers). The presented method is validated in a diverse selection of materials, obtaining accurate results for a large range of porosities, wider than the ranges previously reported. Ongoing work includes the estimation of other effective properties of porous media.

Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y = f(x, y, y ') on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form g(i)(y(i),y ' (i)) = 0, i = 0, 1, assuming S satisfies one-sided growth bounds with respect to y '. (C) 2001 Elsevier Science Ltd. All rights reserved.

A novel self consistent calculation approach for the capacitance-voltage characteristics of semiconductor quantum wire transistors based on a split-gate configuration

Purpose - The purpose of this paper is to develop an efficient numerical algorithm for the self-consistent solution of Schrodinger and Poisson equations in one-dimensional systems. The goal is to compute the charge-control and capacitance-voltage characteristics of quantum wire transistors. Design/methodology/approach - The paper presents a numerical formulation employing a non-uniform finite difference discretization scheme, in which the wavefunctions and electronic energy levels are obtained by solving the Schrodinger equation through the split-operator method while a relaxation method in the FTCS scheme ("Forward Time Centered Space") is used to solve the two-dimensional Poisson equation. Findings - The numerical model is validated by taking previously published results as a benchmark and then applying them to yield the charge-control characteristics and the capacitance-voltage relationship for a split-gate quantum wire device. Originality/value - The paper helps to fulfill the need for C-V models of quantum wire device. To do so, the authors implemented a straightforward calculation method for the two-dimensional electronic carrier density n(x,y). The formulation reduces the computational procedure to a much simpler problem, similar to the one-dimensional quantization case, significantly diminishing running time.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

A generalized alternating-direction implicit scheme for incompressible magnetohydrodynamic viscous flows at low magnetic Reynolds number

This paper presents numerical simulations of incompressible fluid flows in the presence of a magnetic field at low magnetic Reynolds number. The equations governing the flow are the Navier-Stokes equations of fluid motion coupled with Maxwell's equations of electromagnetics. The study of fluid flows under the influence of a magnetic field and with no free electric charges or electric fields is known as magnetohydrodynamics. The magnetohydrodynamics approximation is considered for the formulation of the non-dimensional problem and for the characterization of similarity parameters. A finite-difference technique is used to discretize the equations. In particular, an extension of the generalized Peaceman and Rachford alternating-direction implicit (ADI) scheme for simulating two-dimensional fluid flows is presented. The discretized conservation equations are solved in stream function-vorticity formulation. We compare the ADI and generalized ADI schemes, and show that the latter is more efficient in simulating low Reynolds number and magnetic Reynolds number problems. Numerical results demonstrating the applicability of this technique are also presented. The simulation of incompressible magneto hydrodynamic fluid flows is illustrated by numerical solution for two-dimensional cases. (c) 2007 Elsevier B.V. All rights reserved.

Stabilized Petrov-Galerkin methods for the convection-diffusion-reaction and the Helmholtz equations

We present two new stabilized high-resolution numerical methods for the convection–diffusion–reaction (CDR) and the Helmholtz equations respectively. The work embarks upon a priori analysis of some consistency recovery procedures for some stabilization methods belonging to the Petrov–Galerkin framework. It was found that the use of some standard practices (e.g. M-Matrices theory) for the design of essentially non-oscillatory numerical methods is not feasible when consistency recovery methods are employed. Hence, with respect to convective stabilization, such recovery methods are not preferred. Next, we present the design of a high-resolution Petrov–Galerkin (HRPG) method for the 1D CDR problem. The problem is studied from a fresh point of view, including practical implications on the formulation of the maximum principle, M-Matrices theory, monotonicity and total variation diminishing (TVD) finite volume schemes. The current method is next in line to earlier methods that may be viewed as an upwinding plus a discontinuity-capturing operator. Finally, some remarks are made on the extension of the HRPG method to multidimensions. Next, we present a new numerical scheme for the Helmholtz equation resulting in quasi-exact solutions. The focus is on the approximation of the solution to the Helmholtz equation in the interior of the domain using compact stencils. Piecewise linear/bilinear polynomial interpolation are considered on a structured mesh/grid. The only a priori requirement is to provide a mesh/grid resolution of at least eight elements per wavelength. No stabilization parameters are involved in the definition of the scheme. The scheme consists of taking the average of the equation stencils obtained by the standard Galerkin finite element method and the classical finite difference method. Dispersion analysis in 1D and 2D illustrate the quasi-exact properties of this scheme. Finally, some remarks are made on the extension of the scheme to unstructured meshes by designing a method within the Petrov–Galerkin framework.

Approximation algorithms for two-state anti-ferromagnetic spin systems on bounded degree graphs

In a seminal paper [10], Weitz gave a deterministic fully polynomial approximation scheme for counting exponentially weighted independent sets (which is the same as approximating the partition function of the hard-core model from statistical physics) in graphs of degree at most d, up to the critical activity for the uniqueness of the Gibbs measure on the innite d-regular tree. ore recently Sly [8] (see also [1]) showed that this is optimal in the sense that if here is an FPRAS for the hard-core partition function on graphs of maximum egree d for activities larger than the critical activity on the innite d-regular ree then NP = RP. In this paper we extend Weitz's approach to derive a deterministic fully polynomial approximation scheme for the partition function of general two-state anti-ferromagnetic spin systems on graphs of maximum degree d, up to the corresponding critical point on the d-regular tree. The main ingredient of our result is a proof that for two-state anti-ferromagnetic spin systems on the d-regular tree, weak spatial mixing implies strong spatial mixing. his in turn uses a message-decay argument which extends a similar approach proposed recently for the hard-core model by Restrepo et al [7] to the case of general two-state anti-ferromagnetic spin systems.

Analysis of a finite volume method for a cross-diffusion model in population dynamics

The main goal of this paper is to propose a convergent finite volume method for a reactionâeuro"diffusion system with cross-diffusion. First, we sketch an existence proof for a class of cross-diffusion systems. Then the standard two-point finite volume fluxes are used in combination with a nonlinear positivity-preserving approximation of the cross-diffusion coefficients. Existence and uniqueness of the approximate solution are addressed, and it is also shown that the scheme converges to the corresponding weak solution for the studied model. Furthermore, we provide a stability analysis to study pattern-formation phenomena, and we perform two-dimensional numerical examples which exhibit formation of nonuniform spatial patterns. From the simulations it is also found that experimental rates of convergence are slightly below second order. The convergence proof uses two ingredients of interest for various applications, namely the discrete Sobolev embedding inequalities with general boundary conditions and a space-time $L^1$ compactness argument that mimics the compactness lemma due to Kruzhkov. The proofs of these results are given in the Appendix.

Time Discretization of the SST-generalized Navier-Stokes Equations: Positive and Negative Results

In the theory of the Navier-Stokes equations, the proofs of some basic known results, like for example the uniqueness of solutions to the stationary Navier-Stokes equations under smallness assumptions on the data or the stability of certain time discretization schemes, actually only use a small range of properties and are therefore valid in a more general context. This observation leads us to introduce the concept of SST spaces, a generalization of the functional setting for the Navier-Stokes equations. It allows us to prove (by means of counterexamples) that several uniqueness and stability conjectures that are still open in the case of the Navier-Stokes equations have a negative answer in the larger class of SST spaces, thereby showing that proof strategies used for a number of classical results are not sufficient to affirmatively answer these open questions. More precisely, in the larger class of SST spaces, non-uniqueness phenomena can be observed for the implicit Euler scheme, for two nonlinear versions of the Crank-Nicolson scheme, for the fractional step theta scheme, and for the SST-generalized stationary Navier-Stokes equations. As far as stability is concerned, a linear version of the Euler scheme, a nonlinear version of the Crank-Nicolson scheme, and the fractional step theta scheme turn out to be non-stable in the class of SST spaces. The positive results established in this thesis include the generalization of classical uniqueness and stability results to SST spaces, the uniqueness of solutions (under smallness assumptions) to two nonlinear versions of the Euler scheme, two nonlinear versions of the Crank-Nicolson scheme, and the fractional step theta scheme for general SST spaces, the second order convergence of a version of the Crank-Nicolson scheme, and a new proof of the first order convergence of the implicit Euler scheme for the Navier-Stokes equations. For each convergence result, we provide conditions on the data that guarantee the existence of nonstationary solutions satisfying the regularity assumptions needed for the corresponding convergence theorem. In the case of the Crank-Nicolson scheme, this involves a compatibility condition at the corner of the space-time cylinder, which can be satisfied via a suitable prescription of the initial acceleration.

An efficient numerical scheme for the Euler equations

A finite difference scheme based on flux difference splitting is presented for the solution of the Euler equations for the compressible flow of an ideal gas. A linearised Riemann problem is defined, and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to the usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. The scheme is applied to a shock tube problem and a blast wave problem. Each approximate solution compares well with those given by other schemes, and for the shock tube problem is in agreement with the exact solution.

An efficient numerical scheme for the two-dimensional shallow water equations using arithmetic averaging

A finite difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gas dynamics is defined, and a scheme, based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem, and incorporates the technique of operator splitting. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging. This is in contrast to usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. An extension to the two-dimensional equations with source terms is included. The scheme is applied to the one-dimensional problems of a breaking dam and reflection of a bore, and in each case the approximate solution is compared to the exact solution of ideal fluid flow. The scheme is also applied to a problem of stationary bore generation in a channel of variable cross-section. Finally, the scheme is applied to two other dam-break problems, this time in two dimensions with one having cylindrical symmetry. Each approximate solution compares well with those given by other authors.

A shock capturing scheme for steady, supercritical, free-surface flow

Abstract A finite difference scheme is presented for the solution of the two-dimensional shallow water equations in steady, supercritical flow. The scheme incorporates numerical characteristic decomposition, is shock capturing by design and incorporates space-marching as a result of the assumption that the flow is wholly supercritical in at least one space dimension. Results are shown for problems involving oblique hydraulic jumps and reflection from a wall.

Shock capturing scheme for steady, supersonic, isentropic flow

A finite difference scheme is presented for the solution of the two-dimensional equations of steady, supersonic, isentropic flow. The scheme incorporates numerical characteristic decomposition, is shock-capturing by design and incorporates space marching as a result of the assumption that the flow is wholly supersonic in at least one space dimension. Results are shown for problems involving oblique hydraulic jumps and reflection from a wall.