Mixing times with applications to perturbed Markov chains

A measure of the “mixing time” or “time to stationarity” in a finite irreducible discrete time Markov chain is considered. The statistic , where {π_j} is the stationary distribution and m_ij is the mean first passage time from state i to state j of the Markov chain, is shown to be independent of the initial state i (so that η_i = η for all i), is minimal in the case of a periodic chain, yet can be arbitrarily large in a variety of situations. An application considering the effects perturbations of the transition probabilities have on the stationary distributions of Markov chains leads to a new bound, involving η, for the 1-norm of the difference between the stationary probability vectors of the original and the perturbed chain. When η is large the stationary distribution of the Markov chain is very sensitive to perturbations of the transition probabilities.     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

On the balancing principle for some problems of Numerical Analysis

We discuss a choice of weight in penalization methods. The motivation for the use of penalization in computational mathematics is to improve the conditioning of the numerical solution. One example of such improvement is a regularization, where a penalization substitutes an ill-posed problem for a well-posed one. In modern numerical methods for PDEs a penalization is used, for example, to enforce a continuity of an approximate solution on non-matching grids. A choice of penalty weight should provide a balance between error components related with convergence and stability, which are usually unknown. In this paper we propose and analyze a simple adaptive strategy for the choice of penalty weight which does not rely on a priori estimates of above mentioned components. It is shown that under natural assumptions the accuracy provided by our adaptive strategy is worse only by a constant factor than one could achieve in the case of known stability and convergence rates. Finally, we successfully apply our strategy for self-regularization of Volterra-type severely ill-posed problems, such as the sideways heat equation, and for the choice of a weight in interior penalty discontinuous approximation on non-matching grids. Numerical experiments on a series of model problems support theoretical results.     

Two-scale composite finite element method for Dirichlet problems on complicated domains

In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.     

On the super-approximation property of Galerkin's method with finite elements

Summary For Galerkin's method with finite elements as trial functions for strongly elliptic operator equations in the Hilbert scaleH ^t the super-approximation property and the optimal convergence rate are obtained by using the Aubin-Nitsche lemma. This applies in particular to spline collocation methods for a wide class of pseudodifferential equations.Dedicated to the memory of Professor Lothar Collatz     

On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem with H a Hankel matrix and the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Padé approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled. B. Beckermann was supported in part by INTAS research network NaCCA 03-51-6637. G. H. Golub was supported in part by DOE grant DE-FC-02-01ER41177. G. Labahn was supported in part by NSERC and MITACS Canada grants.     

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h^1/4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

Analysis of one-dimensional Helmholtz equation with PML boundary

In this paper, the linear conforming finite element method for the one-dimensional Bérenger's PML boundary is investigated and well-posedness of the given equation is discussed. Furthermore, optimal error estimates and stability in the L²$L^2$ or H¹$H^1$-norm are derived under the assumption that h$h$, h²ω²$h^2{\omega }^2$ and h²ω³$h^2{\omega }^3$ are sufficiently small, where h$h$ is the mesh size and ω$\omega$ denotes a fixed frequency. Numerical examples are presented to validate the theoretical error bounds.     

On extremum properties of orthogonal quotients matrices

In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.     

A collocation method with cubic B-splines for solving the MRLW equation

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.     

A trapping principle and convergence result for finite element approximate solutions of steady reaction/diffusion systems

We consider nonlinear elliptic systems, with mixed boundary conditions, on a convex polyhedral domain Ω ⊂ R ^N . These are nonlinear divergence form generalizations of Δu = f(·, u), where f is outward pointing on the trapping region boundary. The motivation is that of applications to steady-state reaction/diffusion systems. Also included are reaction/diffusion/convection systems which satisfy the Einstein relations, for which the Cole-Hopf transformation is possible. For maximum generality, the theory is not tied to any specific application. We are able to demonstrate a trapping principle for the piecewise linear Galerkin approximation, defined via a lumped integration hypothesis on integrals involving f, by use of variational inequalities. Results of this type have previously been obtained for parabolic systems by Estep, Larson, and Williams, and for nonlinear elliptic equations by Karátson and Korotov. Recent minimum and maximum principles have been obtained by Jüngel and Unterreiter for nonlinear elliptic equations. We make use of special properties of the element stiffness matrices, induced by a geometric constraint upon the simplicial decomposition. This constraint is known as the non-obtuseness condition. It states that the inward normals, associated with an arbitrary pair of an element’s faces, determine an angle with nonpositive cosine. Drăgănescu, Dupont, and Scott have constructed an example for which the discrete maximum principle fails if this condition is omitted. We also assume vertex communication in each element in the form of an irreducibility hypothesis on the off-diagonal elements of the stiffness matrix. There is a companion convergence result, which yields an existence theorem for the solution. This entails a consistency hypothesis for interpolation on the boundary, and depends on the Tabata construction of simple function approximation, based on barycentric regions. This work was supported by the National Science Foundation under grant DMS-0311263.     

Superconvergence of new mixed finite element spaces

In this paper we prove some superconvergence of a new family of mixed finite element spaces of higher order which we introduced in [ETNA, Vol. 37, pp. 189-201, 2010]. Among all the mixed finite element spaces having an optimal order of convergence on quadrilateral grids, this space has the smallest unknowns. However, the scalar variable is only suboptimal in general; thus we have employed a post-processing technique for the scalar variable. As a byproduct, we have obtained a superconvergence on a rectangular grid. The superconvergence of a velocity variable naturally holds and can be shown by a minor modification of existing theory, but that of a scalar variable requires a new technique, especially for k=1. Numerical experiments are provided to support the theory.     

Worst case scenario analysis for elliptic problems with uncertainty

This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.     

A unifying theory of a posteriori finite element error control

Summary Residual-based a posteriori error estimates are derived within a unified setting for lowest-order conforming, nonconforming, and mixed finite element schemes. The various residuals are identified for all techniques and problems as the operator norm |||| of a linear functional of the formin the variable of a Sobolev space V. The main assumption is that the first-order finite element space is included in the kernel Ker of . As a consequence, any residual estimator that is a computable bound of |||| can be used within the proposed frame without further analysis for nonconforming or mixed FE schemes. Applications are given for the Laplace, Stokes, and Navier-Lamè equations.Supported by the DFG Research Center Matheon Mathematics for key technologies in Berlin.     

On computingINV block preconditionings for the conjugate gradient method

The INV(k) and MINV(k) block preconditionings for the conjugate gradient method require generation of selected elements of the inverses of symmetric matrices of bandwidth 2k+1. Generalizing the previously describedk=1 (tridiagonal) case tok=2, explicit expressions for the inverse elements of a symmetric pentadiagonal matrix in terms of Green's matrix of rank two are given. These expressions are found to be seriously ill-conditioned; hence alternative computational algorithms for the inverse elements must be used. Behavior of thek=1 andk=2 preconditionings are compared for some discretized elliptic partial differential equation test problems in two dimensions.Presented by the first author at the Joint U.S.-Scandinavian Symposium on Scientific Computing and Mathematical Modeling, January 1985, in honor of Germund Dahlquist on the occasion of his 60th birthday. This work was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE AC03-76SF00098.     

The behavior of the best Sobolev trace constant and extremals in thin domains

In this paper, we study the asymptotic behavior of the best Sobolev trace constant and extremals for the immersion W1,p(Ω)?Lq(∂Ω) in a bounded smooth domain when it is contracted in one direction. We find that the limit problem, when rescaled in a suitable way, is a Sobolev-type immersion in weighted spaces over a projection of Ω, W1,p(P(Ω),α)?Lq(P(Ω),β).For the special case p=q, this problem leads to an eigenvalue problem with a nonlinear boundary condition. We also study the convergence of the eigenvalues and eigenvectors in this case.     

Pointwise a posteriori error estimates for monotone semi-linear equations

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.     

A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation

In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient.