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Let F be a compact set of a Banach space X. This paper analyzes the "generalized empirical interpolation method," which, given a function f ∊ F, builds an interpolant Jn[f] in an n dimensional subspace Xn ⊂ X with the knowledge of n outputs (σi(f))ni=1, where σi ∊ X' and X' is the dual space of X. The space Xn is built with a greedy algorithm that is adapted to F in the sense that it is generated by elements of F itself. The algorithm also selects the linear functionals (σi)ni=1 from a dictionary Σ ⊂ X'. In this paper, we study the interpolation error maxf∊F ‖f - Jn[f]‖X by comparing it with the best possible performance on an n dimensional space, i.e., the Kolmogorov n-width of F in X, dn(F,X). For polynomial or exponential decay rates of dn(F,X), we prove that the interpolation error has the same behavior modulo the norm of the interpolation operator. Sharper results are obtained in the case where X is a Hilbert space. [ABSTRACT FROM AUTHOR]

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing ø(Bu) + H(u), where H and ø are convex functions, and ø is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this paper, the convergence rate of BOSVS is analyzed. When H(u) = ‖Au - f ‖2, where A is a matrix, it is shown that for an ergodic approximation uk obtained by averaging k BOSVS iterates, the error in the objective value ø(Buk)+H(uk) is O(1/k). When the optimization problem has a unique solution u*, we obtain the estimate ‖uk-u*‖ = O(1/ √ k). The theoretical analysis is compared to observed convergence rates for partially parallel magnetic resonance image reconstruction problems where A is a large dense ill-conditioned matrix. [ABSTRACT FROM AUTHOR]

Threshold dynamics and its extensions have proven useful in computing interfacial motions in applications as diverse as materials science and machine learning. Certain desirable properties of the algorithm, such as unconditional monotonicity in two-phase ows and gradient stability more generally, hinge on positivity properties of the convolution kernel and its Fourier transform. Recent developments in the analysis of this class of algorithms indicate that sometimes, as in the case of certain anisotropic curvature ows arising in materials science, these properties of the convolution kernel cannot be expected. Other applications, such as machine learning, would benefit from as great a level of exibility in choosing the convolution kernel as possible. In this paper, we establish certain desirable properties of threshold dynamics, such as gamma convergence of its associated energy, for a substantially wider class of kernels than has been hitherto possible. We also present variants of the algorithm that extend some of these properties to even wider classes of convolution kernels. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development of well-balanced high order numerical schemes for systems of balance laws with a linear flux function, whose coefficients may be variable. First, well-balanced first order numerical schemes are obtained based on the use of exact solvers of Riemann problems that include both the flux and the source terms. Godunov's methods so obtained are extended to higher order schemes by using a technique of reconstruction of states. The main contribution of this paper is to introduce a reconstruction technique that preserves the well-balanced property of Godunov's methods. Some numerical experiments are presented to verify in practice the properties of the developed numerical schemes. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of discrete boundary integral equations on polyhedral surfaces in three dimensions. When the solution contains typical edge singularities, highly stretched meshes are preferred to uniform meshes, since they reduce the number of degrees of freedom needed to obtain a fixed accuracy. The classical panel-clustering method can still be applied in the presence of such highly stretched meshes. However, we will show that the savings in computation time and storage become suboptimal because the nearfield matrix arising in the panel-clustering algorithm is no longer as sparse as it is in the case of uniform meshes. Hence, a natural question arises as to whether a new enhanced panel-clustering algorithm can be designed which performs efficiently even in the presence of highly stretched meshes. The main result of this paper is to formulate such an enhanced version of the panel-clustering algorithm. The key features of the algorithm are (i) the employment of partial analytic integration in the direction of stretching, yielding a new kernel function on a one-dimensional manifold where the influence of high aspect ratios in the stretched elements is removed, and (ii) the introduction of a generalized admissibility condition with respect to the partially integrated kernel, which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible. In the context of a model problem, we prove that our algorithm yields an accurate (up to the discretization error) matrix-vector multiplication which requires ...(NlogκN) operations, where N is the number of degrees of freedom and k is small and independent of the aspect ratio. The generalized admissibility condition can be viewed as an addition to the classical method which may be useful in general when stretched meshes are present. We also have performed a numerical experiment which shows that the sparsity of the nearfield matrix for the enhanced panel-clustering method is not negatively affected by stretched elements, and the method will perform optimally. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in [Y. Li and J. Zhou, SIAM J. Sci. Comput. 23 (2001), pp. 840--865; Y. Li and J. Zhou, SIAM J. Sci. Comput., 24 (2002), pp. 840--865]. First an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization. Then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations. Simple averaging formulas are used for the orthogonal projections. Numerical computations of examples with various symmetries, of which some can and others cannot be characterized by a compact group of linear isomorphisms, are carried out to confirm the theory and to illustrate applications. The mathematical features of various problems demonstrated in these examples fall into two categories: nodal solutions of saddle-point type with large Morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems. The new numerical algorithm generates these rather unstable solutions in an efficient and stable way. The existence of many unstable solutions and their behavior found in this paper remain to be investigated. [ABSTRACT FROM AUTHOR]

Local mesh-refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady-state convection-diffusion-reaction equations. The paper establishes residual-type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global H1- and L2-norms. Reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh-refining algorithms over uniform mesh refining. A discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper. [ABSTRACT FROM AUTHOR]

This paper discusses convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle-point problems presented in a recent paper of Bramble, Pasciak, and Vassilev (SIAM J. Numer. Anal., 34 (1997), pp. 1072­1092). It is shown that this algorithm converges under a condition weaker than that stated in their paper. Key words: saddle-point problems; nonlinear inexact Uzawa algorithm; convergence behavior [ABSTRACT FROM AUTHOR]

This paper reports the multiscale analysis and numerical algorithms for a class of distributed elliptic optimal control problems with constraints. Multiscale asymptotic expansions are presented, and an explicit rate of convergence is derived. A multiscale numerical method is developed. Numerical experiments are carried out to validate the predicted convergence results. [ABSTRACT FROM AUTHOR]

This paper investigates backscatter data for the inverse obstacle problem in impedance tomography, when the obstacles are small. It is shown that under this circumstance the backscatter data are close approximations of a rational function that has second or fourth order poles at the locations of the obstacles. Furthermore, a numerical method is presented to locate the obstacles via the poles of certain Laurent-Padé approximations. Numerical experiments explore the potential of this algorithm also for extended obstacles, taking into consideration that the problem is severely ill-posed. [ABSTRACT FROM AUTHOR]

In this paper, a FETI-DP formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered. To resolve the nonconformity of the finite elements, a mortar matching condition is imposed on the subdomain interfaces (faces). A FETI-DP algorithm is then built by enforcing the mortar matching condition in dual and primal ways. In order to make the FETI-DP algorithm scalable, a set of primal constraints, which include average and momentum constraints over interfaces, are selected from the mortar matching condition. A condition number bound, C(1 + log(H/h))², is then proved for the FETI-DP formulation for the elasticity problems with discontinuous material parameters. Only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed. Numerical results are included. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin (DG) and mixed finite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar finite elements to impose weak continuity of fluxes and pressures on the interface. The subdomain grids need not match, and the mortar grid may be much coarser, giving a two-scale method. Convergence results in terms of the fine subdomain scale h and the coarse mortar scale H are established for both types of couplings. In addition, a nonoverlapping parallel domain decomposition algorithm is developed, which reduces the coupled system to an interface mortar problem. The properties of the interface operator are analyzed. [ABSTRACT FROM AUTHOR]

A well-known generalization of Shannon's sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the Nyquist rate. Analogous to Shannon's sampling theorem this generalization requires that an infinite number of samples be available, which, however, is never the case in practice. Most existing reconstruction methods for periodic nonuniform sampling yield very low order (often not even first order) accuracy when only a finite number of samples is given. In this paper we propose a fast, numerically robust, root-exponential accurate reconstruction method. The efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized Fourier analysis. We discuss applications in analog-to-digital conversion where nonuniform periodic sampling arises in various situations. Finally, we demonstrate the performance of our algorithm by numerical examples. [ABSTRACT FROM AUTHOR]

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a bounded geometric domain such that the generating points of the tessellations are also the centroids (mass centers) of the corresponding Voronoi regions with respect to a given density function. Centroidal Voronoi tessellations may also be defined in more abstract and more general settings. Due to the natural optimization properties enjoyed by CVTs, they have many applications in diverse fields. The Lloyd algorithm is one of the most popular iterative schemes for computing the CVTs but its theoretical analysis is far from complete. In this paper, some new analytical results on the local and global convergence of the Lloyd algorithm are presented. These results are derived through careful utilization of the optimization properties shared by CVTs. Numerical experiments are also provided to substantiate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider two-dimensional elliptic problems with discontinuous coefficients discretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a dual-primal FETI method. In this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds. We show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters. Extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations (distribution of subregions, grid assignment, grid ratios, number of subregions) for both continuous and discontinuous problems. [ABSTRACT FROM AUTHOR]

Multigrid algorithms for Co interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper. It is shown that V -cycle, F-cycle and W-cycle algorithms are contractions if the number of smoothing steps is sufficiently large. The contraction numbers of these algorithms are bounded by Cm-α, where m is the number of presmoothing (and postsmoothing) steps, α is the index of elliptic regularity, and the positive constant C is mesh-independent. These estimates are established for a smoothing scheme that uses a Poisson solve as a preconditioner, which can be easily implemented because the Co finite element spaces are standard spaces for second order problems. Furthermore the variable V -cycle algorithm is also shown to be an optimal preconditioner. [ABSTRACT FROM AUTHOR]

In this paper we study the stability for all positive time of the fully implicit Euler scheme for the two-dimensional Navier—Stokes equations. More precisely, we consider the time discretization scheme and with the aid of the discrete Gronwall lemma and the discrete uniform Gronwall lemma we prove that the numerical scheme is stable. [ABSTRACT FROM AUTHOR]

We present and study a nonconforming domain decomposition method for the discretization of the three-dimensional Stokes problem in the velocity-pressure formulation. The approximation is based on some local mixed finite elements for nonmatching tetrahedral grids. The aim pursued is a systematic construction of the mortared discrete velocity space, the pressure being not subjected to any matching constraints across the interfaces. Using the bubble stabilization techniques, applied in Brezzi and Marini's paper to the three fields method [Math. Cornp., 70 (2001), pp. 911-934], allows us to define an algorithm which is easy to implement. The numerical analysis relies on the pressure-splitting argument of Boland and Nicolaides and allows us to establish an inf-sup condition with a constant that does not depend on the mesh size or on the total number of the subdomains. Then, by the Berger-Scott-Strang lemma written down for our saddle point system we derive optimal accuracy results. [ABSTRACT FROM AUTHOR]

Accurate numerical modeling of complex physical, chemical, and biological systems requires numerical simulation capability over a large range of length scales, with the ability to capture rapidly varying phenomena localized in space and/or time. Adaptive mesh refinement (AMR) is a numerical process for dynamically introducing local fine resolution on computational grids during the solution process, in response to unresolved error in a computation. Fast adaptive composite-grid (FAC) methods are a class of algorithms that exploit the multilevel structure of AMR grids to solve elliptic problems efficiently. This paper develops a theoretical foundation for AFACx, an asynchronous FAC method. A new multilevel condition number estimate establishes that the convergence rate of the AFACx algorithm does not degrade as the number of refinement levels in the AMR hierarchy increases. [ABSTRACT FROM AUTHOR]

Finding approximate solutions to systems of it nonlinear equations in it real variables is a much studied problem in numerical analysis. Somewhat more recently, researchers have developed numerical methods to provide mathematically rigorous error bounds on such solutions. (We say that we "verify" existence of the solution within those bounds on the variables.) However, when the Jacobi matrix is singular at the solution, no computational techniques to verify existence can handle the general case. Nonetheless, computational verification that one or more solutions exists within a region in complex space containing the real bounds is possible by computing the topological degree. In a previous paper, we presented theory and algorithms for the simplest case, when the rank-defect of the Jacobian matrix at the solution is 1 and the topological index is 2. Here, we generalize that result to arbitrary topological index d ≥ 2: We present theory, algorithms, and experimental results. We also present a heuristic for determining the degree, obtaining a value that we can subsequently verify with our algorithms. Although execution times are slow compared to corresponding bound verification processes for nonsingular systems, the order with respect to system size is still cubic. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give numerical algorithms and the error analysis for delay quadratic problems in the calculus of variations. These methods are new, efficient, and accurate and have a global a priori error of O(h²), where h is the distance between any two successive node points. We also derive the results for the general, numerical, delay problem, but we focus on proving our results in the simpler quadratic case since the extra technical numerical details have been given previously by the second author. In addition, the authors have previously shown how to reformulate general delay constrained problems in optimal control theory/constrained calculus of variations as unconstrained delay problems. Thus our numerical results and methods will hold for these general constrained problems also. Finally, we note that our algorithm, which solves the stationary condition(s) numerically, avoids the more difficult problems of piecing the solution of second order equations together and requires less smoothness in the solution. Thus we replace difficult second order boundary value problems with the easier task of approximating definite integrals involving first order derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper we show that the use of spectral Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with infinite-order accuracy momentum and energy. Consistency of the method is also proved, and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we introduce an overlapping mortar finite element method for solving two-dimensional elliptic problems discretized on overlapping nonmatching grids. We prove an optimal error bound and estimate the condition numbers of certain overlapping Schwarz preconditioned systems for the two-subdomain case. We show that the error bound is independent of the size of the overlap and the ratio of the mesh parameters. In the second part, we introduce three additive Schwarz preconditioned conjugate gradient algorithms based on the trivial and harmonic extensions. We provide estimates for the spectral bounds on the condition numbers of the preconditioned operators. We show that although the error bound is independent of the size of the overlap, the condition number does depend on it. Numerical examples are presented to support our theory. [ABSTRACT FROM AUTHOR]

The mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains. The resulting methods are nonconforming but still yield optimal approximations. In this paper, we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric, second-order, elliptic equations in two dimensions. Both spectral and finite element methods, for geometrically conforming as well as nonconforming domain decompositions, are studied. In each case, we obtain a polylogarithmic bound on the condition number of the preconditioned matrix. [ABSTRACT FROM AUTHOR]

In this paper, we study the additive Schwarz method for the h-p version of the finite element method in three dimensions. The main idea is to treat separately the h-version (linear) components and the p-version (high-order) components by a vertex-based method. It can also be viewed as a three-level method with the level being the linear finite element approximation on the coarse mesh, the linear finite element approximation on the fine mesh, and the high-order finite element approximation on the fine mesh, respectively. The resulting algorithm can be implemented in parallel on the subdomain level for the h-version components and on the element level for the p-version components. The condition number is of order [This symbol cannot be presented in ASCII format](1 + ln Hipi/hi)⇔2, where Hi stands for the diameter of the subdomain Ωi, hi is the diameter of the elements in Ωi, and pi is the maximum of the polynomial degrees used in Ωi. [ABSTRACT FROM AUTHOR]

In this paper the problem of computing bifurcation diagrams for large-scale nonlinear parameter-dependent steady state systems which arise following the spatial discretization of semilinear PDEs is investigated. A continuation algorithm which employs a preconditioned version of the recursive projection method (RPM) is presented. The RPM is often expensive when it is used in conjunction with the numerical method of lines. Preconditioning the Jacobian of the underlying fixed point operator results in an algorithm (the preconditioned recursive projection method (PRPM)) which is capable of efficiently computing equilibrium solution diagrams of large stiff systems. For many PDE problems the PRPM is a fast and effective means of detecting both steady state and Hopf bifurcation along a branch of solutions. A description of the performance of the PRPM when applied to two numerical examples is given. [ABSTRACT FROM AUTHOR]