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This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "P. L. Chebyshev -- 200," and it describes the results of their joint studies with Manasa Mandava (1984--2019). [ABSTRACT FROM AUTHOR]

In this paper, we propose an embedded low-regularity integrator (ELRI) under a new framework for solving the modified Korteweg-de Vries (mKdV) equation under rough data. Different from the previous work [Wu and Zhao, BIT, Number. Math., (2021)], the present ELRI scheme is constructed based on an approximation of a scaled Schrödinger operator and a new strategy of iterative regularizing through the inverse Miura transform. Moreover, the ELRI scheme is explicitly defined in the physical space, and it is efficient under the Fourier pseudospectral discretization. By rigorous error analysis, we show that ELRI achieves first-order accuracy by requiring the boundedness of one additional spatial derivative of the solution. Numerical results are presented to show the accuracy and efficiency of ELRI. [ABSTRACT FROM AUTHOR]

The article discusses research featured in the Expository Research Papers section including one by Laurent Demanet and Lexing Ying on the efficient representations for functions of elliptic operators and another by Jeffrey Blanchard, Coralia Cartis and Jared Tanner on restricted isometry property.

In this paper, we analyze a class of reaction-diffusion systems with delay and a Neumann condition from the dynamical behavior of the map that determines the equilibria. For scalar equations, a similar analysis was given by Yi and Zou in [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), pp. 2955--2973]. There are strong differences between the scalar equations and the systems. We show with an example that the direct extension of the results by Yi and Zou in [Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), pp. 2955--2973] to systems is not possible. This forces us to introduce additional assumptions. However, our approach is sufficiently general to be applicable to a wide variety of models, including neural networks and population dynamics. In some cases, our technique gives the optimal conditions for the global attraction. [ABSTRACT FROM AUTHOR]

This erratum clarifies that equations (4.21) and (4.23) in Böttcher and Porter [SIAM J. Appl. Math., 81 (2021), pp. 2704-2724] assume that the underlying evolution operator has an orthonormal basis. The results and conclusions of the paper are unaffected by this clarification. [ABSTRACT FROM AUTHOR]

This paper mainly proves the generic properties of the Camassa-Holm equation and the two-component Camassa-Holm equation in a sense of Baire category. We reveal their differences in generic regularity and singular behavior. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we study domain decomposition preconditioners for multiscale flowsin high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent ofthe contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper. [ABSTRACT FROM AUTHOR]

In this paper we analyze iterative regularization with the Bregman distance of the total variation seminorm. Moreover, we prove existence of a solution of the corresponding flow equation as introduced in [M. Burger, G. Gilboa, S. Osher, and J. Xu, Commun. Math. Sci., 4 (2006), pp. 179-212] in a functional analytical setting using methods from convex analysis. The results are generalized to variational denoising methods with Lp-norm fit-to-data terms and Bregman distance regularization terms. For the associated flow equations well-posedness is derived using recent results on metric gradient flows from [L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005]. In contrast to previous work the results of this paper apply for the analysis of variational denoising methods with the Bregman distance under adequate noise assumptions. Aside from the theoretical results we introduce a level set technique based on Bregman distance regularization for denoising of surfaces and demonstrate the efficiency of this method. [ABSTRACT FROM AUTHOR]

In the first part of this paper we study the regularity properties of solutions of initial- or boundary-value problems of linear Fredholm integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid. [ABSTRACT FROM AUTHOR]

This paper extends the derivation of the Lagrangian averaged Euler (LAE-$\alpha$) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Along the way, the derivation of the isotropic and anisotropic LAE-$\alpha$ equations is simplified and clarified. The derivation in this paper involves averaging over a tube of trajectories $\eta^\epsilon$ centered around a given Lagrangian flow $\eta$. With this tube framework, the LAE-$\alpha$ equations are derived by following a simple procedure: start with a given action, Taylor expand in terms of small-scale fluid fluctuations $\xi$, truncate, average, and then model those terms that are nonlinear functions of $\xi$. Closure of the equations is provided through the use of flow rules, which prescribe the evolution of the fluctuations along the mean flow. [ABSTRACT FROM AUTHOR]

In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg--Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg--Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: (1) it deals with two coupled partial differential equations, and (2) under certain circumstances handles equations defined on a semi-infinite domain. [ABSTRACT FROM AUTHOR]

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X½ SX½ = vW, where W is a fixed positive definite matrix and v>0 is a parameter, under the assumption that the problem has a strictly complementary primal-dual optimal solution. It is shown that a weighted central path as a function of √v can be extended analytically beyond 0 and hence that the path converges as v↓0. Characterization of the limit points of the path and its normalized first-order derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as &Tetha;(v) or asT &Tetha;(√v) depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal-dual optimal solutions. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primal-dual interior-point SDP algorithms. [ABSTRACT FROM AUTHOR]

In this paper we apply the generalized Taylor­Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rig- orously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic dis- continuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also con- sidered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver. [ABSTRACT FROM AUTHOR]

In this paper we consider domain decomposition methods for solving the radial basis function interpolation equations. There are three interwoven threads to the paper. The first thread provides good ways of setting up and solving small- to medium-sized radial basis function interpolation problems. These may occur as subproblems in a domain decomposition solution of a larger interpolation problem. The usual formulation of such a problem can suffer from an unfortunate scale dependence not intrinsic in the problem itself. This scale dependence occurs, for instance, when fitting polyharmonic splines in even dimensions. We present and analyze an alternative formulation, available for all strictly conditionally positive definite basic functions, which does not suffer from this drawback, at least for the very important example previously mentioned. This formulation changes the problem into one involving a strictly positive definite symmetric system, which can be easily and efficiently solved by Cholesky factorization. The second section considers a natural domain decomposition method for the interpolation equations and views it as an instance of von Neumann's alternating projection algorithm. Here the underlying Hilbert space is the reproducing kernel Hilbert space induced by the strictly conditionally positive definite basic function. We show that the domain decomposition method presented converges linearly under very weak nondegeneracy conditions on the possibly overlapping subdomains. The last section presents some algorithmic details and numerical results of a domain decomposition interpolatory code for polyharmonic splines in 2 and 3 dimensions. This code has solved problems with 5 million centers and can fit splines with 10,000 centers in approximately 7 seconds on very modest hardware. [ABSTRACT FROM AUTHOR]

This paper addresses the qualitative behavior of a nonlinear convection-diffusion equation on a smooth bounded domain in Rn , in which the strength of the convection grows superlinearly as the density increases. While the initial-boundary value problem is guaranteed to have a local-in-time solution for smooth initial data, it is possible for this solution to be extinguished in finite time. We demonstrate that the way this may occur is through finite-time "blow up," i.e., the unboundedness of the solution in arbitrarily small neighborhoods of one or more points in the closure of the spatial domain. In special circumstances, such as the presence of radial symmetry, the set of blowup points can be identified; these points may be either on the boundary or on the interior of the domain. Furthermore, criteria can be established that guarantee that blowup occurs. In this paper, such criteria are presented, involving the dimension of the space, the growth rate of the nonlinearity, the strength of the imposed convection field, the diameter of the domain, and the mass of the initial data. Furthermore, the temporal rate of blowup is estimated. [ABSTRACT FROM AUTHOR]

In this paper, we consider the finite-dimensional approximations of Tikhonov regularization for nonlinear ill-posed problems with approximately given right-hand sides. We propose an a posteriori parameter choice strategy, which is a modified form of Morozov's discrepancy principle, to choose the regularization parameter. Under certain assumptions on the nonlinear operator, we obtain the convergence and rates of convergence for Tikhonov regularized solutions. This paper extends the results, which were developed by Plato and Vainikko in 1990 for solving linear ill-posed equations, to nonlinear problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with refinement equations of the type [This symbol cannot be presented in ASCII format] where f is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported sequence on Zs, and M is an s × s dilation matrix with m := | det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach. [ABSTRACT FROM AUTHOR]

To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century, when Carl Friedrich Gauss developed the method of least squares of a system of linear equations--its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful generalization of this problem is to find fixed points of averaged resolvents (i.e., firmly nonexpansive mappings). This paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents. It partially extends recent work for two mappings on a question of C. Byrne. The analysis suggests a reformulation in a product space. Algorithmic consequences are also presented. [ABSTRACT FROM AUTHOR]

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

We consider the problem ε²Δu + (u - a(x))(1 - u²) = 0 in Ω, ∂u/ ∂v = 0 on ∂Ω, where Ω is a smooth and bounded domain in R², -1 < a(x) < 1. Assume that Γ = {x ∈ Ω, a(x) = 0} is a closed, smooth curve contained in Ω in such a way that Ω = Ω+ ∪ Γ ∪ Ω - and ∂a/∂n < 0 on Γ, where n is the outer normal to Ω+. Fife and Greenlee [Russian Math. Surveys, 29 (1974), pp. 103-131] proved the existence of an interior transition layer solution uε which approaches -1 in Ω- and +1 in Ω+, for all ε sufficiently small. A question open for many years has been whether an interior transition layer solution approaching 1 in Ω- and -1 in Ω+ exists. In this paper, we answer this question affirmatively when n = 2, provided that ε is small and away from certain critical numbers. A main difficulty is a resonance phenomenon induced by a large number of small critical eigenvalues of the linearized operator. [ABSTRACT FROM AUTHOR]

In this paper we present a unified constructive method to establish the local exact boundary controllability for 1-D quasilinear wave equations with boundary conditions of various types. [ABSTRACT FROM AUTHOR]

This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian-Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included. [ABSTRACT FROM AUTHOR]

The additive Schwarz preconditioned inexact Newton (ASPIN) method was recently introduced [X.-C. Cai and D. E.\ Keyes, {\it SIAM J. Sci.\ Comput.}, 24 (2002), pp. 183--200] to solve the systems of nonlinear equations with nonbalanced nonlinearities. Although the ASPIN method has successfully been used to solve some difficult nonlinear equations, its convergence property has not been studied since it was proposed. In this paper, the convergence property of the ASPIN method is studied, and the obtained result shows that this method is locally convergent. Furthermore, the convergence rate for the ASPIN method is discussed and the obtained result is similar to that of the inexact Newton method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the doubling transformation, a structure-preserving transformation for symplectic pencils, and present its basic properties. Based on these properties, a unified convergence theory for the structure-preserving doubling algorithms for a class of Riccati-type matrix equations is established, using only elementary matrix theory. [ABSTRACT FROM AUTHOR]

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem. [ABSTRACT FROM AUTHOR]

This paper focuses on upscaling of the transport equation for heterogeneous porous media with random flow. We develop an upscaling by the coarse graining method which considers the local flow field being a stationary random field and which is based on filtering procedures in Fourier space. The coarse graining method is used to obtain an upscaled dispersion tensor which depends on the given length scale of the upscaling. We give explicit results for the scale-dependent dispersion coefficient in lowest-order perturbation theory. For a global upscaling the upscaled value agrees with the well-known macrodispersion coefficient. Furthermore, we develop a numerical upscaling scheme based on coarse graining which allows us to test the theoretical results. The numerical coarse graining also allows us to compute the upscaled dispersion for higher variances. The theoretical results are in agreement with the numerical simulations. [ABSTRACT FROM AUTHOR]

Multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper. After showing how to convert a general multirate periodic system to an equivalent linear time invariant (LTI) system with a structural constraint, we formulate some analytic function interpolation problems with such a constraint that can find various applications in the study of multirate and periodic systems. Both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems. [ABSTRACT FROM AUTHOR]

The main objective of this paper consists in discussing the concept of weak solutions of a certain type of backward stochastic differential equations. Using weak convergence in the Meyer--Zheng topology, we shall give a general existence result. The terminal condition $H$ depends in functional form on a driving cadlag process $X$, and the coefficient $f$ depends on time $t$ and in functional form on $X$ and the solution process $Y$. The functional $f(t,x,y),(t,x,y)\in [0,T]\times D([0,T];{\bf R}^{d+m})$ is assumed to be bounded and continuous in $(x,y)$ on the Skorokhod space $D([0,T]\,;{\bf R}^{d+m})$ in the Meyer--Zheng topology. By several examples of Tsirelson type, we will show that there are, indeed, weak solutions which are not strong, i.e., are not solutions in the usual sense. We will also discuss pathwise uniqueness and uniqueness in law of the solution and conclude, similar to the Yamada--Watanabe theorem, that pathwise uniqueness and weak existence ensure the existence of a (uniquely determined) strong solution. Applying these concepts, we are able to state the existence of a (unique) strong solution if, additionally to the assumptions described above, $f$ satisfies a certain generalized Lipschitz-type condition. [ABSTRACT FROM AUTHOR]

In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94-111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577-606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of sup erconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions. [ABSTRACT FROM AUTHOR]

The stress-driven grain boundary diffusion problem is a continuum model of mass transport phenomena in microelectronic circuits due to high current densities (electromigration) and gradients in normal stress along grain boundaries. The model involves coupling many different equations and phenomena, and difficulties such as nonlocality, complex geometry, and singularities in the stress tensor have left open such mathematical questions as existence of solutions and compatibility of boundary conditions. In this paper and its companion, we address these issues and establish a firm mathematical foundation for this problem. We study the properties of a type of Dirichlet-to-Neumann map that involves solving the Lamé equations with interesting interface boundary conditions. We identify a new class of degenerate grain boundary networks that lead to unsuppressed linear growth modes that are suggestive of continental drift in plate tectonics. We use techniques from semigroup theory to prove that the problem is well posed and that the stress field relaxes to a steady state distribution which may or may not completely balance the electromigration force. In the latter (degenerate) case, the displacements continue to grow without bound along stress-free modes. [ABSTRACT FROM AUTHOR]

The stress-driven grain boundary diffusion problem is a continuum model of mass transport phenomena in microelectronic circuits due to high current densities (electromigration) and gradients in normal stress along grain boundaries. The model involves coupling many different equations and phenomena, and difficulties such as nonlocality, complex geometry, and singularities in the stress tensor have left open such mathematical questions as existence of solutions and compatibility of boundary conditions. In this paper and its companion, we address these issues and establish a firm mathematical foundation for this problem. We use techniques from semigroup theory to prove that the problem is well posed and that the stress field relaxes to a steady state distribution which, in the nondegenerate case, balances the electromigration force along grain boundaries. Our analysis shows that while the role of electromigration is important, it is the interplay among grain growth, stress generation, and mass transport that is responsible for the diffusive nature of the problem. Electromigration acts as a passive driving force that determines the steady state stress distribution, but it is not responsible for the dynamics that drive the system to steady state. We also show that stress singularities may develop near grain boundary junctions; however, stress components directly involved in the diffusion process remain finite for all time. Thus, we have identified a mechanism by which large "hidden" stresses may develop that are not directly involved in the diffusion process but may play a role in void nucleation and stress-induced damage. [ABSTRACT FROM AUTHOR]

In this paper we develop numerical methods for integrating general evolution equations ut = F(u), u(0) = u0, where F is defined on a dense subspace of some Banach space (generally infinite-dimensional) and is equivariant with respect to the action of a finite-dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDEs on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time-dependent group element with a "frozen solution" in the given Banach space. We keep the frozen solution as constant as possible by introducing a set of algebraic constraints (phase conditions), the number of which is given by the dimension of the Lie group. The resulting PDAE (partial differential algebraic equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Euler methods in time, and finite differences in space. We provide applications to reaction-diffusion systems that have traveling wave or spiral solutions in one and two space dimensions. [ABSTRACT FROM AUTHOR]

This paper is about the construction of a BGK Navier-Stokes (BGK-NS) solver in the discontinuous Galerkin (DG) framework. Since in the DG formulation the conservative variables and their slopes can be updated simultaneously, the flow evolution in each element involves only the flow variables in the nearest neighboring cells. Instead of using the semidiscrete approach in the Runge-Kutta discontinuous Galerkin (RKDG) method, the current DG-BGK method integrates the governing equations in time as well. Due to the coupling of advection and dissipative terms in the gas-kinetic formulation, the DG-BGK method solves the viscous governing equations directly. Numerical examples for the one-dimensional compressible Navier-Stokes solutions will be presented. [ABSTRACT FROM AUTHOR]

In this paper, we develop a numerical scheme for the Smoluchowski coagulation equation, which relies on a conservative formulation and a finite volume approach. Several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model. In particular the gelation phenomenon and the long time behavior of the solution are numerically studied. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient branch-following procedure that can be used not only to compute branches of periodic solutions of periodically forced dynamical systems but also to determine the stability of the periodic solutions. The procedure combines Broyden's method with a subspace iteration method to determine the dominant eigenvalues. The method has connections with the hybrid Newton-Picard methods developed by Lust et al. in [SIAM 1. Sci. Comput., 19 (1998) pp. 1188-12091. A convergence analysis of the procedure is presented. The method is applied to the computation of periodic states of a reverse flow reactor, and its performance is compared with two variants of the hybrid Newton-Picard method. [ABSTRACT FROM AUTHOR]

The central result of this paper is a new nonlinear equation which describes the evolution of the oriented distance function b... of a set ... with thin boundary under the influence of a velocity field. We relate it to equations and constructions used in the context of level set methods. We further introduce a new moving narrow-band method which not only can be readily implemented to solve our evolution equation, but could also be used for equations of motion by curvatures. In the process we review and sharpen the characterization of smooth sets and manifolds and sets of positive reach (e.g., local semiconvexity in an extended sense of the oriented distance function of the closure of the set). For W2,p-Sobolev domains a new characterization and a compactness theorem are given in terms of the Laplacian of the oriented distance function rather than its whole Hessian matrix. [ABSTRACT FROM AUTHOR]

This paper studies the numerical solution of two-dimensional nonlinear degenerate reaction-diffusion differential equations with singular forcing terms over rectangular domains. The equations considered may generate strong quenching singularities. This investigation focuses on a variable time step Peaceman­Rachford splitting method for the aforementioned problem. The time adaptation is implemented based on arc-length estimations of the first time derivative of the solution. The two-dimensional problem is split into several one-dimensional problems so that the computational cost is significantly reduced. The monotonicity and localized linear stability of the variable step scheme are investigated. We give some numerical examples to illustrate our results as well as to demonstrate the viability and efficiency of the method over existing methods for the quenching problem. It is also shown that the numerical solution obtained preserves important properties of the physical solution of the given problem. [ABSTRACT FROM AUTHOR]

In this paper we study the large time behavior of two decoupled solitary waves of the generalized KdV equations u t + (u xx +f (u))x = 0, where f (u) = Ι u Ι p -1 u/p (3 =p≶5). We prove that if the speeds of the solitary waves are sufficiently close at the initial time, the leading wave becomes larger and the trailing wave becomes smaller, and the distance between two solitary waves becomes larger as t → ∝. [ABSTRACT FROM AUTHOR]

This paper describes a new O(n log3(n)) solver for the positive definite Toeplitz system Tx = b. Instead of computing generators for the inverse of T, the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The generators of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases. [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 present an efficient transformation technique for accurate numerical evaluation of weakly singular integrals with interior singularities. The present transformation technique does not require any division of the integration interval. It is composed of two parts of a sigmoidal transformation whose tails coincide with a singular point smoothly up to the order of the sigmoidal transformation employed. Therefore, in using the standard Gauss quadrature rule, the present transformation has a feature in which the integration points are clustered into the interior singular point very closely, as in the case of endpoint singular integrals. The results of some numerical examples show the superiority of the present method over the existing methods. [ABSTRACT FROM AUTHOR]

In this paper we describe a simple algorithm for the approximation of an invariant curve T of a planar map f. We are particularly interested in the case where T is attracting but not smooth. Results for the delayed logistic map illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

This paper is concerned with monotone iterative methods for numerical solutions of a class of nonlinear fourth-order elliptic boundary value problems in a two-dimensional domain. The boundary value problem is discretized by the finite difference method, and two iterative processes, called block Jacobi and block Gauss­Seidel monotone iterations, are presented for the computation of solutions of the finite difference system using either an upper solution or a lower solution as the initial iteration. It is shown that the sequence of iterations converges monotonically to a maximal solution or a minimal solution if the nonlinear function is quasi-monotone nondecreasing. A sufficient condition is given to ensure that the maximal and minimal solutions coincide and their common value is the unique solution of the finite difference system. Similar results are obtained for quasi-monotone nonincreasing functions. An analytical comparison relation between the block Jacobi and block Gauss­Seidel monotone iterations is obtained. It is also shown that the finite difference solution converges to the continuous solution as the mesh size tends to zero. Numerical results by the block monotone iterative schemes are given for two model problems and are compared with the known analytical solutions for accuracy. Also compared are the rates of convergence of the two types of block monotone iterations as well as similar types of pointwise monotone iterations. [ABSTRACT FROM AUTHOR]

We introduce spectral element-based algebraic multigrid (ρAMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritz-type finite element discretizations of partial differential equations. The method requires access to the element stiffness matrices, which enables accurate approximation of algebraically “smooth” vectors (i.e., error components that relaxation cannot effectively eliminate). Most other algebraic multigrid methods are based in some manner on predefined concepts of smoothness. Coarse-grid selection and prolongation, for example, are often defined assuming that smooth errors vary slowly in the direction of “strong” connections (relatively large coefficients in the operator matrix). One aim of ρAMGe is to broaden the range of problems to which the method can be successfully applied by avoiding any implicit premise about the nature of the smooth error. ρAMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically smooth error components. This provides a foundation for generating the coarse level and for defining effective interpolation. This paper presents a theoretical foundation for ρAMGe along with numerical experiments demonstrating its robustness. [ABSTRACT FROM AUTHOR]

In this paper, a discrete singular convolution (DSC) algorithm is introduced to solve Fisher’s equation, to which obtaining an accurate and reliable traveling wave solution is a challenging numerical problem. Two novel numerical treatments, a moving frame scheme and a new asymptotic scheme, are designed to overcome the subtle difficulties involved in the numerical solution of Fisher’s equation. The moving frame scheme is proposed to provide accurate and efficient spatial resolution of the problem. The new asymptotic scheme is introduced to facilitate the correct prediction of the wave speed after long-time integrations. The resulting DSC algorithm is able to correctly predict long-time traveling wave behavior. The performance of the present algorithm is demonstrated through extensive numerical experiments as well as through a comparison with three other standard approaches, i.e., methods of the Fourier pseudospectral, the accurate spatial derivatives, and the Crank­Nicolson schemes. The spatial and temporal accuracies of all four methods are analyzed and numerically verified. [ABSTRACT FROM AUTHOR]

If, this paper, we study traveling waves if, a suspension bridge system governed by the coupled nonlinear wave and beam equations describing oscillations in the supporting cable and roadbed. By applying the variational method, it is proved that the suspension bridge system has at least one nontrivial traveling wave. [ABSTRACT FROM AUTHOR]

In this paper, by means of the concept of the working set, which is an estimate of the active set, we propose a feasible sequential linear equation algorithm for solving inequality constrained optimization problems. At each iteration of the proposed algorithm, we first solve one system of linear equations with a coefficient matrix of size m x m (where m is the number of constraints) to compute the working set; we then solve a subproblem which consists of four reduced systems of linear equations with a common coefficient matrix. Unlike existing QP-free algorithms, the subproblem is concerned with only the constraints corresponding to the working set. The constraints not in the working set are neglected. Consequently, the dimension of each subproblem is not of full dimension. Without assuming the isolatedness of the stationary points, we prove that every accumulation point of the sequence generated by the proposed algorithm is a KKT point, of the problem. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. In other words, after finitely many steps, only those constraints which are active at the solution will be involved in the subproblem. Under some additional conditions, we show that the convergence rate is two-step superlinear or even Q-superlinear. We also report some preliminary numerical experiments to show that the proposed algorithm is practicable and effective for the test problems. [ABSTRACT FROM AUTHOR]

In this paper we derive optimal a priori L[SUP&infi;](L[SUP2]) error estimates for mixed finite element displacement formulations of the acoustic wave equation. The computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure. However, the displacement formulations with the physical variables of interest, displacement and pressure, requires less regularity on the displacement. [ABSTRACT FROM AUTHOR]