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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.

It is shown that bubbling on wine bottle labels is due to absorption of water from the glue, with subsequent hygroscopic expansion. Contrary to popular belief, most of the glue's water must be lost to the atmosphere rather than to the paper. A simple lubrication model is developed for spreading glue piles in the pressure chamber of the labeling machine. This model predicts a maximum rate for application of labels. Buckling theory shows that the current arrangement of periodic glue strips can indeed accommodate paper expansion. This project provides interesting applications of various areas of undergraduate mathematics, such as trigonometry, Maclaurin series, dimensional analysis, and fluid mechanics. It illustrates that simple mathematical modeling may provide insight into complicated real-world problem. [ABSTRACT FROM AUTHOR]

Introduces a series of articles on industrial and applied mathematics.

The article discusses the paper "Some New Aspects of the Coupon Collectors Problem," by researchers Amy Myers and Herbert Wilf. It mentions that the number of trials that it will take for one collector to obtain a full set of d coupons if at each trial the collector has an equal chance of selecting any one of the d possible coupons is the problem of the classical coupon collector.

In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the tilne-harnlonic scattering problems ill two-layered media. The exterior region of the scatterer is divided into two half spaces by all infinite plane, on two sides of which the wave number takes different vahms. We surround the conlputational donmin where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational dolnain as either the PML absorbing coefficient or the thickness of the PML tends to infinity. [ABSTRACT FROM AUTHOR]

In their 2006 paper, Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. prove that a sufficiently smooth strong solution of the 3D Navier-Stokes equations is robust with respect to small enough changes in initial conditions and forcing function. They also show that if a regular enough strong solution exists, then Galerkin approximations converge to it. They then use these results to conclude that the existence of a sufficiently regular strong solution can be verified using sufficiently refined numerical computations. In this paper we study the strong solutions with less regularity than those considered in Chernyshenko et al. [J. Math. Phys., 48 (2007), 065204, 15 pp]. We prove a similar robustness result and show the validity of the results relating convergent numerical computations and the existence of the strong solutions. [ABSTRACT FROM AUTHOR]

Over the last few years a great effort has been made to develop monotone high order explicit Runge--Kutta methods by means of their Shu--Osher representations. In this context, the stepsize restriction to obtain numerical monotonicity is normally computed using the optimal representation. In this paper we extend the Shu--Osher representations for any Runge--Kutta method giving sufficient conditions for monotonicity. We show how optimal Shu--Osher representations can be constructed from the Butcher tableau of a Runge--Kutta method. The optimum stepsize restriction for monotonicity is given by the radius of absolute monotonicity of the Runge--Kutta method [L. Ferracina and M. N. Spijker, SIAM J. Numer. Anal., 42 (2004), pp. 1073--1093], and hence if this radius is zero, the method is not monotone. In the Shu--Osher representation, methods with zero radius require negative coefficients, and to deal with them, an extra associate problem is considered. In this paper we interpret these schemes as representations of perturbed Runge--Kutta methods. We extend the concept of radius of absolute monotonicity and give sufficient conditions for monotonicity. Optimal representations can be constructed from the Butcher tableau of a perturbed Runge--Kutta method. [ABSTRACT FROM AUTHOR]

We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where $H$ is the Heaviside function and $f$ and $g$ are smooth with the assumptions that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with $f$ convex and $k$ of indefinite sign. [ABSTRACT FROM AUTHOR]

We consider processes in which a focused laser beam is used to induce the melting of silicium. The first goal of this paper is to propose a simple three-dimensional (3D) model of this melting process. Our model is partly based on an energy balance equation. This model leads to a nontrivial ODE describing the evolution in time of the dimension of the melt region. The second goal of this paper is to obtain approximate analytical solutions of this ODE. After using basic solution methods, we propose an original geometrical method to derive asymptotic solutions for time → ∞. These solutions turn out to be the most useful for the description of this process. [ABSTRACT FROM AUTHOR]

This paper continues the analysis started in the first part of this article (cf. [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198-1223]). It was seen there, using the method of matched asymptotics, that a regularized version of the Keller-Segel system admits, for a suitable asymptotic limit, solutions with some regions of high concentrations for the cell density. This paper considers the relation between the phenomenon of blow-up for the limit problem and the dynamics of the concentration regions described in [J. J. L. Velázquez, SIAM J. Appl. Math., 64 (2004), pp. 1198- 1223]. In particular, this paper analyzes the precise way in which the regularization introduced in the Keller-Segel system stops the aggregation process and yields the formation of concentration regions. [ABSTRACT FROM AUTHOR]

Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of tile vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004 there have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS--exact reconstruction from seemingly undersampled measurements--it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry. [ABSTRACT FROM AUTHOR]

This paper is concerned with delay-independent asymptotic stability of linear neutral delay differential-algebraic equations and linear multistep methods. We first give some sufficient conditions for the delay-independent asymptotic stability of these equations. Then we study and derive a sufficient and necessary condition for the delay-independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with Lagrange interpolation. Finally, one numerical example is performed to confirm our theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we study the relation between the error estimate of the bilinear interpolation on a general quadrilateral and the geometric characters of the quadrilateral. Some explicit bounds of the interpolation error are obtained based on some sharp estimates of the integral over 1/∣J∣p-1 for 1 ≤ p≤∞ on the reference element, where J is the Jacobian of the nonaffine mapping. This allows us to introduce weak geometric conditions (depending on p) leading to interpolation error estimates in the W1,p norm, for any p ϵ [1,∞), which can be regarded as a generalization of the regular decomposition property (RDP) condition introduced in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 38 (2000), pp. 1073-1088] for p = 2 and new RDP conditions (NRDP) for p ≠ 2. We avoid the use of the reference family elements, which allows us to extend the results to a larger class of elements and to introduce the NRDP condition in a more unified way. As far as we know, the mesh condition presented in this paper is weaker than any other mesh conditions proposed in the literature for any p with 1 ≤ p≤∞. [ABSTRACT FROM AUTHOR]

In this paper, the frequency domain-based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation. First, the stability boundary curve is exactly determined for these methods. Then, this boundary is analyzed and compared with an accurate (ideal) boundary in different frequency ranges. Also, the critical regions in which the stability does not preserve are determined. Finally, the analytical achievements are confirmed via some numerical illustrations. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior. [ABSTRACT FROM AUTHOR]

In this paper, we consider the singularly perturbed reaction-diffusion problem in one and two dimensions. The boundary value problem is decomposed into a first-order system to which a suitable weighted least-squares formulation is proposed. A robust, stable, and efficient approach is developed based on local discontinuous Galerkin (LDG) discretization for the weak form. Uniform error estimates are derived. Numerical examples are presented to illustrate the method and the theoretical results. Comparison studies are made between the proposed method and other methods. [ABSTRACT FROM AUTHOR]

This paper presents analysis of a weighted-norm least squares finite element method for elliptic problems with boundary singularities. We use H(div) conforming Raviart-Thomas elements and continuous piecewise polynomial elements. With only a rough estimate of the power of the singularity, we employ a simple, locally weighted L² norm to eliminate the pollution effect and recover better rates of convergence. Theoretical results are carried out in weighted Sobolev spaces and include ellipticity bounds of the homogeneous least-squares functional, new weighted Raviart-Thomas interpolation results, and error estimates in both weighted and nonweighted norms. Numerical tests are given to confirm the theoretical estimates and to illustrate the practicality of the method. [ABSTRACT FROM AUTHOR]

The classical error analysis for the Raviart-Thomas interpolation on triangular elements requires the so-called regularity of the elements, or equivalently, the minimum angle condition. However, in the lowest order case, optimal order error estimates have been obtained in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 37 (2000), pp. 18-36] replacing the regularity hypothesis by the maximum angle condition, which was known to be sufficient to prove estimates for the standard Lagrange interpolation. In this paper we prove error estimates on triangular elements for the Raviart-Thomas interpolation of any order under the maximum angle condition. Also, we show how our arguments can be extended to the three-dimensional case to obtain error estimates for tetrahedral elements under the regular vertex property introduced in [G. Acosta and R. G. Durán, SIAM J. Numer. Anal., 37 (2000), pp. 18-36]. [ABSTRACT FROM AUTHOR]

The Lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer. Reducing average distortion at every step, it constructs a Voronoi partition of the domain and replaces each generator with the centroid of the corresponding Voronoi cell. Optimal quantization is obtained in the case of a centroidal Voronoi tessellation (CVT), which is a special Voronoi tessellation of a domain Ω ϵ ℝd having the property that the generators of the Voronoi diagram are also the centers of mass, with respect to a given density function ? ⩾ 0, of the corresponding Voronoi cells. The Lloyd iteration is currently the most popular and elegant algorithm for computing CVTs and optimal quantizers, but many questions remain about its convergence, especially in d-dimensional spaces (d > 1). In this paper, we prove that any limit point of the Lloyd iteration in any dimensional spaces is nondegenerate provided that Ω is a convex and bounded set and ? belongs to L¹(Ω) and is positive almost everywhere. This ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the Lloyd iteration to the set of nondegenerate fixed point quantizers. While previously only conjectured, the convergence properties of the Lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional. The results presented in this paper go beyond existing convergence theories for CVTs and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities. [ABSTRACT FROM AUTHOR]

In this paper several Nyström methods for the van der Pol equation are considered. In an earlier study by Cai, Aoyagi, and Abe it was shown that the second order Nyström, or leapfrog, method fails to approximate the limit cycle of the van der Pol equation, exhibiting a periodic modulation of the amplitude and sporadic high frequency noise instead. Cai et al. did a linear analysis and concluded that the spurious behavior was due to the interaction of the main part of the solution with a high frequency computational mode. In this paper we also apply a third and fourth order Nyström method to the van der Pol equation. Numerical experiments show that in these cases the high frequency mode causes blowup after some time. The onset of the instability can be delayed by decreasing the time step. We also improve on their analysis of the second order scheme by doing a nonlinear analysis, to wit a discrete multiple scales analysis. By this means we are able to explain the spurious behavior of this system completely. [ABSTRACT FROM AUTHOR]

For the first time, we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard C¹ finite element method (FEM). We consider, throughout the paper, two cases in parallel: For convex, bounded, polyhedral domains in Rn, or for C² bounded domains in R², we prove stability and convergence for the corresponding conforming or nonconforming C¹ FEM, respectively. The results for equations and systems of orders 2 and 2m and quadrature approximations appear elsewhere. The classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator. The consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated. The stability has to be proved in an unusual way. This is the hard core of the paper. Essential tools are linearization, a compactness argument, the interplay between the weak and strong form of the linearized operator, and a new regularity result for solutions of finite element equations. An essential basis for our proofs are Davydov's results for C¹ FEs on polyhedral domains in Rn or of local degree 5 for C² domains in R². Better convergence and extensions to Rn for C² domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in Rn, includes the first essentially as a special case. The method applies to quasi-linear elliptic problems not in divergence form as well. A discrete Newton method is shown to converge locally quadratically, essentially independently of the actual grid size by the mesh independence principle. [ABSTRACT FROM AUTHOR]

In this paper, we derive an equation for the growth of a flame ball for a free boundary combustion model with radiative transfer. The equation for the radiative field is given by the linearized Eddington equation. We then study the mathematical properties of this equation of growth and carry out numerical computations in order to discuss the stability or instability of steady flame balls. [ABSTRACT FROM AUTHOR]

Beginning with a proof of the existence of a discrete-time trajectory, this paper establishes the convergence of a time-stepping method for solving continuous-time, boundary-value problems for dynamic systems with frictional contacts characterized by local compliance in the normal and tangential directions. Our investigation complements the analysis of the initial-value rigid-body model with one frictional contact encountering inelastic impacts by Stewart [Arch. Ration. Mech. Anal., 145 (1998), pp. 215-260] and the recent analysis by Anitescu Optimization-Based Simulation for Nonsmooth Rigid Multibody Dynamics, Argonne National Laboratory, Argonne, IL, 2004] using the framework of measure differential inclusions. In contrast to the measure-theoretic approach of these authors, we follow a differential variational approach and address a broader class of problems with multiple elastic or inelastic impacts. Applicable to both initial and affine boundary-value problems, our main convergence result pertains to the case where the compliance in the normal direction is decoupled from the compliance in the tangential directions and where the friction coefficients are sufficiently small. [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]

We present the first part of a theory of monotone implicit methods for scalar conservation laws. In this paper, we focus on the implicit upwind scheme. The theoretical investigation of this method is centered around a rigorously verified implicit monotonicity criterion. The relation between the upwind scheme and a discrete entropy inequality is constructed analogously to the classical approach of Crandall and Majda [M. G. Crandall and A. Majda, Math. Comp., 34 (1980), pp. 1--21]. A proof of convergence is given which does not rely on a classical compactness argument. The theoretical results are complemented by a discussion of numerical aspects. [ABSTRACT FROM AUTHOR]

In this paper we investigate numerical methods for solving hyperbolic conservation laws based on finite volumes and optimal recovery. These methods can, for example, be applied in certain ENO schemes. Their approximation properties depend in particular on the reconstruction from cell averages. Hence, this paper is devoted to prove convergence results for such reconstruction processes from cell averages. [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]

In our papers [Inverse Problems, 15 (1999), pp. 309--327] and [Numer. Math., 88 (2001), pp. 347--365] we proposed algorithm {\tt REGINN}, an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. {\tt REGINN} consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by solving the linearized system. The convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration, especially Landweber iteration, $\nu$-methods, and Tikhonov--Phillips regularization. In the present paper we prove convergence rates for {\tt REGINN} when the conjugate gradient method, which is nonlinear, serves as inner iteration. Thereby we add to a convergence analysis of {Hanke}, who had previously investigated {\tt REGINN} furnished with the conjugate gradient method [Numer. Funct. Anal. Optim., 18 (1997), pp. 971--993]. By numerical experiments we illustrate that the conjugate gradient method outperforms the $\nu$-method as inner iteration. [ABSTRACT FROM AUTHOR]

In this paper, we analyze convergence rates of wavelet schemes for time-dependent convection-reaction equations within the framework of the Eulerian--Lagrangian localized adjoint method (ELLAM). Under certain minimal assumptions that guarantee $ H^1 $-regularity of exact solutions, we show that a generic ELLAM scheme has a convergence rate $ \mathcal{O}(h/\sqrt{\Delta t} + \Delta t) $ in $ L^2 $-norm. Then, applying the theory of operator interpolation, we obtain error estimates for initial data with even lower regularity. Namely, it is shown that the error of such a scheme is $ \mathcal{O}((h/\sqrt{\Delta t})^\theta + (\Delta t)^\theta) $ for initial data in a Besov space $ \displaystyle B^\theta_{2,q} (0 < \theta < 1, 0 < q <= infinity) $. The error estimates are {a priori} and optimal in some cases. Numerical experiments using orthogonal wavelets are presented to illustrate the theoretical estimates. [ABSTRACT FROM AUTHOR]

The aim of this work is to solve hyperbolic conservation laws by means of a finite volume method for both spatial and time discretization. We extend the ideas developed in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779; X.-D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397--425] to fourth-order upwind and central schemes. In order to do this, once we know the cell-averages of the solution, $\overline {u}_j ^n$, in cells $I_{j}$ at time $T=t^n$, we define a new three-degree reconstruction polynomial that in each cell, $I_{j}$, presents the same shape as the cell-averages $\{ {\overline {u}_{j-1} ^n,\overline {u}_j ^n,\overline {u}_{j+1} ^n}\}$. By combining this reconstruction with the nonoscillatory property and the maximum principle requirement described in [X.-D. Liu and S. Osher, SIAM J. Numer. Anal., 33 (1996), pp. 760--779] we obtain a fourth-order scheme that satisfies the total variation bounded (TVB) property. Extension to systems is carried out by componentwise application of the scalar framework. Numerical experiments confirm the order of the schemes presented in this paper and their nonoscillatory behavior in different test problems. [ABSTRACT FROM AUTHOR]

In this paper we show that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable. In particular, we show that the problem is PSPACE-complete. We begin with a review of NP-completeness and polynomial-time reductions, introduce the class PSPACE, and motivate the significance of PSPACE-complete problems. Afterwards, we prove that determining whether a given instance of a generalized Lunar Lockout puzzle is solvable is PSPACE-complete. [ABSTRACT FROM AUTHOR]

In this paper we obtain explicit nontrivial stationary patterns in the one-dimensional Gray-Scott model for cubic autocatalysis. Involved in the reaction are two chemicals, A and B, whose diffusion coefficients are denoted by D[sub A] and D[sub B], respectively. The chemical A is fed into the system at a rate k[sub f], reacts with the catalyst B at a rate k[sub 1], and the catalyst decays at a rate k[sub 2]. If these parameters obey the relation (*) k[sub f]/D[sub A] = k[sub 2]/D[sub B], then, for appropriate values of the rate constants, we present explicit expressions for two families of pulses and one kink. We also show that if (*) is only satisfied approximately, these families of pulses are preserved, and there exists a smooth branch of kinks leading from the explicit one obtained when (*) is satisfied. We determine the local behavior of this branch near the explicit kink. [ABSTRACT FROM AUTHOR]

In this paper, we provide a maximum norm analysis of a finite difference scheme defined on overlapping nonmatching grids for second order elliptic equations. We consider a domain which is the union of p overlapping subdomains where each subdomain has its own independently generated grid. The grid points on the subdomain boundaries need not match the grid points from adjacent subdomains. To obtain a global finite difference discretization of the elliptic problem, we employ standard stable finite difference discretizations within each of the overlapping subdomains and the different subproblems are coupled by enforcing continuity of the solutions across the boundary of each subdomain, by interpolating the discrete solution on adjacent subdomains. If the subdomain finite difference schemes satisfy a strong discrete maximum principle and if the overlap is sufficiently large, we show that the global discretization converges in optimal order corresponding to the largest truncation errors of the local interpolation maps and discretizations. Our discretization scheme and the corresponding theory allows any combination of lower order and higher order finite difference schemes in different subdomains. We describe also how the resulting linear system can be solved iteratively by a parallel Schwarz alternating method or a Schwarz preconditioned Krylov subspace iterative method. Several numerical results are included to support the theory. [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]

In this paper, we first note that the proof of the quadratic convergence of the quasi-Newton method as given in Friedland, Nocedal, and Overton [SIAM J. Numer. Anal., 24 (1987), pp. 634­667] is incorrect. Then we give a correct proof of the convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge­Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown. [ABSTRACT FROM AUTHOR]

We present in this paper numerical simulations of reactive flows interacting with shock waves. We argue that spectral methods are suitable for these problems and review the recent developments in spectral methods that have made them a powerful numerical tool appropriate for long-term integrations of complicated flows, even in the presence of shock waves. A spectral code is described in detail, and the theory that leads to each of its components is explained. Results of interactions of hydrogen jets with shock waves are presented and analyzed, and comparisons with ENO finite difference schemes are carried out. [ABSTRACT FROM AUTHOR]

The wavelet transform is a widely used time-frequency tool for signal processing. However, with some rare exceptions, its use in signal processing is limited to discrete-time critically sampled transforms, which are particular cases of subband coding. On the other hand, interest in continuous wavelet analyses has been repeatedly demonstrated in the literature. However, implementation challenges limit their practical uses: continuous analyses are time consuming, and current fast algorithms are often restricted to particular generating analysis wavelets; syntheses are even more time consuming and usually too approximate. This paper formalizes the quasi-continuous wavelet transform. A review of algorithms proposed in the literature is presented. Thereafter, a fast quasi-continuous algorithm for analysis is proposed using filter banks. It is valid for almost any generating analysis wavelet. Another version that minimizes the redundancies between subbands is also presented. Different synthesis algorithms are described with filter bank implementations. Different methods answer different needs. The "fair synthesis" algorithm gives the same weight to each point of the time-scale plane. It is important for selective reconstructions of portions of this plane. On the other hand, the "closest takes most" method allows a hierarchical approach. Finally, the direct summation method often gives fair approximations. The proposed algorithms are compatible with hybrid wavelet transforms where, for example, the wavelet can change from one scale level to another. [ABSTRACT FROM AUTHOR]

This paper develops a least-squares approach to the solution of the incompressible Navier­Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier­Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle based on L2 norms applied to this system yields optimal discretization error estimates in the H1 norm in each variable, including the velocity flux. An analogous principle based on the use of an H-1 norm for the reduced system (with no curl or trace constraints) is shown to yield similar estimates, but now in the L2 norm for velocity-flux and pressure. Although the H-1 least-squares principle does not allow practical implementation, these results are critical to the analysis of a practical least-squares method for the reduced system based on a discrete equivalent of the negative norm. A practical method of this type is the subject of a companion paper. Finally, we establish optimal multigrid convergence estimates for the algebraic system resulting from the L2 norm approach. [ABSTRACT FROM AUTHOR]

A new technique to solve elliptic linear PDEs, called ultra weak variational formulation (UWVF) in this paper, is introduced in [B. Després, C. R. Acad. Sci. Paris, 318 (1994), pp. 939–944]. This paper is devoted to an evaluation of the potentialities of this technique. It is applied to a model wave problem, the two-dimensional Helmholtz problem. The new method is presented in three parts following the same style of presentation as the classical one of the finite elements method, even though they are definitely conceptually different methods. The first part is committed to the variational formulation and to the continuous problem. The second part defines the discretization process using a Galerkin procedure. The third part actually studies the efficiency of the technique from the order of convergence point of view. This is achieved using theoretical proofs and a series of numerical experiments. In particular, it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom. An application to scattering problems is presented in a fourth part. [ABSTRACT FROM AUTHOR]

In this paper, we study the link between a certain class of discrete-velocity models (DVMs) and the Boltzmann equation. Those models possess an infinite number of velocities laying on a regular grid of step h in R3. Our aim is to prove that it is possible to construct models consistent with the Boltzmann equation, i.e., such that the discrete collision term can be seen as an approximation of the collision integral of the Boltzmann equation. [ABSTRACT FROM AUTHOR]

Introduces a series of articles on mathematical modeling.

Introduces a series of articles on the different applications of mathematical modeling, such as the use of partial differential equations and the basic principles in control theory.

Introduces a series of articles that features numerical methods and techniques for scientific computation. Definition of adaptive finite element methods; Concept of load balancing; Implications of adaptivity.

Presents an overview of the Education section of the periodical "Siam Review." Application of mathematics and scientific computation in schools; Description of web-based software package in applied mathematics; Techniques in information retrieval.

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]

In two earlier articles, we provided sufficient conditions on (mass action) reaction network structure for the preclusion of multiple positive steady states in the context of what chemical engineers call the continuous flow stirred tank reactor. In such reactors, all species are deemed to be present in the effluent stream, a fact which played a strong role in the proofs. When certain species are deemed to be entrapped within the reactor, the questions that must be asked are more subtle, and the mathematics becomes substantially more difficult. Here we extend results of the earlier papers to semiopen reactors and show that very similar results obtain, provided that the network of chemical reactions satisfies certain weak structural conditions; weak reversibility is sufficient but not necessary. [ABSTRACT FROM AUTHOR]

Most probability and statistics textbooks are loaded with dice, coins, and balls in urns. These are perfect metaphors for actual phenomena where uncertainty plays a role. However, students will greatly appreciate a real-life example. In this paper we examine the mathematics of an implementation of an iris recognition system, We show that the determination of a crucial spread parameter is made on implicit assumptions that are not fulfilled. Some elementary probability theory calculations show that this leads in general to an optimistic assessment of the reliability of the identification system. Then we use a famous inequality to quantify this optimism. [ABSTRACT FROM AUTHOR]

Hat problems have become a popular topic in recreational mathematics. In a typical hat problem, each of n players tries to guess the color of the hat, he or she is wearing by looking at the colors of the hats worn by some of the other players. In this paper we consider several variants of the problem, united by the common theme that the guessing strategies are required to be deterministic and the objective is to maximize the number of correct answers in the worst case. We also summarize what is currently known about the worst-case analysis of deterministic hat guessing problems with a finite number of players. [ABSTRACT FROM AUTHOR]