A Nyström Method for the Eigenvalue Problem of a Class of Noncompact Operators

We consider the eigenvalue problem of certain kind of noncompact linear operators given as the sum of a multiplication and a kernel operator. Under the assumption that there is a unique (up to normalization) positive eigenfunction f _p , we propose a combination of the finite section and Nyström methods for approximation of f _p and the corresponding eigenvalue. It is proved that the proposed method is convergent. Some examples of the problem are solved numerically using the proposed method.     

A Spectrum Perturbation Problem and Its Applications to Waves Above an Underwater Ridge

We consider the problem of perturbing the spectrum of a pseudodifferential operator of a real variable in Hardy-type spaces by a compact operator with a small norm. Under some very general requirements on the operators, we prove the existence theorem for an eigenfunction of multiplicity one and prove that the problem is Fredholm in the L ₂ (R) space. Illustrating this theory, we discuss the linear problem of gravitational-capillary surface waves running along an underwater ridge. Assuming the liquid ideal, incompressible, and vortex-free, we show that the waves along the underwater ridge propagate so that their amplitude decays exponentially with a small positive exponent in the direction transverse to the ridge. Moreover, capillarity plays no essential role in a linear approximation.     

Convergence acceleration of series through a variational approach

By means of a variational approach we find new series representations both for well-known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions.     

Numerical approximations and solution techniques for the space-time Riesz–Caputo fractional advection-diffusion equation

In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β ₁ ∈ (0,1) and β ₂ ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.     

A null-space primal-dual interior-point algorithm for nonlinear optimization with nice convergence properties

We present a null-space primal-dual interior-point algorithm for solving nonlinear optimization problems with general inequality and equality constraints. The algorithm approximately solves a sequence of equality constrained barrier subproblems by computing a range-space step and a null-space step in every iteration. The ℓ₂ penalty function is taken as the merit function. Under very mild conditions on range-space steps and approximate Hessians, without assuming any regularity, it is proved that either every limit point of the iterate sequence is a Karush-Kuhn-Tucker point of the barrier subproblem and the penalty parameter remains bounded, or there exists a limit point that is either an infeasible stationary point of minimizing the ℓ ₂ norm of violations of constraints of the original problem, or a Fritz-John point of the original problem. In addition, we analyze the local convergence properties of the algorithm, and prove that by suitably controlling the exactness of range-space steps and selecting the barrier parameter and Hessian approximation, the algorithm generates a superlinearly or quadratically convergent step. The conditions on guaranteeing that all slack variables are still positive for a full step are presented.     

On a method for computing waveguide scattering matrices in the presence of point spectrum

A waveguide occupies a domain G in ? ⁿ⁺¹, n ? 1, having several cylindrical outlets to infinity. The waveguide is described by a general elliptic boundary value problem that is self-adjoint with respect to the Green formula and contains a spectral parameter µ. As an approximation to a row of the scattering matrix S(µ) we suggest a minimizer of a quadratic functional J ^R (·, µ). To construct such a functional, we solve an auxiliary boundary value problem in the bounded domain obtained by cutting off, at a distance R, the waveguide outlets to infinity. It is proved that, if a finite interval [µ₁, µ₂] of the continuous spectrum contains no thresholds, then, as R → ∞, the minimizer tends to the row of the scattering matrix at an exponential rate uniformly with respect to µ ∈ [µ₁, µ₂]. The interval may contain some waveguide eigenvalues whose eigenfunctions exponentially decay at infinity.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

On the Numerical Stability of Fourier Extensions

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308–318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.     

Approximate multiplication in adaptive wavelet methods

Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l ₂-problem, finding the convergent iteration process for the l ₂-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.     

The convergence of two numerical schemes for the Navier-Stokes equations

Summary This paper deals with some convergence/stability results concerning two numerical methods for solving the incompressible nonstationary Navier-Stokes equations. The algorithms are of a particular kind in what regards time discretization (more precisely, of the Peaceman-Rachford and the Strang type resp.), and have been obtained by modifying slightly the numerical treatment of the nonlinear terms in other schemes due to Glowinski et al. (1980). We first describe the full discretization of the homogeneous Dirichlet problem using a (general) external approximation of the spatial functional spaces involved (a particular and simple choice of such an approximation is the standardP ₂-Lagrange finite element for the velocity field when the fluid is bidimensional). Then we establish and prove convergence and stability and make some comments on the numerical treatment of other (generally nonhomogeneous) boundary conditions. The theoretical results show that the schemes are (at least) conditionally stable and convergent, which justifies the success of Glowinski's methods.     

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

A regularized Newton method for the efficient approximation of tensors represented in the canonical tensor format

In the present survey, we consider a rank approximation algorithm for tensors represented in the canonical format in arbitrary pre-Hilbert tensor product spaces. It is shown that the original approximation problem is equivalent to a finite dimensional ? ₂ minimization problem. The ? ₂ minimization problem is solved by a regularized Newton method which requires the computation and evaluation of the first and second derivative of the objective function. A systematic choice of the initial guess for the iterative scheme is introduced. The effectiveness of the approach is demonstrated in numerical experiments.     

An Algorithm for Linear L1 Approximation of Continuous Functions

The problem of solving linear continuous L₁ approximation problemsis considered. An algorithm is given for approximation on aninterval of the real line, which is globally convergent, usuallyat a second-order rate. The method is illustrated by numericalexamples.     

An infeasible-interior-point predictor-corrector algorithm for theP *-geometric LCP

AP _*-geometric linear complementarity problem (P _*GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a “natural” way (i.e., without any change of variables) asP _*GP. It is shown that the algorithm of Mizunoet al. [6] can be extended to solve theP _*GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution. The work of F. A. Potra was supported in part by NSF Grant DMS 9305760     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

Alternating direction method for bi-quadratic programming

Bi-quadratic programming (Bi-QP for short) was studied systematically in Ling et al. (SIAM J. Optim. 20:1286–1320, 2009) due to its various applications in engineering as well as optimization. Several approximation methods were given in the same paper since it is NP-hard. In this paper, we introduce a quadratic SDP relaxation of Bi-QP and discuss the approximation ratio of the method. In particular, by exploiting the favorite structure of the quadratic SDP relaxation, we propose an alternating direction method for solving such a problem and show that the method is globally convergent without any assumption. Some preliminary numerical results are reported which show the effectiveness of the method proposed in this paper.     

Minimum genus embeddings of the complete graph

In this paper,the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied.There are three approaches to solve this problem.The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths;the second approach is to find a current assignment of the current graph by the theory of current graph;the third approach is to find exponentially many embedding(or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph.According to this three approaches,we can construct exponentially many minimum genus embeddings of complete graph K_(12s+8) in orientable surfaces,which show that there are at least 10/3×(200/9)~s distinct minimum genus embeddings for K_(12s+8) in orientable surfaces.We have also proved that K_(12s+8) has at least 10/3×(200/9)~s distinct minimum genus embeddings in non-orientable surfaces.     

Zeroth order H ∞ norm approximation of multivariable systems

In this paper the H _∞ norm approximation of a given stable, proper, rational transfer function by a constant matrix is considered (Zeroth order H _∞ norm approximation problem). The solution method is based on the observation that the H _∞ norm approximation problem can be put into an allpass imbedding problem.