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In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random models of an objective function $f(x)$, obtained from stochastic observations of the function or its gradient. Our method also utilizes estimates of function values to gauge progress that is being made. The convergence analysis relies on requirements that these models and these estimates are sufficiently accurate with sufficiently high, but fixed, probability. Beyond these conditions, no assumptions are made on how these models and estimates are generated. Under these general conditions we show an almost sure global convergence of the method to a first order stationary point. In the second part of the paper, we present examples of generating sufficiently accurate random models under biased or unbiased noise assumptions. Lastly, we present some computational results showing the benefits of the proposed method compared to existing approaches that are based on sample averaging or stochastic gradients., Comment: Revised version posted September 23, 2016. Originally posted April 17, 2015

Regularized empirical risk minimization including support vector machines plays an important role in machine learning theory. In this paper regularized pairwise learning (RPL) methods based on kernels will be investigated. One example is regularized minimization of the error entropy loss which has recently attracted quite some interest from the viewpoint of consistency and learning rates. This paper shows that such RPL methods have additionally good statistical robustness properties, if the loss function and the kernel are chosen appropriately. We treat two cases of particular interest: (i) a bounded and non-convex loss function and (ii) an unbounded convex loss function satisfying a certain Lipschitz type condition., Comment: 36 pages, 1 figure

International audience; We work on a low-order finite element approximation of the vorticity, velocity and pressure formulation of the bidimensional Stokes problem. In a previous paper, we have introduced the adequate space in which to look for the vorticity in order to have a well-posed problem. In this paper, we deal with the numerical approximation of this space, prove optimal convergence of the scheme and show numerical experiments in good accordance with the theory. We remark that despite one supplementary unknown(the vorticity), results of the scheme are much better than the ones obtained with the P1 plus bubble−P1 element in the velocity–pressure formulation.

The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.

The main goal of this paper is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. In Section 1, we discuss a relation between results on cubature formulas and on discrepancy. In particular, we show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. In Section 2, we present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Sections 3 and 4 contain some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this paper is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative. We have included in the paper both new results (Section 2) and known results. We included some known results with their proofs for the following two reasons. First of all we want to make the paper self-contained (within reasonable limits). Secondly, we selected the proofs which demonstrate different methods and are not very much technically involved. Section 5 contains historical notes on discrepancy and cubature formulas, some further comments and remarks. Historical remarks on nonlinear approximation are included in Section 2. We want to point out that this paper is not a complete survey in any of the above-mentioned areas. We did not even try to provide a complete list of results in those areas. We rather wanted to highlight the most typical results in cubature formulas (Sections 3 and 4) and show their relation to the discrepancy theory.

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence $$O\left( k^{-(3p+1)/ 2}\right) $$ O k - ( 3 p + 1 ) / 2 , where $$p \ge 1$$ p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15 % erasures, and asked whether there exist frames that are robust against erasures of more than 50 %. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least \(1+\delta _0\) times the dimension for some \(\delta _0>0\). This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.

This paper deals with the numerical integration of well-posed multiscale systems of ODEs or evolutionary PDEs. As these systems appear naturally in engineering problems, time-subcycling techniques are widely used every day to improve computational efficiency. These methods rely on a decomposition of the vector field in a fast part and a slow part and take advantage of that decomposition. This way, if an unconditionnally stable (semi-)implicit scheme cannot be easily implemented, one can integrate the fast equations with a much smaller time step than that of the slow equations, instead of having to integrate the whole system with a very small time-step to ensure stability. Then, one can build a numerical integrator using a standard composition method, such as a Lie or a Strang formula for example. Such methods are primarily designed to be convergent in short-time to the solution of the original problems. However, their longtime behavior rises interesting questions, the answers to which are not very well known. In particular, when the solutions of the problems converge in time to an asymptotic equilibrium state, the question of the asymptotic accuracy of the numerical longtime limit of the schemes as well as that of the rate of convergence is certainly of interest. In this context, the asymptotic error is defined as the difference between the exact and numerical asymptotic states. The goal of this paper is to apply that kind of numerical methods based on splitting schemes with subcycling to some simple examples of evolutionary ODEs and PDEs that have attractive equilibrium states, to address the aforementioned questions of asymptotic accuracy, to perform a rigorous analysis, and to compare them with their counterparts without subcycling. Our analysis is developed on simple linear ODE and PDE toy-models and is illustrated with several numerical experiments on these toy-models as well as on more complex systems. Lie and, Comment: IMA Journal of Numerical Analysis, Oxford University Press (OUP): Policy A - Oxford Open Option A, 2015

We study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function ρ. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weight ρ and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of [7], where weighted approximation over R was studied.

We consider the problem of approximating fixed points of contractive functions whose contraction factor is close to 1. In a previous paper (1993, K. Sikorski et al. , J. Complexity 9 , 181–200), we proved that for the absolute error criterion, the upper bound on the number of function evaluations to compute e -approximations is O( n 3 (ln(1/ e )+ln(1/(1− q ))+ln n )) in the worst case, where 0 q n is the dimension of the problem. This upper bound is achieved by the circumscribed ellipsoid (CE) algorithm combined with a dimensional deflation process. In this paper we present an inscribed ellipsoid (IE) algorithm that enjoys O( n (ln(1/ e )+ln(1/(1− q )))) bound. For q close to 1, the IE algorithm thus runs in many fewer iterations than the simple iteration method, that requires ⌈ln(1/ e )/ln(1/ q )⌉ function evaluations. Our analysis also implies that: (1) The dimensional deflation procedure in the CE algorithm is not necessary and that the resulting “plain” CE algorithm enjoys O( n 2 (log(1/ e )+log(1/(1− q )))) upper bound on the number of function evaluations. (2) The IE algorithm solves the problem in the residual sense, i.e., computes x such that ‖ f ( x )− x ‖⩽ δ , with O( n ln(1/ δ )) function evaluations for every q ⩽1.

In this paper, we establish the optimal convergence for a second-order exponential-type integrator from Hofmanová & Schratz (2017, An exponential-type integrator for the KdV equation. Numer. Math., 136, 1117–1137) for solving the Korteweg–de Vries equation with rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides second-order accuracy in $H^\gamma $ for initial data in $H^{\gamma +4}$ for any $\gamma \geq 0$, where the regularity requirement is lower than for classical methods. The result is confirmed by numerical experiments, and comparisons are made with the Strang splitting scheme.

We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell’s equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.

This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

The conjugate gradient (CG) method is a method to solve unconstrained optimization problems. Moreover CG method can be applied in medical science, industry, neural network, and many others. In this paper a new three term CG method is proposed. The new CG formula is constructed based on DL and WYL CG formulas to be non-negative and inherits the properties of HS formula. The new modification satisfies the convergence properties and the sufficient descent property. The numerical results show that the new modification is more efficient than DL, WYL, and CG-Descent formulas. We use more than 200 functions from CUTEst library to compare the results between these methods in term of number of iterations, function evaluations, gradient evaluations, and CPU time.

In this paper the concept of symmetrized convex stochastic processes is introduced. Some characterizations involving Hermite–Hadamard type inequalities and a stability result for symmetrized convex stochastic processes are proved.

By using superquadracity, the new inequalities in this paper generalize a result of Hardy–Littlewood–Polya and refine Jensen’s type inequalities. Also, inequalities related to rearrangements of sets are obtained.

PurposeThis study aims to use new formula derived based on the shifted Jacobi functions have been defined and some theorems of the left- and right-sided fractional derivative for them have been presented.Design/methodology/approachIn this article, the authors apply the method of lines (MOL) together with the pseudospectral method for solving space-time partial differential equations with space left- and right-sided fractional derivative (SFPDEs). Then, using the collocation nodes to reduce the SFPDEs to the system of ordinary differential equations, which can be solved by the ode45 MATLAB toolbox.FindingsApplying the MOL method together with the pseudospectral discretization method converts the space-dependent on fractional partial differential equations to the system of ordinary differential equations.Originality/valueThis paper contributes to gain choosing the shifted Jacobi functions basis with special parameters a, b and give the authors this opportunity to obtain the left- and right-sided fractional differentiation matrices for this basis exactly. The results of the examples are presented in this article. The authors found that the method is efficient and provides accurate results, and the authors found significant implications for success in the science, technology, engineering and mathematics domain.

In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

The goal of this paper is threefold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work in proving lower bounds for recently developed new type of discrepancy—the smooth discrepancy. Third, we consider numerical integration in classes, for which we do not impose any smoothness assumptions. We illustrate how nonlinear approximation, in particular greedy approximation, allows us to guarantee some rate of decay of errors of numerical integration even in such a general setting with no smoothness assumptions.

In Donnay’s and Van Brummelen’s monographs on spherical trigonometry, the Cesaro method is revitalized to derive various results on spherical triangles. Using Cesaro’s triangles, we derive in this paper some further results on spherical triangles that are not given in these books. Among these are results concerning the relation of sides of a spherical triangle and their opposite angles, Lexell’s theorem, and a result concerning the area of spherical triangles with one variable side-length. These results are applied to derive theorems of isoperimetric type for spherical polygons and the corresponding extremal properties of regular spherical polygons.

"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

In this paper, we inspect the global asymptotic stability (GAS) of Ebola–epidemic using a modified Susceptible-Infected-Deceased-Pathogen (SIDP) model with perturbation. The feasible region is obtained, and we determine the basic reproduction number using the next-generation matrix method. The GAS of the disease-free and endemic equilibria are discuss and provide parameter conditions for which the stability exists. We show theoretically that the contaminated environmental Ebola human–deceased transmission model is GAS with and without probability. Numerical simulation to support our theoretical findings are presented.

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence., Published in Mathematical Programming

It is well-known that the Kadec-Klee property is an important property in the geometry of Banach spaces. It is closely connected with the approximation compactness and fixed point property of non-expansive mappings. In this paper, a criterion for Musielak-Orlicz function spaces equipped with the Orlicz norm to have the Kadec-Klee property are given. As a corollary, we obtain that a class of non-reflexive Musielak-Orlicz function spaces have the Fixed Point property.

This paper deals with functional equations in the form of $f(x) + g(y) = h(x,y)$ where $h$ is given and $f$ and $g$ are unknown. We will show that if $h$ is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation., Comment: 7 pages, to appear in Aequationes mathematicae

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space, and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.

In this paper, we study a system of nonlinear Riemann–Liouville fractional differential equations with delays. First, we define in an appropriate way initial conditions which are deeply connected with the fractional derivative used. We introduce an appropriate generalization of practical stability which we call practical stability in time. Several sufficient conditions for practical stability in time are obtained using Lyapunov functions and the modified Razumikhin technique. Two types of derivatives of Lyapunov functions are used. Some examples are given to illustrate the introduced definitions and results.

In this paper, Modified Crank–Nicolson method is combined with Richardson extrapolation to solve the 1D heat equation. The method is found to be unconditionally stable, consistent and hence the convergence of the method is guaranteed. The method is also found to be second-order convergent both in space and time variables. When combined with the Richardson extrapolation, the order of the method is improved from second-to fourth-order. To validate the proposed method, two model examples are considered and solved for different values of spatial and temporal step lengths.

In recent work it has been established that deep neural networks (DNNs) are capable of approximating solutions to a large class of parabolic partial differential equations without incurring the curse of dimension. However, all this work has been restricted to problems formulated on the whole Euclidean domain. On the other hand, most problems in engineering and in the sciences are formulated on finite domains and subjected to boundary conditions. The present paper considers an important such model problem, namely the Poisson equation on a domain $D\subset \mathbb {R}^d$ subject to Dirichlet boundary conditions. It is shown that DNNs are capable of representing solutions of that problem without incurring the curse of dimension. The proofs are based on a probabilistic representation of the solution to the Poisson equation as well as a suitable sampling method.

This paper focuses on the stability analysis of a general class of linear multistep methods for decoupled forward–backward stochastic differential equations (FBSDEs). The general linear multistep methods we consider contain many well-known linear multistep methods from the ordinary differential equation framework, such as Adams, Nyström, Milne--Simpson and backward differentiation formula methods. Under the classical root condition, we prove that general linear multistep methods are mean-square (zero) stable for decoupled FBSDEs with generator function related to both y and z. Based on the stability result, we further establish a fundamental convergence theorem.

In this paper we consider the Sturm-Liouville transform ℱ(f) on ℝ+. We analyze the concentration of this transform on sets of finite measure. In particular, Donoho-Stark and Benedicks-type uncertainty principles are given.

In this paper, we examine the existence and uniqueness of integral boundary value problem for implicit fractional differential equations (IFDE’s) involving Hadamard and Caputo-Hadamard fractional derivative. We prove the existence and uniqueness results by utilizing Banach and Schauder’s fixed point theorem. Finally, examples are introduced of our results.

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g., approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.

The Gaver–Stehfest algorithm is widely used for numerical inversion of the Laplace transform. In this paper we provide the first rigorous study of the rate of convergence of the Gaver–Stehfest algorithm. We prove that Gaver–Stehfest approximations converge exponentially fast if the target function is analytic in a neighbourhood of a point and they converge at a rate $o(n^{-k})$ if the target function is $(2k+3)$-times differentiable at a point.

The aim of this paper is to present a reproducing kernel method for solving system of integro-differential equations with nonlocal boundary conditions. The solution obtained by using the present method takes the form of a convergent series with easily computable components. The analysis of convergence shows that the approximate solution and its derivatives converge to the exact solution and its derivatives, respectively. The numerical examples are given to demonstrate the accuracy of the present method. Results obtained by using the present method are compared with those of the exact solution of each example and are found to be in good agreement.

In this paper, I have proposed a method for finding an approximate function for Bratu differential equations (BDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, I define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the approximate function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get an approximate solution function with discrete derivatives of the solution function, we have determined the approximate solution function which satisfies in the Bratu differential equations (BDEs). In the end, the algorithm of the method is elaborated with several examples. In one example, I have presented an absolute error comparison of some approximate methods.

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

The joint spectral radius of a finite set of real d x d matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain one-parameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic ratio which depends continuously upon the parameter. In this paper we answer some open questions from that paper by showing that the dependence of the ratio function upon the parameter takes the form of a Devil's staircase. We show in particular that this Devil's staircase attains every rational value strictly between 0 and 1 on some interval, and attains irrational values only in a set of Hausdorff dimension zero. This result generalises to include certain one-parameter families considered by other authors. We also give explicit formulas for the preimages of both rational and irrational numbers under the ratio function, thereby establishing a large family of pairs of matrices for which the joint spectral radius may be calculated exactly., 40 pages

In this paper we discuss spectral properties of operators associated with the least-squares finite-element approximation of elliptic partial differential equations. The convergence of the discrete eigenvalues and eigenfunctions towards the corresponding continuous eigenmodes is studied and analyzed with the help of appropriate $L^2$ error estimates. A priori and a posteriori estimates are proved.

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation., To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thanks

Chaotic dynamical systems exhibit sensitive dependence to initial conditions. So, because of round-off error, a computed orbit diverges at an exponential rate from the true orbit with the same initial condition. Nevertheless, we are able to exploit the hyperbolicity of the dynamical system to prove a “finite time” shadowing lemma, from which we deduce that a true orbit shadows the computed orbit for a large number of iterates. An algorithm for the computation of the shadowing error is given and, furthermore, the effect of round-off error on these computations is analyzed in detail. The algorithm is applied to the Hénon map. This paper is a continuation of an earlier paper on one-dimensional maps.

In this paper, we present several equivalent conditions related to the forward order law for the core inverse of two matrices, i.e., $$(AB)^{{\tiny {\textcircled {\tiny \#}}}}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ . In addition, we consider problems when $$(AB)^{\dagger }=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ , $$(AB)^{\#}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ and $$(AB)_{{\tiny {\textcircled {\tiny \#}}}}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ , respectively. Thus, we study some hybrid forward order laws too.

This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. To the best of our knowledge, the presented result is the first strong convergence result for a constraint-preserving scheme in the considered setting., Comment: 39 pages, 1 figure

The purpose of this paper is to present an appropriate numerical method for solving linear two-dimensional weakly singular stochastic integral equations. This method is based on bicubic B-spline interpolation, Gauss–Legendre quadrature formula, and two-dimensional Ito approximation. The proposed method is used to transform the solution of mentioned problem to the linear system of algebraic equations which can be solved by using an appropriate method. Convergence analysis of the approach has been investigated. In the end, two examples are given to show efficiency and accuracy of the method and their numerical results are reported.

"In this paper we present an heuristic introduction to Bratu problem and we give some variants of Bratu's theorem (G. Bratu, Sur les \'equations int\'egrales non lin\'eaires, Bulletin Soc. Math. France, 42(1914), 113-142). Using the positivity of Green's function, the monotone iterations technique and the contraction principle, some generalizations of Bratu's result are also given. Numerical aspects are also considered."

In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation with additional stochastic proximal gradient steps to guarantee convergence. Based on suitable bounds on the step sizes, we establish global convergence to stationary points in expectation and an extension of the approach using variance reduction techniques is discussed. Motivated by large-scale and big data applications, we investigate a stochastic coordinate-type quasi-Newton scheme that allows to generate cheap and tractable stochastic higher order directions. Finally, numerical results on large-scale logistic regression and deep learning problems show that our proposed algorithm compares favorably with other state-of-the-art methods.

The cosine function is a classical tool for measuring angles in inner product spaces, and it has various extensions to normed linear spaces. In this paper, we investigate a cosine function for the convex angle formed by two nonzero elements of a complex normed linear space, in connection with recent results on the Birkhoff-James approximate orthogonality sets.

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve C is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C . Finally, a numerical application is given for dealing with financial data.