In this paper, we present a respectively scaled splitting (RSS) iteration method for the block 4-by-4 linear system from eddy current electromagnetic problems. Unconditional convergence properties of the RSS iteration method are established. Theoretical results show that the quasi-optimal iterative parameter that minimizes the spectral radius is \(\alpha _{opt}=1\) and the corresponding convergence

The original article has been corrected.

In this paper, we study the invading phenomenon of an alien predator to the habitat of two aborigine preys by traveling waves connecting the predator-free state to the co-existence state. We characterize the minimal wave speed of this invading process based on an application of Schauder’s fixed point theorem with the help of (generalized) upper-lower-solutions. New form of upper-lower-solutions are

This paper aims to survey fast methods of verifying the accuracy of a numerical solution of a linear systems. For the last decade, a number of fast verification algorithms have been proposed to obtain an error bound of a numerical solution of a dense or sparse linear system. Such fast algorithms rely on the verified numerical computation using floating-point arithmetic defined by IEEE standard 754

A numerical verification, method of steady state solutions for a system of reaction-diffusion equations is described. Using a decoupling technique, the system is reduced to a single nonlinear equation and a computer-assisted method for second-order elliptic boundary value problems based on the infinite dimensional fixed-point theorem can be applied. Some numerical examples confirm the effectiveness

In this aricle, the author considers mathematical formulation and numerical solutions of distributed and Neumann boundary optimal control problems associated with the stationary Bénard problem. The solution of the optimal control problem is obtained by controlling of the source term of the equations and/or Neumann boundary conditions. Then the author considers the approximation, by finite element methods

We propose four conservative schemes for the regularized long-wave (RLW) equation. The RLW equation has three invariants: mass, momentum, and energy. Our schemes are designed by using the discrete variational derivative method to inherit appropriate conservation properties from the equation. Two of our schemes conserve mass and momentum, while the other two schemes conserve mass and energy. With one

In this paper the sinc-Galerkin method, as well as the sinc-collocation method, based on the double exponential transformation (DE transformation) for singularly perturbed boundary value problems of second order ordinary differential equation is considered. A large merit of the present method exists in that we can apply the standard sinc method with only a small care for perturbation parameter. Through

We give a survey on direct methods for interval linear systems. We also consider various kinds of solution sets and show how the interval hull can be computed.

This paper is concerned with a robust geometric predicate for the 2D orientation problem. Recently, a fast and accurate floating-point summation algorithm is investigated by Rump, Ogita and Oishi, which provably outputs a result faithfully rounded from the exact value of the summation of floating-point numbers. We optimize their algorithm for applying it to the 2D orientation problem which requires

After the introduction basic properties of interval arithmetic are discussed and different approaches are repeated by which one can compute verified numerical approximations for a solution of a nonlinear equation.

Let ann xn matrixA of floating-point numbers in some format be given. Denote the relative rounding error unit of the given format by eps. AssumeA to be extremely ill-conditioned, that is cond(A) ≫ eps−1. In about 1984 I developed an algorithm to calculate an approximate inverse ofA solely using the given floating-point format. The key is a multiplicative correction rather than a Newton-type additive

An explicit scheme for time-dependent convection-diffusion problems is presented. It is shown that convenient bounds for the time step value ensureL ∞ stability, in both space and time, for piecewise linear finite element discretizations in any space dimension. Convergence results in the same sense are also demonstrated under certain conditions. Numerical results certify the good performance of the

The construction of sphericalt-designs with (t+1)2 points on the unit sphereS 2 in ℝ3 can be reformulated as an underdetermined system of nonlinear equations. This system is highly nonlinear and involves the evaluation of a degreet polynomial in (t+1)4 arguments. This paper reviews numerical verification methods using the Brouwer fixed point theorem and Krawczyk interval operator for solutions of the

This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV)M which approximately minimizes ∥/ −M A∥F or ∥I −AM∥F. Then, we also discuss the theoretical issues such as the equivalence between the

We survey a class of algorithms to evaluate polynomials with floating point coefficients and for computation performed with IEEE-754 floating point arithmetic. The principle is to apply, once or recursively, an error-free transformation of the polynomial evaluation with the Horner algorithm and to accurately sum the final decomposition. These compensated algorithms are as accurate as the Horner algorithm

This paper surveys applications of Krawczyk-Moore-Jones’s algorithm and presents its further developments. The application is focused on nonlinear electric circuit analysis. The further developments are described on parallel KMJ algorithm, Gray code KMJ algorithm and KMJ processor.

The authors give short survey on validated computation of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods on PDEs and apply it to initial value problems for ODEs with some numerical experiments.

In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of π/2 or 3π/2, we have obtained an explicit a prioriH 10 error estimation ofO(h) and anL 2 error estimation ofO(h 2) for such

This paper treats a linear equation $$Av = b,$$ where\(A \in {\mathbb{F}}^{n \times n} \) and\(b \in {\mathbb{F}}^n \). Here,\({\mathbb{F}}\) is a set of floating point numbers. Letu be the unit round-off of the working precision and κ(A)=‖A‖∞‖A −1‖∞ be the condition number of the problem. In this paper, ill-conditioned problems with $$1< u\kappa ({\rm A})< \infty $$ are considered and an iterative

We highlight selected results of recent development in the area of rigorous computations which use interval arithmetic to analyse dynamical systems. We describe general ideas and selected details of different ways of approach and we provide specific sample applications to illustrate the effectiveness of these methods. The emphasis is put on a topological approach, which combined with rigorous calculations

In this paper, we are concerned with the analysis of two-transistor circuits. Applying technique for the numerical verification, we prove rigorously the existence of five solutions in a two-transistor circuit. The system of equations for a transistor circuit is obtained as nonlinear equations, therefore Krawczyk’s method is applied for proving the existence of a solution.

We first summarize the general concept of our verification method of solutions for elliptic equations. Next, as an application of our method, a survey and future works on the numerical verification method of solutions for heat convection problems known as Rayleigh-Bénard problem are described. We will give a method to verify the existence of bifurcating solutions of the two-dimensional problem and

For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a “close” and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach’s fixed-point theorem to an equivalent problem for the error

Today simulation technologies (based on numerical computation) are definitely vital in many fields of science and engineering. The accuracy of numerical simulations grows in importance as simulation technologies develop and prevail, and many researches have been carried out for establishing numerically stable algorithms. In recent years, combining computer algebra and other guaranteed accuracy approaches

We consider an eigenvalue problem for differential operators, and show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region can be assured. Some examples for several types of operators will be presented.

This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations

Recently, stochastic differential equation (SDE) has been used for many applications in option pricing models which satisfy the non-negativity. So, constructing new numerical method preserves non-negativity for solving SDE is very important. This paper investigates the numerical analyses; convergence, non-negativity and stability of the multi-step Milstein method for SDE. We derive the new general

We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution. We also consider the hedging problem in bond

This article points out how a driver might misperceive the curvature of a road curve if the curve is combined with a slope change that the driver does not recognize. We investigated the relationship between the apparent curvature of a road and the change of the slope, and clarified the difference between the perceived curvature and the actual curvature when the driver mistakenly believes that the slope

We consider the radial symmetric stationary solutions of \(u_{t}=\varDelta u-|x|^{q}u^{-p}\). We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for \(p \in 2{\mathbb{N}}\), \(q\in{\mathbb{R}}\). Moreover, we give the asymptotic behavior of u(r) and \(u'(r)\) at both ends of the finite interval. Second, we

We propose a two-dimensional fast Fourier transform (FFT) network to retrieve the prices of options that depend on two Lévy processes. Applications include, but are not limited to, the valuation of options on two stocks under the Lévy processes, and options on a single stock under a random time-change Lévy process. The proposed numerical scheme can be applied to different multivariate Lévy constructions

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t. Using martingale duality approach and Feynman–Kac formula, we derive

A thick-restart Lanczos type algorithm is proposed for Hermitian J-symmetric matrices. Since Hermitian J-symmetric matrices possess doubly degenerate spectra or doubly multiple eigenvalues with a simple relation between the degenerate eigenvectors, we can improve the convergence of the Lanczos algorithm by restricting the search space of the Krylov subspace to that spanned by one of each pair of the

We investigate the optimal consumption, portfolio, and life insurance decisions problem of a liquidity constrained household whose preference is given by the CES (constant elasticity of substitution) utility function. By applying the martingale and duality method, we obtain the closed-form solution for the household’s value function and optimal strategies. We provide a rigorous proof for the optimality

The reflection principle for Brownian motion gives a way to calculate the joint distribution of a hitting time and a one dimensional marginal. The Carr–Nadtochiy transform is a formulation that generalizes the reflection principle in this respect. The transform originated from a way to hedge so-called barrier options in the literature of financial mathematics. The existence of the transform has been

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an \(\alpha\)-stable process with \(\alpha \in (1,2)\). More precisely, we find an upper bound for the \(L^{\alpha -1}(\varOmega ,{\mathbb {P}})\) distance between two solutions in terms of the \(L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)\) distance of the coefficients

We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents

Stationary waves of constant shape and constant propagation speed on rotational flows of two layers are computed numerically. Two layers are assumed to be of distinct constant vorticity distributions. Three different kinds of waves of finite depth are considered: pure capillary, capillary-gravity, and gravity waves. The problem is formulated as a bifurcation problem, which involves many parameters

The shortest bibranching problem is a common generalization of the minimum-weight edge cover problem in bipartite graphs and the minimum-weight arborescence problem in directed graphs. For the shortest bibranching problem, an efficient primal-dual algorithm is given by Keijsper and Pendavingh (J Comb Theory Ser B 73:130–145, 1998), and the tractability of the problem is ascribed to total dual integrality

In simulating physical systems, conservation of the total energy is often essential, especially when energy conversion between different forms of energy occurs frequently. Recently, a new fourth order energy-preserving integrator named MB4 was proposed based on the so-called continuous stage Runge–Kutta methods (Miyatake and Butcher in SIAM J Numer Anal 54(3):1993–2013, 2016). A salient feature of

NIST SP800-22 is one of the most widely used statistical testing tools for pseudorandom number generators (PRNGs). This tool consists of 15 tests (one-level tests) and two additional tests (two-level tests). Each one-level test provides one or more p-values. The two-level tests measure the uniformity of the obtained p-values for a fixed one-level test. One of the two-level tests categorizes the p-values

We present a numerical approach for determination of the spectral stability of solitary waves by computing eigenvalues and eigenfunctions of the corresponding eigenvalue problems, along with their continuation, for nonlinear wave equations in one space dimension. We illustrate the approach for the nonlinear Schrödinger (NLS) equation with a small potential, and numerically determine the spectral stability

Mathematical models for self-propelled motions are often utilized for understanding the mechanism of collective motions observed in biological systems. Indeed, several patterns of collective motions of camphor disks have been reported in experimental systems. In this paper, we show the existence of asymmetrically rotating solutions of a two-camphor model and give necessary conditions for their existence

The purpose of the paper is to establish higher-order Karush–Kuhn–Tucker higher-order necessary optimality conditions for set-valued optimization where the derivatives of objective and constraint functions are separated. We first introduce concepts of higher-order weak lower inner epiderivatives for set-valued maps and discuss some useful properties about new epiderivatives, for instance, convexity

In this paper, a new preconditioner for a class of \(2\times 2\) block linear systems is proposed. The proposed new preconditioner is a better approximation to the original coefficient matrix than the previous ones. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the

This paper presents a new projected Barzilai–Borwein method for the complementarity problem over symmetric cone by applying the Barzilai–Borwein-like steplengths to the projected method. A new descent direction is employed and a non-monotone line search is used in the method in order to guarantee the global convergence. The projected Barzilai–Borwein method is proved to be globally convergent under

For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents

We propose a mathematical model to understand the transmission dynamics of HIV/AIDS in an environment. In addition to previous approaches, we incorporate two classes of isolated. By isolated we do not mean physical separation, but commitment to keep its status. We establish the well-posedness of our model and fully analyze the asymptotic behavior of the solutions which depends on the basic reproduction

The nonlinear matrix equation \(X^p=R+M^T(X^{-1}+B)^{-1}M\), where p is a positive integer, M is an arbitrary \(n\times n\) real matrix, R and B are symmetric positive semidefinite matrices, is considered. When \(p=1\), this matrix equation is the well-known discrete-time algebraic Riccati equation (DARE), we study the convergence rate of an iterative method which was proposed in Meng and Kim (J Comput

In this paper we will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction–diffusion equations, and we will apply our nonlinear bounds to the Lotka–Volterra system of two and four competing species as examples. The idea used in a series of papers by the first author et al. for the establishment

This paper considers a bidimensional continuous-time renewal risk model, in which the two components of each pair of claim sizes are linked via the strongly asymptotic independence structure and the two claim-number processes from different lines of business are (almost) arbitrarily dependent. Precise asymptotic formulas for three kinds of finite-time ruin probabilities are established when the claim

Inner product functional encryption (IPFE) is one class of functional encryption supporting only inner product functionality. All previous IPFE schemes are bounded schemes, meaning that the vector length that can be handled in the scheme is fixed in the setup phase. In this paper, we propose the first unbounded IPFE schemes, in which we do not have to fix the lengths of vectors in the setup phase and

In this paper, a new iterative refinement for ill-conditioned linear systems is derived based on discrete gradient methods for gradient systems. It is proved that the new method is convergent for any initial values irrespective of the choice of the stepsize h. Some properties of the new iterative refinement are presented. It is shown that the condition number of the coefficient matrix in the linear

For continuous functions, midpoint convexity characterizes convex functions. By considering discrete versions of midpoint convexity, several types of discrete convexities of functions, including integral convexity, L\(^\natural\)-convexity and global/local discrete midpoint convexity, have been studied. We propose a new type of discrete midpoint convexity that lies between L\(^\natural\)-convexity

The original article has been corrected.

In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel

In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is to prove the stable convergence of a sequence of such

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space \(\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)\), where \(\varOmega _{\lambda , r_2} (T)\) is the

In this paper, we address the weighted linear matroid intersection problem from computation of the degree of the determinant of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an \(O(mn^3 \log n)\) time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column