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We investigate the error of the randomized Milstein algorithm for solving scalar jump–diffusion stochastic differential equations. We provide a complete error analysis under substantially weaker assumptions than those known in the literature. In case the jump-commutativity condition is satisfied, we prove optimality of the randomized Milstein algorithm by establishing matching lower bounds. Moreover, we give some insight into the multidimensional case by investigating the optimal convergence rate for the approximation of jump–diffusion type Lévys' areas. Finally, we report numerical experiments that support our theoretical findings. [ABSTRACT FROM AUTHOR]

We study the approximation capabilities of two families of univariate polynomials that arise in applications of quantum signal processing. Although approximation only in the domain [ 0 , 1 ] is physically desired, these polynomial families are defined by bound constraints not just in [ 0 , 1 ] , but also with additional bound constraints outside [ 0 , 1 ]. One might wonder then if these additional constraints inhibit their approximation properties within [ 0 , 1 ]. The main result of this paper is that this is not the case — the additional constraints do not hinder the ability of these polynomial families to approximate arbitrarily well any continuous function f : [ 0 , 1 ] → [ 0 , 1 ] in the supremum norm, provided f also matches any polynomial in the family at 0 and 1. We additionally study the specific problem of approximating the step function on [ 0 , 1 ] (with the step from 0 to 1 occurring at x = 1 2 ) using one of these families, and propose two subfamilies of monotone and non-monotone approximations. For the non-monotone case, under some additional assumptions, we provide an iterative heuristic algorithm that finds the optimal polynomial approximation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present an efficient algorithm for the computation of triangular Shepard interpolation method. More precisely, it is well known that the triangular Shepard method reaches an approximation order better than the Shepard one (DellAccio et al., 2016), but it needs to identify useful general triangulation of the node set. Here we propose a searching technique used to detect and select the nearest neighbor points in the interpolation scheme (Cavoretto et al., 2016, 2017). It consists in determining the closest points belonging to the different neighborhoods and subsequently applies to the triangulation-based approach. Numerical experiments and some geological applications show efficiency and accuracy of the interpolation procedure. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the existence, the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation. The viscoelastic body is assumed to behave according to Kelvin–Voigt constitutive law with added thermal effects under the Green–Lindsay theory. We prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the Faedo–Galerkin method. An energy decay property is also shown under the assumption of radial symmetry. Then, a numerical approximation based on the finite element method is proposed. A stability result is proved from which the decay of the discrete energy is deduced. A priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions. Finally, some numerical experiments are described to support our results. [ABSTRACT FROM AUTHOR]

Most of the existing numerical schemes developed to solve Burgers’ equation cannot exhibit its correct physical behavior for very small values of viscosity. This difficulty can be overcome by using splitting methods derived for near-integrable system. This class of methods has positive real coefficients and can be used for non-reversible systems such as Burgers’ equation. It also has the advantage of being able to account small viscosity in the accuracy. The algorithm is based on the combination of implicit–explicit finite difference schemes to solve each simplified problem and filtering technique to treat nonlinear instability. The resulting algorithm is accurate, efficient and easy to implement. The new numerical results are compared with numerical and exact solutions reported in the literature and found that they are very accurate for small values of the viscosity. [ABSTRACT FROM AUTHOR]

Arnoldi method approximates exterior eigenvalues of a large sparse matrix, but may fail to approximate corresponding eigenvectors. The refined Arnoldi method approximates an eigenpair by solving a related singular value problem. In this paper, we propose a new procedure to extract an approximate eigenpair from a Krylov subspace in Arnoldi method, using a minimization problem. Unlike the refined Arnoldi method, the suggested procedure requires solving a linear system. [ABSTRACT FROM AUTHOR]

In this paper, we propose three modified approximations of the ruin probability and the inverse function of the ruin probability using the inversion of the scaled values of Laplace transform suggested by Mnatsakanov et al. (2015). The problem of evaluating numerically the tail-Value at Risk of an insurance portfolio is also discussed briefly. Performances of the proposed constructions are demonstrated via the graphs and tables using several examples. [ABSTRACT FROM AUTHOR]

In Lagrangian meshfree methods, the underlying spatial discretization, referred to as a point cloud or a particle cloud, moves with the flow velocity. In this paper, we consider different numerical methods of performing this movement of points or particles. The movement is most commonly done by a first order method, which assumes the velocity to be constant within a time step. We show that this method is very inaccurate and that it introduces volume and mass conservation errors. We further propose new methods for the same which prescribe an additional ODE system that describes the characteristic velocity. Movement is then performed along this characteristic velocity. The first new way of moving points is an extension of mesh-based streamline tracing ideas to meshfree methods. In the second way, the movement is done based on the difference in approximated streamlines between two time levels, which approximates the pathlines in unsteady flow. Numerical comparisons show these methods to be vastly superior to the conventionally used first order method. [ABSTRACT FROM AUTHOR]

We present the spatial discretization of the nonlinear convective flux in the incompressible Navier–Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier–Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection–diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations. [ABSTRACT FROM AUTHOR]

A new second-order in space linearized difference scheme on non-uniform grid that approximates the Dirichlet problem for multidimensional quasilinear convection–diffusion equation with unbounded nonlinearity is constructed. Proposed algorithm is a novel nonlinear generalization of difference schemes for linear problems developed earlier by the authors. Nontrivial two-side pointwise estimates of the solution of the scheme fully consistent with the corresponding estimates for the differential problem are established. Such estimates permit to prove the nonnegativity of the exact solution, important in physical problems, and also to find sufficient conditions on the input data when the nonlinear problem is parabolic. As a result a priori estimates of the approximate solution in the grid norm C that depend on the initial and boundary conditions and on the right-hand side only are proved. [ABSTRACT FROM AUTHOR]

The present paper is devoted to recovering the source term and the initial data simultaneously for a time-fractional diffusion equation from additional temperature data at two fixed times t = T 1 and t = T 2 . Firstly, we prove the uniqueness result for the direct problem. And then we apply a non-stationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm. Three numerical examples in both one-dimensional and two-dimensional cases are provided to demonstrate the effectiveness and feasibility of the proposed method. [ABSTRACT FROM AUTHOR]

For the nonlinear Richards equation as an unsaturated flow through heterogeneous media, we build a new coarse-scale approximation algorithm utilizing numerical homogenization. This approach follows deep neural networks (DNNs) to quickly and frequently calculate macroscopic parameters. More specifically, we train neural networks with a training set consisting of stochastic permeability realizations and corresponding computed macroscopic targets (effective permeability tensor, homogenized stiffness matrix, and right-hand side vector). Our proposed deep learning scheme develops nonlinear maps between such permeability fields and macroscopic characteristics, and the treatment for Richards equation's nonlinearity is included in the predicted coarse-scale homogenized stiffness matrix, which is a novelty. This strategy's good performance is demonstrated by several numerical tests in two-dimensional model problems, for predictions of the macroscopic properties and consequently solutions. [ABSTRACT FROM AUTHOR]

In this paper, the properties of quartic linear normal (LN) curves are studied. In particular, we present necessary and sufficient conditions for quartic LN curves to be regular. Using these conditions, we obtain an approximation method for convex curves by quartic regular LN curves which are G2 Hermite interpolation of convex curves. We show that the approximation order of our approximation method is six. In the approximation method, a convex curve is approximated by a piecewise G2 quartic regular LN curve. Each of the pieces approximates the convex curve as long as possible within the tolerance. Consequently, the G2 spline curves consist of the minimum number of quartic regular LN curves that approximate the convex curve. The algorithm for our approximation method has been implemented and has been used to approximate convex curves. [ABSTRACT FROM AUTHOR]

A class of tests for the two-sample problem for count data whose test statistic is an L 2 -norm of the difference between the empirical probability generating functions associated with each sample is considered. The tests can be applied to count data of any arbitrary fixed dimension. Since the null distribution of the test statistic is unknown, some approximations are investigated. Specifically, the bootstrap, permutation and weighted bootstrap estimators are examined. All of them provide consistent estimators. A simulation study analyzes the performance of these approximations for small and moderate sample sizes. This study also includes a comparison with other two-sample tests whose test statistic is a weighted integral of the difference between the empirical characteristic functions of the samples. [ABSTRACT FROM AUTHOR]

In this paper, we modify the general iterative method to approximate a common element of the set of solutions of split variational inclusion problem and the set of common fixed points of a finite family of k -strictly pseudo-contractive nonself mappings. Strong convergence theorem is established under some suitable conditions in a real Hilbert space, which also solves some variational inequality problems. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors. Finally, some examples to study the rate of convergence and some illustrative numerical examples are presented. [ABSTRACT FROM AUTHOR]

Exponentially-fitted and trigonometrically-fitted methods have a long successful history in the solution of initial-value problems, but other functions might be considered in adapted methods. Specifically, this paper aims at the derivation of a new numerical scheme for approximating initial value problems of ordinary differential equations using elliptic functions. The example considered is the undamped Duffing equation where the forcing term is of autonomous type affected by a perturbation parameter. The new scheme is constructed by considering a suitable approximation to the theoretical solution based on elliptic functions. The proposed elliptic fitting procedure has been tested on a variety of problems, showing its good performance. [ABSTRACT FROM AUTHOR]

In this paper, the implicit midpoint method is used to solve the semi-discrete modified anomalous sub-diffusion equation with a nonlinear source term, and the weighted and shifted Grünwald–Letnikov difference operator and the compact difference operator are applied to approximate the Riemann–Liouville fractional derivative and space partial derivative respectively, then the new numerical scheme is constructed. The stability and the convergence of this method are analyzed. Numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results. [ABSTRACT FROM AUTHOR]

This paper continues research initiated in Solodushkin et al. (2015). We develop a finite difference scheme for a first order multidimensional partial differential equation including a time delay. This class of equations is used to model different time lapse phenomena, e.g. birds migration, proliferation of viruses or bacteria and transfer of nuclear particles. For the constructed difference schemes the order of approximation, stability and convergence order are substantiated. To conclude we support the obtained results with some test examples. [ABSTRACT FROM AUTHOR]

In this work, we investigate both analytically and numerically a spatially two-dimensional advection–diffusion–reaction equation that generalizes the Burgers’ and the Fisher’s equations. The partial differential equation of interest is a nonlinear model for which the existence and the uniqueness of positive and bounded solutions are analytically established here. At the same time, we propose an exact finite-difference discretization of the Burgers–Fisher model of interest and show that, as the continuous counterpart, the method proposed is capable of preserving the positivity and the boundedness of the numerical approximations as well as the temporal and spatial monotonicity of the discrete initial–boundary conditions. It is shown that the method is convergent with first order in time and second order in space. We provide some simulations that illustrate the fact that the proposed technique preserves the positivity, the boundedness and the monotonicity. [ABSTRACT FROM AUTHOR]

We propose a numerical method to approximate the solutions of generalized forms of two bi-dimensional models of mathematical physics, namely, the Burgers–Fisher and the Burgers–Huxley equations. In one-dimensional form, the literature in the area gives account of the existence of analytical solutions for both models, in the form of traveling-wave fronts bounded within an interval I of the real numbers. Motivated by this fact, we propose a finite-difference methodology that guarantees that, under certain analytical conditions on the model and computer parameters, estimates within I will evolve discretely into new estimates which are likewise bounded within I . Additionally, we establish the preservation in the discrete domain of the skew-symmetry of the solutions of the models under study. Our computational implementation of the method confirms numerically that the properties of positivity and boundedness are preserved under the analytical constraints derived theoretically. Our simulations show a good agreement between the analytical solutions derived in the present work and the corresponding numerical approximations. [ABSTRACT FROM AUTHOR]

An alternative strategy for solving systems of nonlinear equations when the classical Newton’s method fails is presented. The proposed strategy is an extension for systems based on the idea presented in the article (Ramos and Vigo-Aguiar, 2015). It relies in obtaining the approximate solutions of a given system of equations through solving an associated system obtained through the theory underlying the Newton’s method. In this way, the solutions of the associated system that are not 2-cycles of the Newton iteration function provide solutions of the original system. As it is usual, in most cases, the associated system cannot be solved exactly, and some iterative procedure must be used. For particular starting values, solving the associated system with the Newton’s method results to be more efficient than the application of the Newton’s method to the original system. Some examples are given to illustrate the performance of the proposed strategy. Performance profiles are evaluated in terms of number of iterations, error and CPU time. [ABSTRACT FROM AUTHOR]

In this paper three formulas for recovering the conditional mean and conditional variance based on product moments are proposed. The upper bounds for the uniform rate of approximations of regression and derivatives of some moment-determinate function are derived. Two cases where the support of underlying functions is bounded and unbounded from above are studied. Based on the proposed approximations, novel nonparametric estimates of the distribution function and its density in multiplicative-censoring model are constructed. Simulation study justifies the consistency of the estimates. [ABSTRACT FROM AUTHOR]

A BDDC (Balancing Domain Decomposition by Constraints) algorithm for a staggered discontinuous Galerkin approximation is considered. After applying domain decomposition method, a global linear system on the subdomain interface unknowns is obtained and solved by the conjugate gradient method combined with a preconditioner. To construct a preconditioner that is robust to the coefficient variations, a generalized eigenvalue problem on each subdomain interface is solved and primal unknowns are selected from the eigenvectors using a predetermined tolerance. By the construction of the staggered discontinuous Galerkin methods, the degrees of freedom on subdomain interfaces are shared by only two subdomains, and hence the construction of primal unknowns are simplified. The resulting BDDC algorithm is shown to have the condition number bounded by the predetermined tolerance. A modified algorithm for parameter dependent problems is also introduced, where the primal unknowns are only computed in an offline stage. Numerical results are included to show the performance of the proposed method and to verify the theoretical estimate. [ABSTRACT FROM AUTHOR]

In this paper, the meshless method is presented for numerically solving the 2-dimensional Fredholm integral equation with complex factors. First, the existence and uniqueness of solution are proved by the fixed point theorem. Second, the approximation solution method of the radial basis function interpolation is given. The results of convergence analysis and error estimation are also obtained. Some numerical examples are presented to verify the validity and applicability of the method. [ABSTRACT FROM AUTHOR]

We present an algorithm for generating approximations for the logarithm of Barnes G -function in the half-plane Re (z) ≥ 3 / 2. These approximations involve only elementary functions and are easy to implement. The algorithm is based on a two-point Padé approximation and we use it to provide two approximations to ln (G (z)) , accurate to 3 × 1 0 − 16 and 3 × 1 0 − 31 in the half-plane Re (z) ≥ 3 / 2 ; a reflection formula is then used to compute Barnes G -function in the entire complex plane. A by-product of our algorithm is that it also produces accurate approximations to the gamma function. [ABSTRACT FROM AUTHOR]

This paper analyzes the stability of a two-level coupled explicit MacCormack/Crank–Nicolson method for solving the nonstationary mixed Stokes-Darcy model. The approach is systematically derived based on the monolithic weak formulations of the explicit MacCormack technique and the implicit Crank–Nicolson discretization. The stability of the explicit scheme does not require a time step restriction. The two-step MacCormack provides an approximate solution at the coarse grid level while the implicit Crank–Nicolson algorithm uses this approximation to compute the desired numerical solution at the fine grid stage. The proposed method is unconditionally stable over long time intervals and the computational cost is reduced. This combination is efficient than a wide set of numerical schemes applied to time dependent mixed Stokes-Darcy problem. A large range of numerical examples which confirm the theoretical study are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Numerical approximation of stochastic Stokes–Darcy equations usually requires repeated sampling of the random hydraulic conductivity tensor and then simulating flow ensembles. In this setting, we propose an efficient, second order, ensemble algorithm for fast computation of the whole set of realizations of the stochastic Stokes–Darcy model corresponding to different random hydraulic conductivity tensor samples. The ensemble algorithm only requires the solution of two linear systems that have the same constant coefficient matrices for all realizations. We give a complete long time stability and convergence analysis for the method. Numerical experiments are presented to support theoretical results and demonstrate the application of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new approximation algorithm for solving generalized Lyapunov matrix equations. We also present a convergence analysis for this algorithm. In each step of this algorithm two standard Lyapunov matrix equations with real coefficient matrices should be solved. Then we determine the optimal parameter to minimize the corresponding spectral radius of iteration matrix to obtain fastest speed of convergence. Finally some numerical examples are given to prove the capability of the present algorithm and a comparison is made with the existing results. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of the approximation algorithm for the ground state computation of a Gross–Pitaevskii equation, which is an eigenvalue problem with eigenvector nonlinearity. We will adapt the convergence results of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penalty hp-adaptive algorithm against the performance of the hp-DGCFEM. [ABSTRACT FROM AUTHOR]

Accurate algorithms are proposed for approximation of integrals involving highly oscillatory Bessel function of the first kind over finite and infinite domains. Accordingly, Bessel oscillatory integrals having high oscillatory behavior are transformed into oscillatory integrals with Fourier kernel by using complex line integration technique. The transformed integrals contain an inner non-oscillatory improper integral and an outer highly oscillatory integral. A modified meshfree collocation method with Levin approach is considered to evaluate the transformed oscillatory type integrals numerically. The inner improper complex integrals are evaluated by either Gauss–Laguerre or multi-resolution quadrature. Inherited singularity of the meshfree collocation method at x = 0 is treated by a splitting technique. Error estimates of the proposed algorithms are derived theoretically in the inverse powers of ω and verified numerically. [ABSTRACT FROM AUTHOR]

We deal with pointwise approximation of solutions of scalar stochastic differential equations in the presence of informational noise about underlying drift and diffusion coefficients. We define a randomized derivative-free version of Milstein algorithm A ̄ n d f − R M and investigate its error. We also study the lower bounds on the error of arbitrary algorithm. It turns out that in some case the scheme A ̄ n d f − R M is the optimal one. Finally, in order to test the algorithm A ̄ n d f − R M in practice, we report performed numerical experiments. [ABSTRACT FROM AUTHOR]

We present a novel fast direct solver to simulate 3D large scale elastic inclusion problems. The method combines the isogeometric analysis boundary element method (IGABEM) and the hierarchical off-diagonal low-rank (HODLR) matrix based on non-uniform rational B-splines (NURBS). Hence the 3D geometric surface can be accurately described by the bivariate NURBS basis functions. In order to solve the large scale problems, a stable accelerated algorithm is used to approximate the off-diagonal submatrices by low-rank matrices. Based on the accelerated algorithm, a hybrid approximation algorithm consisting of singular value decomposition (SVD) and adaptive cross approximation (ACA) is proposed to solve the 3D elastic inclusion problems. The validity and accuracy of the method are verified by testing the four methods. Among the numerical results obtained from the four methods, the method proposed in this paper uses less CPU time and storage space to obtain accurate results. [ABSTRACT FROM AUTHOR]

Computing low-rank approximations of a given function is a key step for implementing efficiently numerous algorithms in various fields, including the discretisation of non-local integral operators and Isogeometric Analysis. The adaptive cross approximation (ACA) algorithm is an efficient method requiring few computational resources introduced by Bebendorf. We introduce in the present paper the new paradigm of approximating the given function by a piecewise low-rank function with C 1 -regularity. The proposed approximation is based on the ACA algorithm and our main contribution is the extension of the interpolation property characterising this algorithm to Hermite interpolation. Therefore, we introduce a new method for low-rank Hermite interpolation using a limiting case of the ACA algorithm. The proposed method has full approximation order. We then propose a piecewise low-rank approximation with adaptive refinement using either the ACA algorithm or our new method to compute each piece. We finally compare the results obtained for the two methods. [ABSTRACT FROM AUTHOR]

In this paper we propose linear-time CUR approximation algorithms for admissible matrices obtained from the hierarchical form of Boundary Element matrices. We propose a new approach called geometric sampling to obtain indices of most significant rows and columns using information from the domains where the problem is posed. Our strategy is tailored to Boundary Element Methods (BEM) since it uses directly and explicitly the cluster tree containing information from the problem geometry. Our CUR algorithm has precision comparable with low-rank approximations created with the truncated QR factorization with column pivoting (QRCP) and the Adaptive Cross Approximation (ACA) with full pivoting, which are quadratic-cost methods. When compared to the well-known linear-time algorithm ACA with partial pivoting, we show that our algorithm improves, in general, the convergence error and overcomes some cases where ACA fails. We provide a general relative error bound for CUR approximations created with geometrical sampling. Finally, we evaluate the performance of our algorithms on traditional BEM problems defined over different geometries. [ABSTRACT FROM AUTHOR]