Search

This paper studies the problem of learning an unknown function f from given data about f. The learning problem is to give an approximation f ^ to f that predicts the values of f away from the data. There are numerous settings for this learning problem depending on (i) what additional information we have about f (known as a model class assumption), (ii) how we measure the accuracy of how well f ^ predicts f, (iii) what is known about the data and data sites, (iv) whether the data observations are polluted by noise. A mathematical description of the optimal performance possible (the smallest possible error of recovery) is known in the presence of a model class assumption. Under standard model class assumptions, it is shown in this paper that a near optimal f ^ can be found by solving a certain finite-dimensional over-parameterized optimization problem with a penalty term. Here, near optimal means that the error is bounded by a fixed constant times the optimal error. This explains the advantage of over-parameterization which is commonly used in modern machine learning. The main results of this paper prove that over-parameterized learning with an appropriate loss function gives a near optimal approximation f ^ of the function f from which the data is collected. Quantitative bounds are given for how much over-parameterization needs to be employed and how the penalization needs to be scaled in order to guarantee a near optimal recovery of f. An extension of these results to the case where the data is polluted by additive deterministic noise is also given. [ABSTRACT FROM AUTHOR]

This paper is concerned with the error estimation of the fast multipole method (FMM) for scattering problems in 2-D. The FMM error is caused by truncating Graf’s addition theorem in each step of the algorithm, including two expansions and three translations. We first give a novel bound on the truncation error of Graf’s addition theorem by the limiting forms of Bessel and Neumann functions, and then estimate the error of the FMM. Explicit error bound and its convergence order are derived. The method proposed in this paper can also be used to the FMM for other problems, such as potential problems, elastostatic problems, Stokes flow problems and so on. [ABSTRACT FROM AUTHOR]

In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil S (λ) that is a strong linearization of a rational matrix R (λ) expressed in the form R (λ) = D (λ) + C (λ I ℓ - A) - 1 B , where D (λ) is a polynomial matrix and C (λ I ℓ - A) - 1 B is a minimal state-space realization. We consider the family of block Kronecker linearizations of R (λ) , which have the following structure S (λ) : = M (λ) K ^ 2 T C K 2 T (λ) B K ^ 1 A - λ I ℓ 0 K 1 (λ) 0 0 , where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to S (λ) will compute the exact eigenstructure of a perturbed pencil S ^ (λ) : = S (λ) + Δ S (λ) and the special structure of S (λ) will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil S ~ (λ) = (I - X) S ^ (λ) (I - Y) that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix R ~ (λ) = D ~ (λ) + C ~ (λ I ℓ - A ~) - 1 B ~ , where D ~ (λ) is a polynomial matrix with the same degree as D (λ) . Moreover, we bound appropriate norms of D ~ (λ) - D (λ) , C ~ - C , A ~ - A and B ~ - B in terms of an appropriate norm of Δ S (λ) . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both S (λ) and R (λ) have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix R ~ (λ) that can be expressed in exactly the same form as R (λ) with the parameters defining the representation very near to those of R (λ) . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results. [ABSTRACT FROM AUTHOR]

Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only O (N) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions. [ABSTRACT FROM AUTHOR]

In this paper we consider a strongly coupled flow and nonlinear transport problem arising in sedimentation-consolidation processes in R n , n ∈ { 2 , 3 } , and introduce and analyze a Banach spaces-based variational formulation yielding a new mixed-primal finite element method for its numerical solution. The governing equations are determined by the coupling of a Brinkman flow with a nonlinear advection–diffusion equation, in addition to Dirichlet boundary conditions for the fluid velocity and the concentration. The approach is based on the introduction of the Cauchy fluid stress and the gradient of its velocity as additional unknowns, thus yielding a mixed formulation in a Banach spaces framework for the Brinkman equations, whereas the usual Hilbertian primal formulation is employed for the transport equation. Differently from previous works on this and related problems, no augmented terms are incorporated, and hence, besides becoming fully equivalent to the original physical model, the resulting variational formulation is much simpler, which constitutes its main advantage, mainly from the computational point of view. The well-posedness of the continuous formulation is analyzed firstly by rewriting it as a fixed-point operator equation, and then by applying the Schauder and Banach theorems, along with the Babuška-Brezzi theory and the Lax-Milgram lemma. An analogue fixed-point strategy is employed for the analysis of the associated Galerkin scheme, using in this case the Brouwer theorem instead of the Schauder one. The resulting discrete scheme becomes momentum conservative for the fluid in an approximate sense. Next, a Strang-type lemma and suitable algebraic manipulations are utilized to derive the a priori error estimates, which, along with the approximation properties of the finite element subspaces, yield the corresponding rates of convergence. The paper is ended with several numerical results illustrating the performance of the mixed-primal scheme and confirming the theoretical decay of the error. [ABSTRACT FROM AUTHOR]

This paper provides a qualitative and quantitative study of a second-order Singularly Perturbed Reaction–Diffusion type System of Integro-differential equations with discontinuous source term. To obtain the numerical solution of the problem, an exponentially-fitted method that can be applied to a Shishkin mesh. This method shows that uniform convergence with respect to the perturbation parameter and necessary examples are given. [ABSTRACT FROM AUTHOR]

This paper presents ConvStabNet, a convolutional neural network designed to predict optimal stabilization parameters for each cell in the Streamline Upwind Petrov Galerkin (SUPG) stabilization scheme. ConvStabNet employs a shared parameter approach, allowing the network to understand the relationships between cell characteristics and their corresponding stabilization parameters while efficiently handling the parameter space. Comparative analyses with state-of-the-art neural network solvers based on variational formulations highlight the superior performance of ConvStabNet. To improve the accuracy of SUPG in solving partial differential equations (PDEs) with interior and boundary layers, ConvStabNet incorporates a loss function that combines a strong residual component with a cross-wind derivative term. The findings confirm ConvStabNet as a promising method for accurately predicting stabilization parameters in SUPG, thereby marking it as an advancement over neural network-based PDE solvers. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of a finite volume approximation of a convective diffusive elliptic problem with Neumann boundary conditions and L 1 data. To deal with the non-coercive character of the equation and the low regularity of the right hand-side we mix the finite volume tools and the renormalized techniques. To handle the Neumann boundary conditions we choose solutions having a null median and we prove a convergence result. We present also some numerical experiments in dimension 2 to illustrate the rate of convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the lower and upper bounds for Stokes eigenvalues by finite element schemes. For the schemes studied here, roughly speaking, the loss of the local approximation property of the discrete velocity and pressure spaces may lead to different computed bounds of the eigenvalues. Formally theoretical analysis is constructed based on certain mathematical hypotheses, and numerical experiments are given to illustrate the validity of the theory. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical stability region and the asymptotic stability of linear fractional neutral delay differential equations. Employing boundary locus techniques, the stability region of this problem is analyzed. Furthermore, we derive the fundamental solution of linear fractional neutral delay differential equations, and prove the exponential boundedness, the asymptotic stability and the algebraic decay rate. Finally, numerical tests are conducted to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solution of Fredholm integral equations of the second kind with endpoint algebraic singularities and with a kernel of Green's function type. Novel Nyström schemes employing the Gauss quadrature rule are presented. These methods take into account the lack of smoothness along the diagonal of the kernel and may recover the full convergence rate of smooth kernels. A complete analysis of the stability and convergence is provided, and several numerical tests that illustrate the efficiency and accuracy of various approaches are considered. [ABSTRACT FROM AUTHOR]

This paper presents an extension of the transpose-free quasi-minimal residual (TFQMR) method for solving the generalized coupled Sylvester tensor equations. The new algorithm is based on the tensor format of the TFQMR process. We analyze the convergence behavior of this method and present a bound for the residual norm of the method depending on the specific parameter computed by the algorithm. The numerical experiments demonstrate the efficiency of the new method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a 3 ρ -locally maximum volume submatrix A ^ ∈ R r × r in the matrix A ∈ R M × N can be found in O M N r log r + log ρ r operations, and a ρ -locally maximum volume submatrix for ρ ⩽ 3 can be found in O M N r 3 log ρ r operations. Based on these submatrices, it is possible to construct a rank revealing LU decomposition with guarantees for the approximation accuracy in spectral and Chebyshev norms. [ABSTRACT FROM AUTHOR]

In this paper, a class of quasi-Said-Ball-Vandermonde (q-SBV) matrices belonging to generalized sign regular matrices with signature (1 , ⋯ , 1 , - 1) is introduced. We first accurately compute all the parameters of q-SBV matrices from their bidiagonal decompositions. Furthermore, an algorithm is proposed for computing all the eigenvalues of q-SBV matrices to high relative accuracy by using the accurate parameters. At last, some numerical experiments are presented to show the efficiency and accuracy of our algorithms. [ABSTRACT FROM AUTHOR]

This paper expands the isoparametric framework to construct a stable, H 1 -conforming, and divergence-free method for the Stokes problem in two dimensions based on the Scott-Vogelius pair on Clough-Tocher splits. The pressure space is defined through composition, whereas the velocity space is constructed via a new divergence-preserving mapping that imposes full continuity across shared edges in the isoparametric mesh. Our construction is motivated by operators and spaces found in isoparametric C 1 finite element methods. We prove the method is stable, pressure-robust, and has optimal order convergence. Numerical experiments are provided which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of a new augmented formulation in terms of vorticity, velocity and pressure for the Stokes equations with variable viscosity and mixed boundary conditions. The well-posedness of the continuous problem holds under assumptions on the viscosity. When the domain is a parallelepiped, the spectral discretization is proposed using the Galerkin method with numerical integration. Then, we prove the well-posedness of the obtained discrete problem under the same type of conditions on the viscosity. A priori error estimates is then derived for the three unknowns. Finally, numerical experiments are presented that confirm the interest of the discretization. [ABSTRACT FROM AUTHOR]

This paper treats the properties of polynomials generated by a four-term recurrence relation and linear coefficients. From the eigenvalue theory, we give information on the location of the zeros. Furthermore, we investigate the distribution and simplicity of the zeros in a particular case. [ABSTRACT FROM AUTHOR]

This paper discusses finite elements defined by Ciarlet's triple on grids that consist of general quadrilaterals not limited in parallelograms. Specifically, two finite elements are established for the H 1 and H (r o t) elliptic problems, respectively. An O (h) order convergence rate in energy norm for both of them and an O (h 2) order convergence in L 2 norm for the H 1 scheme are proved under the O (h 2) asymptotic-parallelogram assumption on the grids. Further, these two finite element spaces, together with the space of piecewise constant functions, formulate a discretized de Rham complex on general quadrilateral grids. The finite element spaces consist of piecewise polynomial functions, and, thus, are nonconforming on general quadrilateral grids. Indeed, a rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element by Ciarlet's triple that can formulate a finite element space which consists of continuous piecewise polynomial functions on a grid that may include arbitrary quadrilaterals. [ABSTRACT FROM AUTHOR]

Accelerating slowly convergent sequences is one of the main purposes of extrapolation methods. In this paper, we present a new tensor polynomial extrapolation method, which is based on a modified minimisation problem and some ideas leading to the recent Tensor Global Minimal Extrapolation Method (TG-MPE). We discuss the application of our method to fixed-point iterative process. An efficient algorithm via the higher order Singular Value Decomposition (HOSVD) is proposed for its implementation. The numerical tests show clearly the effectiveness and performance of the proposed method. [ABSTRACT FROM AUTHOR]

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter ε . Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space. [ABSTRACT FROM AUTHOR]

This paper addresses the tensor completion problem, whose task is to estimate missing values with limited information. However, the crux of this problem is how to reasonably represent the low-rank structure embedded in the underlying data. In this work, we consider a new low-rank tensor completion model combined with the multi-directional partial tensor nuclear norm and the total variation (TV) regularization. Specifically, the partial sum of the tensor nuclear norm (PSTNN) is used to narrow the gap between the tensor tubal rank and its lower convex envelop [i.e. tensor nuclear norm (TNN)], and the TV regularization is adopted to maintain the smooth structure along the spatial dimension. In addition, the weighted sum of the tensor nuclear norm (WSTNN) is introduced to replace the traditional TNN to extend the PSTNN to the high-order tensor, which also can flexibly handle different correlations along different modes, resulting in an improved low d-tubal rank approximation. To tackle this new model, we develop the alternating directional method of multipliers (ADMM) algorithm tailored for the proposed optimization problem. Theoretical analysis of the ADMM is conducted to prove the Karush–Kuhn–Tucker (KKT) conditions. Numerical examples demonstrate the proposed method outperforms some state-of-the-art methods in qualitative and quantitative aspects. [ABSTRACT FROM AUTHOR]

In this paper we extend the Banach spaces-based fully mixed approach recently developed for the coupled Stokes and Poisson–Nernst–Planck equations, to cover the coupled Navier–Stokes and Poisson–Nernst–Planck equations. In addition to the velocity and pressure of the fluid, the velocity gradient and the Bernoulli-type stress tensor are added as further unknowns. Similarly, fully mixed formulations for the Poisson and Nernst–Planck sub-problems are achieved by considering, alongside the electrostatic potential and the concentration of ionized particles, the electric current field and total ionic fluxes as new mixed variables. As a consequence, two saddle-point problems, one of them non-linear, and both involving nonlinear source terms depending on the other unknowns, along with a perturbed saddle-point problem that is in turn further perturbed by a bilinear form depending on the remaining unknowns, constitute the resulting variational formulation of the whole coupled system. Fixed-point strategies are then employed to prove, under smallness assumptions on the data, the well-posedness of the continuous and associated Galerkin schemes, the latter for arbitrary finite element subspaces under suitable stability assumptions. The main theoretical tools utilized include the Babuška–Brezzi and Banach–Nečas–Babuška theories in Banach spaces, an abstract result for perturbed saddle-point problems (also in Banach spaces), and the classical Banach and Brouwer fixed-point theorems. Strang-type lemmas are then applied to establish a priori error estimates. Next, specific finite element subspaces (defined by Raviart–Thomas elements of order k ≥ 0 and piecewise polynomials of degree ≤ k ) are shown to satisfy the required hypotheses, and this yields specific convergence rates. Finally, several numerical results are reported, confirming the theoretical findings and illustrating the good performance of the method. [ABSTRACT FROM AUTHOR]

Bernoulli’s method finds a primary solvent of an unilateral quadratic matrix equation A X 2 + B X + C = 0 when some conditions are given on its coefficients. When a stricter condition is given, diagonal update method improves the performances of Bernoulli method, both in terms of number of iterations and in computational time. In this paper, we suggest generalized diagonal update method which improves Bernoulli’s method without the stricter condition. We also define optimal constant for the generalized diagonal update method, which guarantees monotone convergence to the primary solvent. We compare iteration numbers of the generalized diagonal update method with pure Bernoulli’s method, also compare computational time of them with the Newton’s method and the cyclic reduction. Furthermore, we show that this generalized diagonal update method is useful to solve unilateral quadratic matrix equations with the form A X 2 + ϵ B X + C = 0 . [ABSTRACT FROM AUTHOR]

To integrate large systems of nonlinear differential equations in time, we consider a variant of nonlinear waveform relaxation (also known as dynamic iteration or Picard–Lindelöf iteration), where at each iteration a linear inhomogeneous system of differential equations has to be solved. This is done by the exponential block Krylov subspace (EBK) method. Thus, we have an inner-outer iterative method, where iterative approximations are determined over a certain time interval, with no time stepping involved. This approach has recently been shown to be efficient as a time-parallel integrator within the PARAEXP framework. In this paper, convergence behavior of this method is assessed theoretically and practically. We examine efficiency of the method by testing it on nonlinear Burgers, Liouville–Bratu–Gelfand, and nonlinear heat conduction equations and comparing its performance with that of conventional time-stepping integrators. [ABSTRACT FROM AUTHOR]

The nonlinear Kaczmarz method was recently proposed to solve the system of nonlinear equations. In this paper, we first discuss two greedy selection rules, i.e., the maximum residual and maximum distance rules, for the nonlinear Kaczmarz iteration. Then, based on them, two kinds of greedy randomized sampling methods are presented. Furthermore, we also devise four corresponding greedy randomized block methods, i.e., the multiple samples-based methods. The linear convergence in expectation of all the proposed methods is proved. Numerical results show that, in some applications, including brown almost linear function and generalized linear model, the greedy selection rules give faster convergence rates than the existing ones, and the block methods outperform the single sample-based ones. [ABSTRACT FROM AUTHOR]

This paper present an effective numerical approach for solving a class of space-time fractional diffusion equations. By discretizing the time-fractional derivative, the original problem is transformed to a semi-discrete problem, which was solved by using the fractional reproducing kernel collocation method (FRKCM). Additionally, the stability and convergence analysis are given and some numerical examples demonstrate the feasibility and reliability of the method. [ABSTRACT FROM AUTHOR]

In this paper we analyze a virtual element method for the two dimensional elasticity problem allowing small edges. With this approach, the classic assumptions on the geometrical features of the polygonal meshes can be relaxed. In particular, we consider only star-shaped polygons for the meshes. Suitable error estimates are presented, where a rigorous analysis on the influence of the Lamé constants in each estimate is presented. We report numerical tests to assess the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, polynomially based projection and modified projection methods for approximating the solution of Uryshon integral equations with a kernel of Green’s function type are proposed. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and the superconvergence of their iterated versions are analyzed. A numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This work concerns with the numerical approximation for the stochastic Lotka–Volterra model originally studied by Mao et al. (Stoch Process Appl 97(1):95–110, 2002). The natures of the model including multi-dimension, super-linearity of both the drift and diffusion coefficients and the positivity of the solution make most of the existing numerical methods fail. In particular, the super-linearity of the diffusion coefficient results in the explosion of the 1st moment of the analytical solution at a finite time. This becomes one of our main technical challenges. As a result, the convergence framework is to be set up under the θ th moment with 0 < θ < 1 . The idea developed in this paper will not only be able to cope with the stochastic Lotka–Volterra model but also work for a large class of multi-dimensional super-linear SDE models. [ABSTRACT FROM AUTHOR]

In this paper, we address the study of the time-dependent Stokes system with boundary conditions involving the pressure. We obtain existence and uniqueness for a class of Lipschitz-continuous domains. Next, a spectral discretizations of the problem is proposed combined with the backward Euler scheme. The discrete spaces are defined in a way to give exactly divergence-free discrete approximations for the velocity. Then, we prove the associated discrete inf–sup condition and derive a priori error estimates. Finally, some numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Accelerated regularization algorithms for ill-posed problems have received much attention from researchers of inverse problems since the 1980s. The current optimal theoretical results indicate that some regularization algorithms, e.g. the ν -method and the Nesterov method, are such that under conventional source conditions the optimal convergence rates can be obtained with approximately the square root of the iterations of those needed for the benchmark (i.e. the Landweber iteration). In this paper, we propose a new class of regularization algorithms with parameter n, called the Acceleration Regularization of order n (AR n ). Theoretically, we prove that, for an arbitrary number n > - 1 , AR n can attach the optimal convergence rates with approximately the n + 1 root of the iterations needed for the benchmark method. Moreover, unlike the existing accelerated regularization algorithms, AR n s have no saturation restriction. Some symplectic iterative regularizing algorithms are developed for the numerical realization of AR n . Finally, numerical experiments with integral equations and inverse problems in partial differential equations demonstrate that, at least for n ≤ 2 , the numerical behavior of AR n matches our theoretical findings, also breaking the practical acceleration capability of all existing regularization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three series of Jacobi rational basis functions on the half line by using the matrix decomposition technique. The new basis functions are simultaneously orthogonal in both L 2 - and H 1 -inner products, and lead to diagonal systems for second order problems with constant coefficients. We construct efficient space-time spectral methods for parabolic problems on semi-infinite domains using Jacobi rational approximation in space and multi-domain Legendre–Gauss collocation approximation in time, which can be implemented in a synchronous parallel fashion. Some rigorous error estimates are carried out for one-dimensional parabolic equations. Numerical results demonstrate that the suggested approaches possess high-order accuracy and greatly improve the efficiency. [ABSTRACT FROM AUTHOR]

We study in this paper new developments of the Lagrange–Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange–Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange–Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange–Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments. [ABSTRACT FROM AUTHOR]

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in L 2 -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose a gradient recovery method for the direct discontinuous Galerkin (DDG) method. A quadratic polynomial is obtain by using the local discrete least-squares fitting to the gradient of numerical solution at certain sampling points. The recovered gradient is defined on a piecewise continuous space, and it may be discontinuous on the whole domain. Based on the recovered gradient, we introduce a posteriori error estimator which takes the L 2 norm of the difference between the direct and post-processed approximations. Some benchmark test problems with typical difficulties are carried out to illustrate the superconvergence of the recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator. Most of the test problems are from the US National Institute for Standards and Technology (NIST). [ABSTRACT FROM AUTHOR]

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, a new Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the Rational [3/2] Bernstein Bezier polynomial. We call it the [3/2]-scheme. The limit function of the [3/2]-scheme interpolates both the function values and their derivatives. The proposed scheme has three shape parameters w 0 , w 1 and w 2 . It is shown that if w 1 = w 0 + w 2 2 , then the [3/2]-scheme reproduces linear polynomial and is C 1 provided w 0 and w 2 lie in a region of convergence. The scheme also satisfies the shape preserving properties, i.e., monotonicity and convexity. We also compare the [3/2]-scheme with other existing schemes like the [2/2]-scheme and the Merrien scheme introduced recently. An error analysis shows that the [3/2]-scheme is better than the [2/2]-scheme and the Merrien scheme. Further, it is observed that in case w 0 = w 1 = w 2 , the [3/2]-scheme reduces to the Merrien scheme. [ABSTRACT FROM AUTHOR]

For a general third-order tensor A ∈ R n × n × n the paper studies two closely related problems, an SVD-like tensor decomposition and an (approximate) tensor diagonalization. We develop a Jacobi-type algorithm that works on 2 × 2 × 2 subtensors and, in each iteration, maximizes the sum of squares of its diagonal entries. We show how the rotation angles are calculated and prove convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on higher-order tensors. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fully discrete multiphysics finite element method to solve a nonlinear poroelasticity model with finite strain. To reveal the multi-physical processes of deformation and diffusion and propose a stable numerical method, we reformulate the original model into a fluid–fluid coupled problem—a generalized nonlinear Stokes problem of displacement vector field and pseudo pressure field and a diffusion problem of other pseudo pressure field by a new technique. Then, we propose a multiphysics finite element method to approximate the spatial variables and use Newton method to solve the nonlinear problem, prove that the proposed numerical method is stable and has the optimal convergence orders, and give some numerical tests to show that the proposed numerical method is stable and has no oscillation for displacement and pressure. Finally, we draw conclusions to summary the main results of this work. [ABSTRACT FROM AUTHOR]

The approximate linear equation A x ≈ b with some columns of A error-free can be solved via mixed least squares-total least squares (MTLS) model by minimizing a nonlinear function. This paper is devoted to the Gauss–Newton iteration for the MTLS problem. With an appropriately chosen initial vector, each iteration step of the standard Gauss–Newton method requires to solve a smaller-size least squares problem, in which the QR of the coefficient matrix needs a rank-one modification. To improve the convergence, we devise a relaxed Gauss–Newton (RGN) method by introducing a relaxation factor and provide the convergence results as well. The convergence is shown to be closely related to the ratio of the square of subspace-restricted singular values of [A, b]. The RGN can also be modified to solve the total least squares (TLS) problem. Applying the RGN method to a Bursa–Wolf model in parameter estimation, numerical results show that the RGN-based MTLS method behaves much better than the RGN-based TLS method. Theoretical convergence properties of the RGN-MTLS algorithm are also illustrated by numerical tests. [ABSTRACT FROM AUTHOR]

Some popular stabilization techniques, such as nonsymmetric interior penalty Galerkin (NIPG) method, have important application value in computational fluid dynamics. In this paper, we analyze a NIPG method on Shishkin mesh for a singularly perturbed convection diffusion problem, which is a typical simplified fluid model. According to the characteristics of the solution, the mesh and the numerical scheme, a new interpolation is designed for convergence analysis. More specifically, Gauß Lobatto interpolation and Gauß Radau interpolation are introduced inside and outside the layer, respectively. On the basis of that, by selecting special penalty parameters at different mesh points, we establish supercloseness of almost k + 1 order in an energy norm. Here k ≥ 1 is the degree of piecewise polynomials. Then, a simple post-processing operator is constructed, and it is proved that the corresponding post-processing can make the numerical solution achieve higher accuracy. In this process, a new analysis is proposed for the stability analysis of this operator. Finally, superconvergence is derived under a discrete energy norm. These conclusions can be verified numerically. Furthermore, numerical experiments show that the increase of polynomial degree k and mesh parameter N, the decrease of perturbation parameter ε or the use of over-penalty technology may increase the condition number of linear system. Therefore, we need to cautiously consider the application of high-order algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical methods for both the forward and backward problems of a time-space fractional diffusion equation. For the two-dimensional forward problem, we propose a finite difference method. The stability of the scheme and the corresponding Fast Preconditioned Conjugated Gradient algorithm are given. For the backward problem, since it is ill-posed, we use a quasi-boundary-value method to deal with it. Based on the Fourier transform, we obtain two kinds of order optimal convergence rates by using an a-priori and an a-posteriori regularization parameter choice rules. Numerical examples for both forward and backward problems show that the proposed numerical methods work well. [ABSTRACT FROM AUTHOR]

In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples. [ABSTRACT FROM AUTHOR]

Over the several decades, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples m. Our work focuses on providing theoretical approximation guarantees for the class of so-called (b , ε) -holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of m-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds. [ABSTRACT FROM AUTHOR]

In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver. [ABSTRACT FROM AUTHOR]

In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in L 2 (I ; L 2 (Ω) d) and L 2 (I ; H 1 (Ω) d) norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type L 2 (I ; H 1 (Ω)) error estimate are new even for scalar parabolic problems. [ABSTRACT FROM AUTHOR]

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are O (τ 2 + h 2) for the velocity in the L ∞ -norm and the surface elevation in the L 2 -norm, where τ denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, we prove L p estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular, we prove that the discrete Green's functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green's function is of order h - 1 , which is consistent with the behavior of the continuous Green's function. In addition, we also show that the discrete Green's function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green's function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations. [ABSTRACT FROM AUTHOR]

We propose two explicit and structure-preserving exponential wave integrator Fourier pseudo-spectral (SPEWIFP) methods for the Dirac equation in the simultaneously massless and nonrelativistic regime. In this regime, the solution of Dirac equation is highly oscillatory in time because of the small parameter 0 < ε ≪ 1 which is inversely proportional to the speed of light. The proposed methods are proved to be time symmetric, stable only under the condition τ ≲ 1 and preserve the modified energy and modified mass in the discrete level. Although our methods can only preserve the modified energy and modified mass instead of the original energy and mass, our methods are explicit and greatly reduce the computational cost compared to the traditional structure-preserving methods which are often implicit. Through rigorous error analysis, we give the error bounds of the methods at O (h m 0 + τ 2 / ε 2) where h is mesh size, τ is time step and the integer m 0 is determined by the regularity conditions. These error bounds indicate that, to obtain the correct numerical solution in the simultaneously massless and nonrelativistic regime, our methods request the ε -scalability as h = O (1) and τ = O (ε) which is better than the ε -scalability of the finite difference (FD) methods: h = O (ε 1 / 2) and τ = O (ε 3 / 2) . Numerical experiments confirm that the theoretical results in this paper are correct. [ABSTRACT FROM AUTHOR]

As a special UTV decomposition, the QLP decomposition is an effective alternative of the singular value decomposition (SVD) for the low-rank approximation. In this paper, we propose a single-pass randomized QLP decomposition algorithm for computing a low-rank matrix approximation. Compared with the randomized QLP decomposition, the complexity of the proposed algorithm does not increase significantly and the data matrix needs to be accessed only once. Therefore, our algorithm is suitable for a large matrix stored outside of memory or generated by streaming data. In the error analysis, we give the matrix approximation error analysis. We also provide singular value approximation error bounds, which can track the target largest singular values of the data matrix with high probability. Numerical experiments are also reported to verify our results. [ABSTRACT FROM AUTHOR]