Subjects -> MATHEMATICS (Total: 1118 journals)     - APPLIED MATHEMATICS (92 journals)    - GEOMETRY AND TOPOLOGY (23 journals)    - MATHEMATICS (819 journals)    - MATHEMATICS (GENERAL) (45 journals)    - NUMERICAL ANALYSIS (26 journals)    - PROBABILITIES AND MATH STATISTICS (113 journals) MATHEMATICS (819 journals)                  1 2 3 4 5 | Last

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 Advances in Computational MathematicsJournal Prestige (SJR): 0.812 Citation Impact (citeScore): 1Number of Followers: 23      Hybrid journal (It can contain Open Access articles) ISSN (Print) 1572-9044 - ISSN (Online) 1019-7168 Published by Springer-Verlag  [2658 journals]
• Comparison of integral equations for the Maxwell transmission problem with
general permittivities

Abstract: Two recently derived integral equations for the Maxwell transmission problem are compared through numerical tests on simply connected axially symmetric domains for non-magnetic materials. The winning integral equation turns out to be entirely free from false eigenwavenumbers for any passive materials, also for purely negative permittivity ratios and in the static limit, as well as free from false essential spectrum on non-smooth surfaces. It also appears to be numerically competitive to all other available integral equation reformulations of the Maxwell transmission problem, despite using eight scalar surface densities.
PubDate: 2021-10-15

• An a posteriori error estimate for a dual mixed method applied to Stokes
system with non-null source terms

Abstract: In this work, we focus our attention in the Stokes flow with nonhomogeneous source terms, formulated in dual mixed form. For the sake of completeness, we begin recalling the corresponding well-posedness at continuous and discrete levels. After that, and with the help of a kind of a quasi-Helmholtz decomposition of functions in H(div), we develop a residual type a posteriori error analysis, deducing an estimator that is reliable and locally efficient. Finally, we provide numerical experiments, which confirm our theoretical results on the a posteriori error estimator and illustrate the performance of the corresponding adaptive algorithm, supporting its use in practice.
PubDate: 2021-10-15

• Mixed precision path tracking for polynomial homotopy continuation

Abstract: This article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step, it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.
PubDate: 2021-09-29

• A rational RBF interpolation with conditionally positive definite kernels

Abstract: In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method.
PubDate: 2021-09-27

• Accurate singular values of a class of parameterized negative matrices

Abstract: Typically, parametrization captures the essence of a class of matrices, and its potential advantage is to make accurate computations possible. But, in general, parametrization suitable for accurate computations is not always easy to find. In this paper, we introduce a parametrization of a class of negative matrices to accurately solve the singular value problem. It is observed that, given a set of parameters, the associated nonsingular negative matrix can be orthogonally transformed into a totally nonnegative matrix in an implicit and subtraction-free way, which implies that such a set of parameters determines singular values of the associated negative matrix accurately. Based on this observation, a new O(n3) algorithm is designed to compute all the singular values, large and small, to high relative accuracy.
PubDate: 2021-09-10
DOI: 10.1007/s10444-021-09898-z

• Error analysis of the SAV Fourier-spectral method for the
Cahn-Hilliard-Hele-Shaw system

Abstract: In this paper, we construct several efficient scalar auxiliary variable (SAV) schemes based on the Fourier-spectral method in space for the Cahn-Hilliard-Hele-Shaw system. The temporal discretizations are built upon the first-order Euler and second-order BDF method, respectively. We derive the unconditional energy stability for both schemes and also establish the rigorous error estimates for the first-order SAV Fourier-spectral scheme. Finally, various numerical experiments are presented to demonstrate the accuracy and performance for the constructed schemes.
PubDate: 2021-09-10
DOI: 10.1007/s10444-021-09897-0

• Carleman estimates and controllability results for fully discrete
approximations of 1D parabolic equations

Abstract: In this paper, we prove a Carleman estimate for fully discrete approximations of one-dimensional parabolic operators in which the discrete parameters h and △t are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results for fully discrete linear and semilinear parabolic equations.
PubDate: 2021-09-10
DOI: 10.1007/s10444-021-09885-4

• An alternative approach for order conditions of Runge-Kutta-Nyström
methods

Abstract: We present an alternative approach proposed by Albrecht to derive general order conditions for Runge-Kutta-Nyström methods and relate it to the classical RKN-theory. The RKN-methods are treated as composite linear methods to yield the general order conditions as orthogonal relations. We then exploit the orthogonal structure of the order conditions and obtain a simple recursion to generate the order conditions. Implications of this approach on the classical RKN-theory are discussed and it may be worthwhile to generalize the approach to other discretizations.
PubDate: 2021-09-07
DOI: 10.1007/s10444-021-09894-3

• Kernel aggregated fast multipole method

Abstract: Many different simulation methods for Stokes flow problems involve a common computationally intense task—the summation of a kernel function over O(N2) pairs of points. One popular technique is the kernel independent fast multipole method (KIFMM), which constructs a spatial adaptive octree for all points and places a small number of equivalent multipole and local equivalent points around each octree box, and completes the kernel sum with O(N) cost, using these equivalent points. Simpler kernels can be used between these equivalent points to improve the efficiency of KIFMM. Here we present further extensions and applications to this idea, to enable efficient summations and flexible boundary conditions for various kernels. We call our method the kernel aggregated fast multipole method (KAFMM), because it uses different kernel functions at different stages of octree traversal. We have implemented our method as an open-source software library STKFMM based on the high-performance library PVFMM, with support for Laplace kernels, the Stokeslet, regularized Stokeslet, Rotne-Prager-Yamakawa (RPY) tensor, and the Stokes double-layer and traction operators. Open and periodic boundary conditions are supported for all kernels, and the no-slip wall boundary condition is supported for the Stokeslet and RPY tensor. The package is designed to be ready-to-use as well as being readily extensible to additional kernels.
PubDate: 2021-09-06
DOI: 10.1007/s10444-021-09896-1

• Analysis of a Helmholtz preconditioning problem motivated by uncertainty
quantification

Abstract: This paper analyses the following question: let Aj, j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (Aj∇uj) + k2njuj = −f. How small must $$\ A_{1} -A_{2}\ _{L^{q}}$$ and $$\ {n_{1}} - {n_{2}}\ _{L^{q}}$$ be (in terms of k-dependence) for GMRES applied to either $$(\mathbf {A}_1)^{-1}\mathbf {A}_2$$ or A2(A1)− 1 to converge in a k-independent number of iterations for arbitrarily large k' (In other words, for A1 to be a good left or right preconditioner for A2') We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients A and n. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different A and n, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.
PubDate: 2021-09-03
DOI: 10.1007/s10444-021-09889-0

• Robust preconditioning techniques for multiharmonic finite element method
with application to time-periodic parabolic optimal control problems

Abstract: We are concerned with efficient solutions of the time-periodic parabolic optimal control problems. By using the multiharmonic FEM, the linear algebraic equations characterizing the first-order optimality conditions can be decoupled into a series of parallel solvable block 4 × 4 linear systems with respect to the cosine and sine Fourier coefficients of the state and scaled control variables for different frequencies. Parameter robust preconditioners are proposed for solving these linear systems along with information on practical algorithm implementation and detailed spectral analysis. Problem independent eigenvalue bounds and upper bound approximations of the condition numbers of the eigenvector matrices are obtained for the preconditioned matrices. Such results ensure efficient Krylov subspace acceleration methods and a parameter-free Chebyshev acceleration method, which are both robust in view of all discretization and model parameters. Numerical experiments are presented to demonstrate the robustness and effectiveness of the proposed preconditioners within both Krylov subspace and Chebyshev accelerations compared with some already available preconditioned Krylov subspace methods.
PubDate: 2021-09-03
DOI: 10.1007/s10444-021-09887-2

• An O(N) algorithm for computing expectation of N-dimensional truncated
multi-variate normal distribution I: fundamentals

Abstract: In this paper, we present the fundamentals of a hierarchical algorithm for computing the N-dimensional integral $$\phi (\mathbf {a}, \mathbf {b}; A) = {\int \limits }_{\mathbf {a}}^{\mathbf {b}} H(\mathbf {x}) f(\mathbf {x} A) \text {d} \mathbf {x}$$ representing the expectation of a function H(X) where f(x A) is the truncated multi-variate normal (TMVN) distribution with zero mean, x is the vector of integration variables for the N-dimensional random vector X, A is the inverse of the covariance matrix Σ, and a and b are constant vectors. The algorithm assumes that H(x) is “low-rank” and is designed for properly clustered X so that the matrix A has “low-rank” blocks and “low-dimensional” features. We demonstrate the divide-and-conquer idea when A is a symmetric positive definite tridiagonal matrix and present the necessary building blocks and rigorous potential theory–based algorithm analysis when A is given by the exponential covariance model. The algorithm overall complexity is O(N) for N-dimensional problems, with a prefactor determined by the rank of the off-diagonal matrix blocks and number of effective variables. Very high accuracy results for N as large as 2048 are obtained on a desktop computer with 16G memory using the fast Fourier transform (FFT) and non-uniform FFT to validate the analysis. The current paper focuses on the ideas using the simple yet representative examples where the off-diagonal matrix blocks are rank 1 and the number of effective variables is bounded by 2, to allow concise notations and easier explanation. In a subsequent paper, we discuss the generalization of current scheme using the sparse grid technique for higher rank problems and demonstrate how all the moments of kth order or less (a total of O(Nk) integrals) can be computed using O(Nk) operations for k ≥ 2 and $$O(N \log N)$$ operations for k = 1.
PubDate: 2021-09-01
DOI: 10.1007/s10444-021-09888-1

• Computing low-rank rightmost eigenpairs of a class of matrix-valued linear
operators

Abstract: In this article, a new method is proposed to approximate the rightmost eigenpair of certain matrix-valued linear operators, in a low-rank setting. First, we introduce a suitable ordinary differential equation, whose solution allows us to approximate the rightmost eigenpair of the linear operator. After analyzing the behaviour of its solution on the whole space, we project the ODE on a low-rank manifold of prescribed rank and correspondingly analyze the behaviour of its solutions. For a general linear operator we prove that—under generic assumptions—the solution of the ODE converges globally to its leading eigenmatrix. The analysis of the projected operator is more subtle due to its nonlinearity; when ca is self-adjoint, we are able to prove that the associated low-rank ODE converges (at least locally) to its rightmost eigenmatrix in the low-rank manifold, a property which appears to hold also in the more general case. Two explicit numerical methods are proposed, the second being an adaptation of the projector splitting integrator proposed recently by Lubich and Oseledets. The numerical experiments show that the method is effective and competitive.
PubDate: 2021-09-01
DOI: 10.1007/s10444-021-09895-2

• A fast ADI orthogonal spline collocation method with graded meshes for the
two-dimensional fractional integro-differential equation

Abstract: We propose and analyze a time-stepping Crank-Nicolson(CN) alternating direction implicit(ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.
PubDate: 2021-08-31
DOI: 10.1007/s10444-021-09884-5

• An H1 convergence of the spectral method for the time-fractional
non-linear diffusion equations

Abstract: The generalized discrete Gronwall inequality is applied to analyze the optimal H1 error estimate of the time-stepping spectral method for the time-fractional diffusion equations, where the time-fractional derivative is discretized by the second-order fractional backward difference formula or the second-order generalized Newton-Gregory formula. The methodology is extended to analyze the fractional Crank–Nicolson spectral method and the time-stepping spectral method for the multi-term time-fractional differential equations. Numerical simulations are provided to support the theoretical analysis.
PubDate: 2021-08-25
DOI: 10.1007/s10444-021-09892-5

• Convergence rates for boundedly regular systems

Abstract: In this work, we consider a continuous dynamical system associated with the fixed point set of a nonexpansive operator which was originally studied by Boţ and Csetnek (J. Dyn. Diff. Equat. 29(1), pp. 155–168, 2017). Our main results establish convergence rates for the system’s trajectories when the nonexpansive operator satisfies an additional regularity property. This setting is the natural continuous-time analogue to discrete-time results obtained in Bauschke, Noll and Phan (J. Math. Anal. Appl. 421(1), pp. 1–20, 2015) and Borwein, Li and Tam (SIAM J. Optim. 27(1), pp. 1–33, 2017) by using the same regularity properties. Closure properties of the class of Hölder regular operators under taking convex combinations and compositions are also derived.
PubDate: 2021-08-24
DOI: 10.1007/s10444-021-09891-6

• Non-symmetric isogeometric FEM-BEM couplings

Abstract: We present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting of two disjoint domains. We consider the Finite Element Method in the bounded domains to simulate possibly non-linear materials. The Boundary Element Method is applied in unbounded or thin domains where the material behavior is linear. The isogeometric framework allows to combine different design and analysis tools: first, we consider the same type of NURBS parameterizations for an exact geometry representation and second, we use the numerical analysis for the Galerkin approximation. Moreover, it facilitates to perform h- and p-refinements. For the sake of analysis, we consider the framework of strongly monotone and Lipschitz continuous operators to ensure well-posedness of the coupled system. Furthermore, we provide a priori error estimates. We additionally show an improved convergence behavior for the errors in functionals of the solution that may double the rate under certain assumptions. Numerical examples conclude the work which illustrate the theoretical results.
PubDate: 2021-08-19
DOI: 10.1007/s10444-021-09886-3

• Rapid evaluation of the spectral signal detection threshold and Stieltjes
transform

Abstract: Accurate detection of signal components is a frequently-encountered challenge in statistical applications with a low signal-to-noise ratio. This problem is particularly challenging in settings with heteroscedastic noise. In certain signal-plus-noise models of data, such as the classical spiked covariance model and its variants, there are closed formulas for the spectral signal detection threshold (the largest sample eigenvalue attributable solely to noise) for isotropic noise in the limit of infinitely large data matrices. However, more general noise models currently lack provably fast and accurate methods for numerically evaluating the threshold. In this work, we introduce a rapid algorithm for evaluating the spectral signal detection threshold in the limit of infinitely large data matrices. We consider noise matrices with a separable variance profile (whose variance matrix is rank 1), as these arise often in applications. The solution is based on nested applications of Newton’s method. We also devise a new algorithm for evaluating the Stieltjes transform of the spectral distribution at real values exceeding the threshold. The Stieltjes transform on this domain is known to be a key quantity in parameter estimation for spectral denoising methods. The correctness of both algorithms is proven from a detailed analysis of the master equations characterizing the Stieltjes transform, and their performance is demonstrated in numerical experiments.
PubDate: 2021-08-13
DOI: 10.1007/s10444-021-09890-7

• Two-scale finite element discretizations for nonlinear eigenvalue problems
in quantum physics

Abstract: In this paper, some two-scale finite element discretizations are introduced and analyzed for a class of nonlinear elliptic eigenvalue problems on tensor product domains. It is shown that the solution obtained by the standard finite element method on a one-scale fine grid can be numerically replaced with a combination of some solutions on a coarse grid and some univariate fine grids by two-scale finite element discretizations. Compared with the standard finite element solution, the two-scale finite element approximations save computational cost significantly while achieving the same accuracy.
PubDate: 2021-08-12
DOI: 10.1007/s10444-021-09883-6

• Efficient spatial second-/fourth-order finite difference ADI methods for
multi-dimensional variable-order time-fractional diffusion equations

Abstract: Variable-order time-fractional diffusion equations (VO-tFDEs), which can be used to model solute transport in heterogeneous porous media are considered. Concerning the well-posedness and regularity theory (cf., Zheng & Wang, Anal. Appl., 2020), two finite difference ADI and compact ADI schemes are respectively proposed for the two-dimensional VO-tFDE. We show that the two schemes are unconditionally stable and convergent with second and fourth orders in space with respect to corresponding discrete norms. Besides, efficiency and practical computation of the ADI schemes are also discussed. Furthermore, the ADI and compact ADI methods are extended to model three-dimensional VO-tFDE, and unconditional stability and convergence are also proved. Finally, several numerical examples are given to validate the theoretical analysis and show efficiency of the ADI methods.
PubDate: 2021-08-06
DOI: 10.1007/s10444-021-09881-8

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