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Showing papers in "Ima Journal of Numerical Analysis in 2007"


Journal ArticleDOI
TL;DR: In this paper, the authors define a new evolving surface finite element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time, based on approximating t by an evolving interpolated polyhedral surface h(t) consisting of a union of simplices (triangles for n = 2) whose vertices lie on (t).
Abstract: In this article, we define a new evolving surface finite-element method for numerically approximating partial differential equations on hypersurfaces (t) in n+1 which evolve with time. The key idea is based on approximating (t) by an evolving interpolated polyhedral (polygonal if n = 1) surface h(t) consisting of a union of simplices (triangles for n = 2) whose vertices lie on (t). A finite-element space of functions is then defined by taking the set of all continuous functions on h(t) which are linear affine on each simplex. The finite-element nodal basis functions enjoy a transport property which simplifies the computation. We formulate a conservation law for a scalar quantity on (t) and, in the case of a diffusive flux, derive a transport and diffusion equation which takes into account the tangential velocity and the local stretching of the surface. Using surface gradients to define weak forms of elliptic operators naturally generates weak formulations of elliptic and parabolic equations on (t). Our finite-element method is applied to the weak form of the conservation equation. The computations of the mass and element stiffness matrices are simple and straightforward. Error bounds are derived in the case of semi-discretization in space. Numerical experiments are described which indicate the order of convergence and also the power of the method. We describe how this framework may be employed in applications. Key Words: finite elements; evolving surfaces; conservation; diffusion; existence; error estimates; computations

369 citations


Journal ArticleDOI
TL;DR: These methods involve two iteration parameters whose special choices can recover the known preconditioned HSS iteration methods, as well as yield new ones, and show that the new methods converge unconditionally to the unique solution of the saddle-point problem.
Abstract: We establish a class of accelerated Hermitian and skew-Hermitian splitting (AHSS) iteration methods for large sparse saddle-point problems by making use of the Hermitian and skew-Hermitian splitting (HSS) iteration technique. These methods involve two iteration parameters whose special choices can recover the known preconditioned HSS iteration methods, as well as yield new ones. Theoretical analyses show that the new methods converge unconditionally to the unique solution of the saddle-point problem. Moreover, the optimal choices of the iteration parameters involved and the corresponding asymptotic convergence rates of the new methods are computed exactly. In addition, theoretical properties of the preconditioned Krylov subspace methods such as GMRES are investigated in detail when the AHSS iterations are employed as their preconditioners. Numerical experiments confirm the correctness of the theory and the effectiveness of the methods.

345 citations


Journal ArticleDOI
TL;DR: The convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp.
Abstract: A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a (x, co) in a bounded domain D R d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω e Q) variables in a(x,ω) via Karhunen-Loeve or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ω) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.

232 citations


Journal ArticleDOI
TL;DR: It is argued that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditionsers for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation.
Abstract: We revisit real-valued preconditioned iterative methods for the solution of complex linear systems, with an emphasis on symmetric (non-Hermitian) problems. Different choices of the real equivalent formulation are discussed, as well as different types of block preconditioners for Krylov subspace methods. We argue that if either the real or the symmetric part of the coefficient matrix is positive semidefinite, block preconditioners for real equivalent formulations may be a useful alternative to preconditioners for the original complex formulation. Numerical experiments illustrating the performance of the various approaches are presented.

143 citations


Journal ArticleDOI
TL;DR: In this article, a finite-element approximation for a non-linear parabolic-elliptic system is considered, which describes the aggregation of slime moulds resulting from their chemotactic features and is called a simplified Keller-Segel system.
Abstract: Finite-element approximation for a non-linear parabolic-elliptic system is considered. The system describes the aggregation of slime moulds resulting from their chemotactic features and is called a simplified Keller-Segel system. Applying an upwind technique, first we present a finite-element scheme that satisfies both positivity and mass conservation properties. Consequently, if the triangulation is of acute type, our finite-element approximation preserves the Ll norm, which is an important property of the original system. Then, under some assumptions on the regularity of a solution and on the triangulation, we establish error estimates in L P x W 1,∞ with a suitable p > d, where d is the dimension of a spatial domain. Our scheme is well suited for practical computations. Some numerical examples that validate our theoretical results are also presented.

88 citations


Journal ArticleDOI
TL;DR: By using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity and is based on adaptive steepest descent iterations.
Abstract: This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D.

82 citations


Journal ArticleDOI
TL;DR: This paper showed that the Degasperis-procesi equation is well-posed in the class of (discontinuous) entropy solutions and provided several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data.
Abstract: Recent work (COCLITE, G. M. & KARLSEN, K. H. (2006) On the well-posedness of the Degasperis-Procesi equation. J. Funct. Anal., 233, 60-91) has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper, we construct numerical schemes and prove that they converge to entropy solutions. Additionally, we provide several numerical examples accentuating that discontinuous (shock) solutions form independently of the smoothness of the initial data. Our focus on discontinuous solutions contrasts notably with the existing literature on the Degasperis-Procesi equation, which seems to emphasize similarities with the Camassa-Holm equation (bi-Hamiltonian structure, integrability, peakon solutions and H 1 as the relevant functional space).

68 citations


Journal ArticleDOI
TL;DR: This work develops the a posteriori error estimation of interior penalty discontinuous Galerkin discretizations for H(curl)-elliptic problems that arise in eddy current models and validates its reliability and efficiency for a range of test problems.
Abstract: We develop the a posteriori error estimation of interior penalty discontinuous Galerkin discretizations for H(curl)-elliptic problems that arise in eddy current models. Computable upper and lower bounds on the error measured in terms of a natural (mesh-dependent) energy norm are derived. The proposed a posteriori error estimator is validated by numerical experiments, illustrating its reliability and efficiency for a range of test problems.

62 citations


Journal ArticleDOI
TL;DR: It is proved that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns.
Abstract: In this paper, we discuss the application of hierarchical matrix techniques to the solution of Helmholtz problems with large wave number {kappa} in 2D. We consider the Brakhage–Werner integral formulation of the problem discretized by the Galerkin boundary-element method. The dense n x n Galerkin matrix arising from this approach is represented by a sum of an Formula -matrix and an Formula 2-matrix, two different hierarchical matrix formats. A well-known multipole expansion is used to construct the Formula 2-matrix. We present a new approach to dealing with the numerical instability problems of this expansion: the parts of the matrix that can cause problems are approximated in a stable way by an Formula -matrix. Algebraic recompression methods are used to reduce the storage and the complexity of arithmetical operations of the Formula -matrix. Further, an approximate LU decomposition of such a recompressed Formula -matrix is an effective preconditioner. We prove that the construction of the matrices as well as the matrix-vector product can be performed in almost linear time in the number of unknowns. Numerical experiments for scattering problems in 2D are presented, where the linear systems are solved by a preconditioned iterative method.

58 citations


Journal ArticleDOI
TL;DR: The a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations is developed.
Abstract: We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

57 citations


Journal ArticleDOI
TL;DR: In this paper, a variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow, as well as related flows, is presented, covering both the closed-curve case and the case of curves that are connected via triple junction points.
Abstract: We present a variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow, as well as related flows. The proposed scheme covers both the closed-curve case and the case of curves that are connected via triple junction points. On introducing a parametric finite-element approximation, we prove stability bounds and report on numerical experiments, including regularized crystalline mean curvature flow and regularized crystalline surface diffusion. The presented scheme has very good properties with respect to the distribution of mesh points and, if applicable, area conservation.

Journal ArticleDOI
TL;DR: In this paper, the stochastic steady-state diffusion problem is solved using a multigrid algorithm under the mild assumption that the diffusion coefficient takes the form of a finite Karhunen-Loeve expansion.
Abstract: We study multigrid for solving the stochastic steady-state diffusion problem. We operate under the mild assumption that the diffusion coefficient takes the form of a finite Karhunen-Loeve expansion. The problem is discretized using a finite-element methodology using the polynomial chaos method to discretize the stochastic part of the problem. We apply a multigrid algorithm to the stochastic problem in which the spatial discretization is varied from grid to grid while the stochastic discretization is held constant. We then show, theoretically and experimentally, that the convergence rate is independent of the spatial discretization, as in the deterministic case, and the stochastic discretization.

Journal ArticleDOI
TL;DR: This paper analyses two-level Schwarz methods for matrices arising from the p-version finite-element method on triangular and tetrahedral meshes and investigates several decompositions with large or small overlap leading to optimal or close to optimal condition numbers.
Abstract: This paper analyses two-level Schwarz methods for matrices arising from the p-version finite-element method on triangular and tetrahedral meshes. The coarse level consists of the lowest-order finite-element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a simple model problem and an elasticity problem on a complex geometry.

Journal ArticleDOI
TL;DR: This work considers the construction of a special family of Runge-Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev-Gauss-Lobatto type, with A-stability and stiffly accurate characteristics, suitable for solving stiff initial-value problems.
Abstract: We consider the construction of a special family of Runge-Kutta (RK) collocation methods based on intra-step nodal points of Chebyshev-Gauss-Lobatto type, with A-stability and stiffly accurate characteristics. This feature with its inherent implicitness makes them suitable for solving stiff initial-value problems. In fact, the two simplest cases consist in the well-known trapezoidal rule and the fourth-order Runge-Kutta-Lobatto IIIA method. We will present here the coefficients up to eighth order, but we provide the formulas to obtain methods of higher order. When the number of stages is odd, we have considered a new strategy for changing the step size based on the use of a pair of methods: the given RK method and a linear multistep one. Some numerical experiments are considered in order to check the behaviour of the methods when applied to a variety of initial-value problems.

Journal ArticleDOI
TL;DR: A new a posteriori error estimator for the Lame system based on H(div) -conforming elements and equilibrated fluxes is derived and gives rise to an upper bound where the constant is one up to higher-order terms.
Abstract: We derive a new a posteriori error estimator for the Lame system based on H(div) -conforming elements and equilibrated fluxes. It is shown that the estimator gives rise to an upper bound where the constant is one up to higher-order terms. The lower bound is also established using Argyris elements. The reliability and efficiency of the proposed estimator are confirmed by some numerical tests.

Journal ArticleDOI
Abstract: This work is concerned with the preservation of invariants and of volume-forms by numerical methods which can be expanded into B-series. The situation we consider here is that of a split vector field where each sub-field either has the common invariant I or is divergence free. We derive algebraic conditions on the coefficients of the B-series for it either to preserve I or to preserve the volume for generic vector fields and interpret them for additive Runge-Kutta methods. Comparing the two sets of conditions then enables us to state some non-existence results. For a more restrictive class of problems, where the system is partitionned into several components, we nevertheless obtain simplified conditions and show that they can be solved.

Journal ArticleDOI
TL;DR: It is shown that the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form is also optimally convergent in the L 2 -norm, on tetrahedral meshes and for smooth material coefficients.
Abstract: We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L 2 -norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetics.

Journal ArticleDOI
TL;DR: This paper considers the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact inverse iteration with a fixed shift and derives a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditionser.
Abstract: In this paper, we consider the computation of an eigenvalue and the corresponding eigenvector of a large sparse Hermitian positive-definite matrix using inexact inverse iteration with a fixed shift. For such problems, the large sparse linear systems arising at each iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomplete Cholesky factorization and derive a new tuned Cholesky preconditioner which shows considerable improvement over the standard preconditioner. This improvement is analysed using the convergence theory for MINRES. We also compare the spectral properties of the tuned preconditioned matrix with those of the standard preconditioned matrix. In particular, we provide both a perturbation result and an interlacing result, and these results show that the spectral properties of the tuned preconditioner are similar to those of the standard preconditioner. For Rayleigh quotient shifts, comparison is also made with a technique introduced by Simoncini & Elden (2002, BIT, 42, 159–182) which involves changing the right-hand side of the inverse iteration step. Several numerical examples are given to illustrate the theory described in the paper.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed efficient numerical algorithms for a class of forward-backward stochastic differential equations (FBSDEs) and related quasi-linear parabolic partial differential equations.
Abstract: Efficient numerical algorithms for a class of forward-backward stochastic differential equations (FBSDEs) and related quasi-linear parabolic partial differential equations are proposed. The quasi-linear parabolic equation is solved by new layer methods which are constructed by means of a probabilistic approach. The proposed algorithms for solving FBSDEs are based on the four-step scheme of Ma, Protter and Yong. Convergence theorems are proved. Results of some numerical experiments are presented.

Journal ArticleDOI
TL;DR: A new semi-discrete model based on the space discretization of the wave equation using a mixed ¯nite element method with two di®erent basis functions for the position and velocity is introduced and the main theoretical result is a uniform observability inequality which allows for a sequence of approximations converging to the minimal L2inorm control of the continuous wave equation.
Abstract: This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models ob- tained by discretizing the wave equation with the usual ¯nite di®erence or ¯nite element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation, as the mesh size goes to zero (see [7, 15]). Here we introduce and analyze a new semi-discrete model based on the space discretization of the wave equa- tion using a mixed ¯nite element method with two di®erent basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L2inorm control of the continuous wave equation. We also introduce a fully-discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and ¢t, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.

Journal ArticleDOI
TL;DR: In this article, a finite-volume discretization for multidimensional nonlinear drift-diffusion system is proposed, which is composed of two parabolic equations and an elliptic one.
Abstract: In this paper, we propose a finite-volume discretization for multidimensional nonlinear drift-diffusion system. Such a system arises in semiconductors modelling and is composed of two parabolic equations and an elliptic one. We prove that the numerical solution converges to a steady state when time goes to infinity. Several numerical tests show the efficiency of the method.

Journal ArticleDOI
TL;DR: This paper introduces a sparse approximation of the system matrix by cutoff, in order to reduce the storage costs, and introduces a panel clustering method to further reduce these costs.
Abstract: We consider the wave equation in a boundary integral formulation. The discretization in time is done by using convolution quadrature techniques and a Galerkin boundary element method for the spatial discretization. In a previous paper, we have introduced a sparse approximation of the system matrix by cut-off, in order to reduce the storage costs. In this paper, we extend this approach by introducing a panel clustering method to further reduce these costs.

Journal ArticleDOI
TL;DR: In this article, a cubic C1 spline-based approximating scheme based on cubic C 1 splines on type-6 tetrahedral partitions using data on volumetric grids is described.
Abstract: We describe an approximating scheme based on cubic C1 splines on type-6 tetrahedral partitions using data on volumetric grids. The quasi-interpolating piecewise polynomials are directly determined by setting their Bernstein-Bezier coefficients to appropriate combinations of the data values. Hence, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. The locality of the method and the uniform boundedness of the associated operator provide an error bound, which shows that the approach can be used to approximate and reconstruct trivariate functions. Simultaneously, we show that the derivatives of the quasi-interpolating splines yield nearly optimal approximation order. Numerical tests with up to 17 x 10 6 data sites show that the method can be used for efficient approximation.

Journal ArticleDOI
TL;DR: In this article, the authors studied the dimension reduction techniques of Brownian bridge (BB) and principal component analysis (PCA) in the context of high-dimensional integration problems.
Abstract: High-dimensional integrals occur in a variety of areas, including mathematical finance. In the classical settings, multivariate integration problems suer from the curse of dimensionality. To vanquish the curse of dimensionality, one may shrink the function class. Here we use weighted function spaces, in which groups of variables are associated with weights, in order to capture the dierent importance of each group of variables. For practical applications, the principal diculty then is in choosing the "right" weights for a given problem or class of problems. We work in weighted reproducing kernel Hilbert spaces (with "general" rather than "product" weights). We first present a principle to find the best weights for a given problem. This general approach is then applied to a simplified high-dimensional problem from finance, in which the dimension is the number of discrete time steps for a price of a risky asset which follows geometric Brownian motion. A second focus of this paper is on the dimension reduction techniques of Brownian bridge (BB) and principal component analysis (PCA). It turns out that the behavior of the model problem is dramatically improved when QMC is used in conjunction with the dimension reduction techniques: if the "right" weights are used in every case, then the integration error can be bounded independently of the dimension, whereas without BB or PCA the error bound depends exponentially on the dimension. Finally, for this model problem we show how to construct shifted lattice rules which, when used in conjunction with BB or PCA, yield integration errors converging as O(n 3/4+ ) or O(n 1+ ) (for arbitrary small > 0) respectively, independently of the dimension. Thus in both cases well-designed algorithms can avoid the curse of dimensionality, with PCA having an advantage over BB with respect to the proved order of convergence.

Journal ArticleDOI
TL;DR: In this article, an adaptive finite element method (AFEM) is proposed to compute strongly converging stress approximations for a class of degenerate convex minimization problems.
Abstract: A class of degenerate convex minimization problems allows for some adaptive finite-element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of successive loops of the form SOLVE → ESTIMATE -> MARK → REFINE and employs the bulk criterion. The convergence in L P' (Ω; R m×n ) relies on new sharp strict convexity estimates of degenerate convex minimization problems with J(v):= ∫ Ω W(Dν)dx-∫ Ω fνdx for ν ∈ V:= W 0 1,p (Ω; R m ). The class of minimization problems includes strong convex problems and allows applications in an optimal design task, Hencky elastoplasticity or relaxation of two-well problems allowing for microstructures.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the numerical analysis of evolution variational inequalities which are derived from Maxwell's equations coupled with a nonlinear constitutive relation between the electric field and the current density and governing the magnetic field around a type-II bulk superconductor located in 3D space.
Abstract: We consider the numerical analysis of evolution variational inequalities which are derived from Maxwell's equations coupled with a nonlinear constitutive relation between the electric field and the current density and governing the magnetic field around a type-II bulk superconductor located in 3D space. The nonlinear Ohm's law is formulated using the subdifferential of a convex energy so the theory is applied to the Bean critical-state model, a power law model and an extended Bean critical-state model. The magnetic field in the nonconducting region is expressed as a gradient of a magnetic Scalar potential in order to handle the curl-free constraint. The variational inequalities ate discretized in time implicitly and in space by Nedelec's curl-conforming finite element of lowest order. The honsmooth energies are smoothed with a regularization parameter so that the fully discrete problem is a system of nonlinear algebraic equations at each time step. We prove various convergence results. Some numerical simulations under a uniform external magnetic field are presented.

Journal ArticleDOI
TL;DR: In this article, a general framework for interpolatory quadrature rules for Hadamard finite-part integrals with a second-order singularity is presented, and the equivalence of some existing formulas which were obtained in different ways.
Abstract: In this paper, we present a general framework for interpolatory quadrature rules for Hadamard finite-part integrals with a second-order singularity. Gaussian quadrature rules are viewed as a special case and many interesting features can be obtained easily from the framework. We prove theoretically the equivalence of some existing formulas which were obtained in different ways. We show the point-wise superconvergence of these interpolatory quadrature rules, i.e. when the singular point coincides with certain a priori known points, the accuracy is better than what is generally possible. The extension of a popular interpolatory quadrature rule for Cauchy principal value integrals is presented. A new quadrature rule of Gaussian type is proposed for the evaluation of integrals simultaneously involving different types of singularities. Numerical examples confirm our theoretical results

Journal ArticleDOI
TL;DR: This paper is concerned with the analysis of the discontinuous Galerkin finite-element method applied to the space semi-discretization of a nonlinear non-stationary convection-diffusion problem and the derivation of an L ∞ (L 2 )-optimal error estimate for the symmetric interior penaltyGalerkin scheme.
Abstract: This paper is concerned with the analysis of the discontinuous Galerkin finite-element method applied to the space semi-discretization of a nonlinear non-stationary convection-diffusion problem. Attention is paid on the derivation of an L ∞ (L 2 )-optimal error estimate for the symmetric interior penalty Galerkin scheme. The error analysis is performed for standard simplicial meshes under the assumption that the exact solution of the problem and the solution of an elliptic dual problem are sufficiently regular. The theoretical results are illustrated by numerical experiments.

Journal ArticleDOI
TL;DR: In this article, the eigenvalues of a second-order operator pencil of the form Q- XP were computed using sinc techniques, where Q and P are self-adjoint differential operators of the second and first order, respectively.
Abstract: In this paper, we use sinc techniques to compute the eigenvalues of a second-order operator pencil of the form Q- XP approximately Here Q and P are self-adjoint differential operators of the second and first order, respectively Also the eigenparameter appears in the boundary conditions linearly

Journal ArticleDOI
TL;DR: In this paper, the decay rates of radial basis functions (RBFs) are derived for cardinal data, in both 1D and 2D, and the authors explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.
Abstract: Many types of radial basis functions (RBFs) are global in terms of having large magnitude across the entire domain. Yet, in contrast, e.g. with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centred in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well-conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1D and 2D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.