Showing papers in "Advances in Computational Mathematics in 2005"

Near-optimal data-independent point locations for radial basis function interpolation

Abstract: The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples.

A least-squares preconditioner for radial basis functions collocation methods

Abstract: Although meshless radial basis function (RBF) methods applied to partial differential equations (PDEs) are not only simple to implement and enjoy exponential convergence rates as compared to standard mesh-based schemes, the system of equations required to find the expansion coefficients are typically badly conditioned and expensive using the global Gaussian elimination (G-GE) method requiring \(\mathcal{O}(N^{3})\) flops We present a simple preconditioning scheme that is based upon constructing least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE matrix elements The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve for the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain Our preconditioner requires \(\mathcal{O}(mN^{2})\) flops to set up, and \(\mathcal{O}(mN)\) storage locations where m is a user define parameter of order of 10 For the 2D MQ-RBF with the shape parameter \(c\sim1/\sqrt{N}\) , the number of iterations required for convergence is of order of 10 for large values of N, making this a very attractive approach computationally As the shape parameter increases, our preconditioner will eventually be affected by the ill conditioning and round-off errors, and thus becomes less effective We tested our preconditioners on increasingly larger c and N A more stable construction scheme is available with a higher set up cost

Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids

Abstract: Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the accuracy of finite difference schemes. We find that the accuracy is directly linked to the decay rate, at large arguments, of the (generalized) Fourier transform of the radial function. Three different types of convergence rates can be distinguished as the node density increases – polynomial, spectral, and superspectral, as exemplified, for example, by thin plate splines, multiquadrics, and Gaussians respectively.

A new method of fundamental solutions applied to nonhomogeneous elliptic problems

Abstract: The classical method of fundamental solutions (MFS) has only been used to approximate the solution of homogeneous PDE problems. Coupled with other numerical schemes such as domain integration, dual reciprocity method (with polynomial or radial basis functions interpolation), the MFS can be extended to solve the nonhomogeneous problems. This paper presents an extension of the MFS for the direct approximation of Poisson and nonhomogeneous Helmholtz problems. This can be done by using the fundamental solutions of the associated eigenvalue equations as a basis to approximate the nonhomogeneous term. The particular solution of the PDE can then be evaluated. An advantage of this mesh-free method is that the resolution of both homogeneous and nonhomogeneous equations can be combined in a unified way and it can be used for multiscale problems. Numerical simulations are presented and show the quality of the approximations for several test examples.

The basis of meshless domain discretization: the meshless local Petrov–Galerkin (MLPG) method

Abstract: The MLPG method is the general basis for several variations of meshless methods presented in recent literature. The interrelation of the various meshless approaches is presented in this paper. Several variations of the meshless interpolation schemes are reviewed also. Recent developments and applications of the MLPG methods are surveyed.

Introduction to the Web-method and its applications

Abstract: The Web-method is a meshless finite element technique which uses weighted extended B-splines (Web-splines) on a tensor product grid as basis functions. It combines the computational advantages of B-splines and standard mesh-based elements. In particular, degree and smoothness can be chosen arbitrarily without substantially increasing the dimension. Hence, accurate approximations are obtained with relatively few parameters. Moreover, the regular grid is well suited for hierarchical refinement and multigrid techniques. This article should serve as an introduction to finite element approximation with B-splines. We first review the construction of Web-bases and discuss their basic properties. Then we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects.

Generalized Strang–Fix condition for scattered data quasi-interpolation

Abstract: Quasi-interpolation is very useful in the study of the approximation theory and its applications, since the method can yield solutions directly and does not require solving any linear system of equations. However, quasi-interpolation is usually discussed only for gridded data in the literature. In this paper we shall introduce a generalized Strang–Fix condition, which is related to nonstationary quasi-interpolation. Based on the discussion of the generalized Strang–Fix condition we shall generalize our quasi-interpolation scheme for multivariate scattered data, too.

Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics

Abstract: It is shown that, depending upon the orientation of the end tangents $\t_0, \t_1$ relative to the end point displacement vector $\Delta\p=\p_1-\p_0$, the problem of $G^1$ Hermite interpolation by PH cubic segments may admit zero, one, or two distinct solutions. For cases where two interpolants exist, the bending energy may be used to select among them. In cases where no solution exists, we determine the minimal adjustment of one end tangent that permits a spatial PH cubic Hermite interpolant. The problem of assigning tangents to a sequence of points $\p_0,\ldots,\p_n$ in $\mathbb{R}^3$, compatible with a $G^1$ piecewise--PH--cubic spline interpolating those points, is also briefly addressed. The performance of these methods, in terms of overall smoothness and shape--preservation properties of the resulting curves, is illustrated by a selection of computed examples.

Material stability analysis of particle methods

Abstract: Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with reasonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz–Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability, so that material instabilities can occur in stress states that should be stable. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material stability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations.

Approximation of parabolic PDEs on spheres using spherical basis functions

Abstract: In this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres S n⊂Rn+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.

Local error analysis for approximate solutions of hyperbolic conservation laws

Abstract: We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted

Matrix-valued radial basis functions: stability estimates and applications

Abstract: Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980’s. Several applications require that certain physical properties are satisfied by the interpolant, for example, being divergence-free in case of incompressible data. In this paper we consider a class of customized (e.g., divergence-free) RBFs that are matrix-valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier–Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution.

Meshfree modeling and analysis of physical fields in heterogeneous media

Abstract: Continuous and discrete variations in material properties lead to substantial difficulties for most mesh-based methods for modeling and analysis of physical fields. The meshfree method described in this paper relies on distance fields to boundaries and to material features in order to represent variations of material properties as well as to satisfy prescribed boundary conditions. The method is theoretically complete in the sense that all distributions of physical properties and all physical fields are represented by generalized Taylor series expansions in terms of powers of distance fields. We explain how such Taylor series can be used to construct solution structures – spaces of functions satisfying the prescribed boundary conditions exactly and containing the necessary degrees of freedom to satisfy additional constraints. Fully implemented numerical examples illustrate the effectiveness of the proposed approach.

On convergence of the method of fundamental solutions for solving the Dirichlet problem of Poisson’s equation

Abstract: In this paper the convergence of using the method of fundamental solutions for solving the boundary value problem of Laplace’s equation in R2 is established, where the boundaries of the domain and fictitious domain are assumed to be concentric circles. Fourier series is then used to find the particular solutions of Poisson’s equation, which the derivatives of particular solutions are estimated under the L 2 norm. The convergent order of solving the Dirichlet problem of Poisson’s equation by the method of particular solution and method of fundamental solution is derived.

A projection technique for incompressible flow in the meshless finite volume particle method

Abstract: The finite volume particle method is a meshless discretization technique, which generalizes the classical finite volume method by using smooth, overlapping and moving test functions applied in the weak formulation of the conservation law. The method was originally developed for hyperbolic conservation laws so that the compressible Euler equations particularly apply. In the present work we analyze the discretization error and enforce consistency by a new set of geometrical quantities. Furthermore, we introduce a discrete Laplace operator for the scheme in order to extend the method to second order partial differential equations. Finally, we transfer Chorin’s projection technique to the finite volume particle method in order to obtain a meshless scheme for incompressible flow.

Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions

Abstract: By extending Wendland’s meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved.

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Abstract: Generalizing a classical idea of Biermann, we study a way of constructing a unisolvent array for Lagrange interpolation in Cn+m out of two suitably ordered unisolvent arrays respectively in Cn and Cm For this new array, important objects of Lagrange interpolation theory (fundamental Lagrange polynomials, Newton polynomials, divided difference operator, vandermondian, etc) are computed

Analysis of a cell boundary element method

Abstract: The CBEM (cell boundary element method) was proposed as a numerical method for second-order elliptic problems by the first author in the earlier paper [10]. In this paper we prove a quasi-optimal order of convergence of the method, O(h1−ɛ) for ɛ>0 in H1-norm for the triangular mesh; also a stability result is obtained. We provide numerical examples and it is observed that the method conserves flux exactly when a certain condition on meshes is satisfied.

Positive definite dot product kernels in learning theory

Abstract: In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=x⋅y are used. In many models of learning theory, polynomial kernels K(x,y)=∑ =0 a l (x⋅y) l generating polynomials of degree N, and dot product kernels K(x,y)=∑ =0 +∞ a l (x⋅y) l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f :R→R such that the matrix [f(x i ⋅x j )] ,=1 is positive semi-definite for any x1,x2,. . .,x m ∈R n , n≥2.

Generalized Fourier transform on an arbitrary triangular domain

Abstract: In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions The orthogonality and completeness of the systems are then proved Some essential convergence properties of the generalized Fourier series are discussed Error estimates are obtained in Sobolev norms Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated

A matrix decomposition MFS algorithm for axisymmetric biharmonic problems

Abstract: We consider the approximate solution of axisymmetric biharmonic problems using a boundary-type meshless method, the Method of Fundamental Solutions (MFS) with fixed singularities and boundary collocation. For such problems, the coefficient matrix of the linear system defining the approximate solution has a block circulant structure. This structure is exploited to formulate a matrix decomposition method employing fast Fourier transforms for the efficient solution of the system. The results of several numerical examples are presented.

A simple method for smoothing functions and compressing hermite data

Abstract: Let τ=(a=x0

Periodic wavelet frames

Abstract: In this paper, starting from any two functions satisfying some simple conditions, using a periodization method, we construct a dual pair of periodic wavelet frames and show their optimal bounds. The obtained periodic wavelet frames possess trigonometric polynomial expressions. Finally, we present two examples to explain our theory.

Shape-preserving interpolation by splines using vector subdivision

Abstract: We give a local convexity preserving interpolation scheme using parametricC2 cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.

Best time localized trigonometric polynomials and wavelets

Abstract: In spaces of trigonometric polynomials, the minimum of the angular variance is determined, which is a time localization measure forL 2∏ 2 . Wavelets and wavelet packets are constructed with the resulting polynomials.

Flexible piecewise approximations based on partition of unity

Abstract: In this paper, we study a flexible piecewise approximation technique based on the use of the idea of the partition of unity. The approximations are piecewisely defined, globally smooth up to any order, enjoy polynomial reproducing conditions, and satisfy nodal interpolation conditions for function values and derivatives of any order. We present various properties of the approximations, that are desirable properties for optimal order convergence in solving boundary value problems.

Solution of elementary equations in the Minkowski geometric algebra of complex sets

Abstract: The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks $\mathcal{A}$ and ℬ with radii a and b, a solution of the linear equation $\mathcal{A}\otimes \mathcal{X}=\mathcal{B}$ in an unknown set $\mathcal{X}$ exists if and only if a≤b. When it exists, the solution $\mathcal{X}$ is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limacon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a

On the finite sum representations of the Lauricella functions F D

Abstract: By using divided differences, we derive two different ways of representing the Lauricella function of n variables F D (n) (a,b 1,b 2, ,b n;c;x 1,x 2, ,x n) as a finite sum, for b 1,b 2, ,b n positive integers, and a,c both positive integers or both positive rational numbers with c−a a positive integer

Maximal approximation order for a box-spline semi-cardinal interpolation scheme on the three-direction mesh

Abstract: Let M be the centred 3-direction box-spline whose direction matrix has every multiplicity 2. A new scheme is proposed for interpolation at the vertices of a semi-plane lattice from a subspace of the cardinal box-spline space generated by M. The elements of this ‘semi-cardinal’ box-spline subspace satisfy certain boundary conditions extending the ‘not-a-knot’ end-conditions of univariate cubic spline interpolation. It is proved that the new semi-cardinal interpolation scheme attains the maximal approximation order 4.

Extrapolation of the Hood–Taylor elements for the Stokes problem

Abstract: An asymptotic error expansion concerning of Hood–Taylor elements for the Stokes problem is established on uniform rectangular meshes. The extrapolation based on such an expansion increases the rate of convergence by one order.