Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by ?(x;α,h)≡exp(-[α²/h²]x²). The only significant numerical parameter is α, the inverse width of the RBF functions relative to h. In the limit α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(π²/[4α²]). However, we also show that the approximation to the constant f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter α are analyzed. For α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10⁴) and the error saturation is smaller than machine epsilon, so this α is the center of a “safe operating range” for Gaussian RBFs.     

Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes

The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N ^?1 = o(ε^v), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.     

Error saturation in Gaussian radial basis functions on a finite interval

Radial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ?(x;α,h)=exp(−α²(x/h)²): in the limit h→0 for fixed α, the error does not converge to zero, but rather to E_S(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases E_S(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α²)‖f‖_∞ — huge compared to the O(2π/α²)exp(−π²/[4α²]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α?1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π²/[4α²])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then .     

The use of solutions on embedded grids for the approximation of singularly perturbed parabolic convection-diffusion equations on adapted grids

The Dirichlet problem on a closed interval for a parabolic convection-diffusion equation is considered. The higher order derivative is multiplied by a parameter ? taking arbitrary values in the semi-open interval (0, 1]. For the boundary value problem, a finite difference scheme on a posteriori adapted grids is constructed. The classical approximations of the equation on uniform grids in the main domain are used; in some subdomains, these grids are subjected to refinement to improve the grid solution. The subdomains in which the grid should be refined are determined using the difference of the grid solutions of intermediate problems solved on embedded grids. Special schemes on a posteriori piecewise uniform grids are constructed that make it possible to obtain approximate solutions that converge almost ?-uniformly, i.e., with an error that weakly depends on the parameter ?: |u(x, t) ? z(x, t)| ≤ M[N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀ + ?^?1 N ₁ ^?K ln ^K?1 N ₁], (x, t) ε ? _h , where N ₁ + 1 and N ₀ + 1 are the numbers of grid points in x and t, respectively; K is the number of refinement iterations (with respect to x) in the adapted grid; and M = M(K). Outside the σ-neighborhood of the outflow part of the boundary (in a neighborhood of the boundary layer), the scheme converges ?-uniformly at a rate O(N ₁ ^?1 ln² N ₁ + N ₀ ^?1 lnN ₀), where σ ≤ MN ₁ ^{?K + 1} ln ^K?1 N ₁ for K ≥ 2.     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

Stability of spline approximation and the restoration of grid functions

For a functionf(x) ∈ H _ω ^r , defined on a uniform grid approximately, we propose a stable method for approximately restoring the function with the aid of polynomial splines. We derive uniform estimates for the deviations of the spline and its derivatives from the function and its derivatives.     

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ? (this is the coefficient of the higher order derivatives of the equation, ? ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S ₁ ^L . In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ?-uniformly at a rate of O(N ^?1lnN + N ₀), where N and N ₀ define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N ^?1 + N ₀ ^?1 ? ?. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S ₁ ^L with respect to x and t, the convergence under the condition N ^?1 + N ₀ ^?1 ≤ ?^1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ?-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ?-uniformly at a rate of O(N ^?1lnN + N ₀).     

Uniform boundary Harnack principle and generalized triangle property

Let D be a bounded open subset in R^d, d?2, and let G denote the Green function for D with respect to (-Δ)^α/2, 0<α?2, α<d. If α<2, assume that D satisfies the interior corkscrew condition; if α=2, i.e., if G is the classical Green function on D, assume—more restrictively—that D is a uniform domain. Let g=G(·,y₀)∧1 for some y₀∈D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that when d(z,x)?d(z,y). An intermediate step is the approximation G(x,y)≈|x-y|^α-dg(x)g(y)/g(A)², where A is an arbitrary point in a certain set B(x,y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.     

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ?-uniformly well conditioned or ?-uniformly stable to perturbations of the data of the grid problem (here, ? is a perturbation parameter, ? ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ?-uniformly in the maximum norm at an O(N ^?1lnN + N ₀ ^?1 ) rate, where N + 1 and N ₀ + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ?-uniformly well conditioned and ?-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ^?2lnδ^?1 + δ ₀ ^?1 ); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N ^?1lnN and δ₀ = N ₀ ^?1 are the accuracies of the discrete solution in x and t, respectively.     

Parameter-uniform error estimate for the implicit four-point scheme for a singularly perturbed heat equation with corner singularities

We study an initial-boundary value problem for a singularly perturbed one-dimensional heat equation on an interval. At the corner points, the input data are subjected to continuity conditions only, which violates the smoothness of the derivatives of the solution in neighborhoods of these points, starting from the derivatives occurring in the equation. To approximate the problem, we use the implicit four-point difference scheme on a Shishkin grid uniform with respect to time and piecewise uniform with respect to the space variable. We prove that the grid solution error is O(τ +N ^?2 ln² N) ln(j +1) uniformly with respect to the parameter, where τ is the grid increment with respect to the time variable, j is the index of the time layer, and N is the number of nodes in the piecewise uniform space grid.     

Solution of the Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square on a Bakhvalov grid

The Dirichlet problem for a singularly perturbed reaction-diffusion equation in a square is solved with the help of the classic five-point difference scheme and a grid that is the tensor product of 1D Bakhvalov grids. Without imposing additional matching conditions in the corners of the domain, it is shown that the grid solution to the problem has the accuracy O(N ⁻²) in the norm L _∞ ^h , where N is the number of grid nodes along each direction. The accuracy is uniform with respect to a small parameter. A simulation confirms the theoretical prediction.     

Strong stability of a scheme on locally uniform meshes for a singularly perturbed ordinary differential convection-diffusion equation

The Dirichlet problem for a singularly perturbed ordinary differential convection-diffusion equation with a small parameter ? (? ?? (0, 1]) multiplying the higher order derivative is considered. For the problem, a difference scheme on locally uniform meshes is constructed that converges in the maximum norm conditionally, i.e., depending on the relation between the parameter ? and the value N defining the number of nodes in the mesh used; in particular, the scheme converges almost ?-uniformly (i.e., its accuracy depends weakly on ?). The stability of the scheme with respect to perturbations in the data and its conditioning are analyzed. The scheme is constructed using classical monotone approximations of the boundary value problem on a priori adapted grids, which are uniform on subdomains where the solution is improved. The boundaries of these subdomains are determined by a majorant of the singular component of the discrete solution. On locally uniform meshes, the difference scheme converges at a rate of O(min[?^?1 N ^?K lnN, 1] + N ^?1lnN), where K is a prescribed number of iterations for refining the discrete solution. The scheme converges almost ?-uniformly at a rate of O(N ^?1lnN) if N ^?1 ?? ?^??, where ?? (the defect of ?-uniform convergence) determines the required number K of iterations (K = K(??) ?? ??^?1) and can be chosen arbitrarily small from the half-open interval (0, 1]. The condition number of the difference scheme satisfies the bound ?? _P = O(?^?1/K ln^1/K ?^?1??^{?(K + 1)/K} ), where ?? is the accuracy of the solution of the scheme in the maximum norm in the absence of perturbations. For sufficiently large K, the scheme is almost ?-uniformly strongly stable.     

Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension

In the present paper we present the tensor-product approximation of a multidimensional convolution transform discretized via a collocation-projection scheme on uniform or composite refined grids. Examples of convolving kernels are provided by the classical Newton, Slater (exponential) and Yukawa potentials, 1/‖x‖, and with x∈R^d. For piecewise constant elements on the uniform grid of size n^d, we prove quadratic convergence O(h²) in the mesh parameter h=1/n, and then justify the Richardson extrapolation method on a sequence of grids that improves the order of approximation up to O(h³). A fast algorithm of complexity O(dR₁R₂nlogn) is described for tensor-product convolution on uniform/composite grids of size n^d, where R₁,R₂ are tensor ranks of convolving functions. We also present the tensor-product convolution scheme in the two-level Tucker canonical format and discuss the consequent rank reduction strategy. Finally, we give numerical illustrations confirming: (a) the approximation theory for convolution schemes of order O(h²) and O(h³); (b) linear-logarithmic scaling of 1D discrete convolution on composite grids; (c) linear-logarithmic scaling in n of our tensor-product convolution method on an n×n×n grid in the range n≤16384.     

Hilbert boundary value problem in the weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>(<Emphasis Type="Italic">ρ</Emphasis>)

The paper studies a Hilbert boundary value problem in L ¹(ρ), where ρ(t) = |1–t|^α and α is a real number. For α > ?1, it is proved that the homogeneous problem has n + κ linearly independent solutions when n + κ ≥ 0, where a(t) is the coefficient of the problem, besides, κ ind a(t) and n = [α] + 1 if α is not an integer, and n = α if α is an integer. Conditions under which the problem is solvable are found for the case when α > ?1 and n+κ < 0. For α ≤ ?1 the number of linearly independent solutions of the homogeneous problem depends on the behavior of the function a(t) at the point t = 1.     

Global singularization and the failure of SCH

We say that κ is μ-hypermeasurable (or μ-strong) for a cardinal μ≥κ⁺ if there is an embedding j:V→M with critical point κ such that H(μ)^V is included in M and j(κ)>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V^∗ where F is realised on all V-regular cardinals and moreover, all F(κ)-hypermeasurable cardinals κ, where F(κ)>κ⁺, with a witnessing embedding j such that either j(F)(κ)=κ⁺ or j(F)(κ)≥F(κ), are turned into singular strong limit cardinals with cofinality ω. This provides some partial information about the possible structure of a continuum function with respect to singular cardinals with countable cofinality.As a corollary, this shows that the continuum function on a singular strong limit cardinal κ of cofinality ω is virtually independent of the behaviour of the continuum function below κ, at least for continuum functions which are simple in that 2^α∈{α⁺,α⁺⁺} for every cardinal α below κ (in this case every κ⁺⁺-hypermeasurable cardinal in the ground model is witnessed by a j with either j(F)(κ)≥F(κ) or j(F)(κ)=κ⁺).     

Hamilton connectedness and the partially square graphs

Let G be a κ-connected graph on n vertices. The partially square graphG^* of G is obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition N_G(x)⊂N_G[u]∪N_G[v]. Clearly G⊆G^*⊆G², where G² is the square of G. In particular G^*=G² if G is quasi-claw-free (and claw-free). In this paper we prove that a κ-connected, (κ?3) graph G is either hamiltonian-connected or the independence number of G^* is at least κ. As a consequence we answer positively two open questions. The first one by Ainouche and Kouider and the second one by Wu et al. As a by-product we prove that a quasi-claw-free (and hence a claw-free) graph satisfying the condition α(G²)<κ is hamiltonian-connected.     

Matrix Bounds on the Spectral Condition Number

The purpose of this paper is to obtain some new lower and upper bounds on κ(A)+κ^-1(A), where κ(A) is the spectral condition number of an n×n nonsingular matrix A, and compare these with previously known bounds.     

The Richardson scheme for the singularly perturbed parabolic reaction-diffusion equation in the case of a discontinuous initial condition

The Dirichlet problem for a singularly perturbed parabolic reaction-diffusion equation with a piecewise continuous initial condition in a rectangular domain is considered. The higher order derivative in the equation is multiplied by a parameter ?², where ? ∈ (0, 1]. When ? is small, a boundary and an interior layer (with the characteristic width ?) appear, respectively, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the reduced equation passing through the discontinuity point of the initial function; for fixed ?, these layers have limited smoothness. Using the method of additive splitting of singularities (induced by the discontinuities of the initial function and its low-order derivatives) and the condensing grid method (piecewise uniform grids that condense in a neighborhood of the boundary layers), a finite difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + n ₀ ^?1 ), where N + 1 and N ₀ + 1 are the numbers of the mesh points in x and t, respectively. Based on the Richardson technique, a scheme that converges ?-uniformly at a rate of O(N ^?3 + N ₀ ^?2 ) is constructed. It is proved that the Richardson technique cannot construct a scheme that converges in ?-uniformly in x with an order greater than three.     

On the enumeration of skew Young tableaux

For fixed n and a fixed partition α of k<n we give an explicit formula for the number N(n;α) of standard skew Young tableaux with n squares and shape λ/α for some λ. From this formula the entire asymptotic expansion of N(n;α) as n→∞ can in principle be computed, generalizing recent work of McKay, Morse, and Wilf. We also give asymptotic formulas for the number f^λ/α of standard skew Young tableaux of shape λ/α for α fixed and λ “large.”     

Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration

Solving large radial basis function (RBF) interpolation problems with non‐customised methods is computationally expensive and the matrices that occur are typically badly conditioned. For example, using the usual direct methods to fit an RBF with N centres requires O(N ²) storage and O(N ³) flops. Thus such an approach is not viable for large problems with N ≥10,000. In this paper we present preconditioning strategies which, in combination with fast matrix–vector multiplication and GMRES iteration, make the solution of large RBF interpolation problems orders of magnitude less expensive in storage and operations. In numerical experiments with thin‐plate spline and multiquadric RBFs the preconditioning typically results in dramatic clustering of eigenvalues and improves the condition numbers of the interpolation problem by several orders of magnitude. As a result of the eigenvalue clustering the number of GMRES iterations required to solve the preconditioned problem is of the order of 10-20. Taken together, the combination of a suitable approximate cardinal function preconditioner, the GMRES iterative method, and existing fast matrix–vector algorithms for RBFs [4,5] reduce the computational cost of solving an RBF interpolation problem to O(N) storage, and O(N \log N) operations. This revised version was published online in June 2006 with corrections to the Cover Date.