The collocation problem for noisy data with a piecewise constant noise variance is considered. An equivalence to the WIENERKOLMOGOROV equations in stationary collocation theory is constructed. A numerical solution based on HAAR wavelets is given.

We consider the problem of numerical inversion of Fredholm integral equations of the first kind via piecewise interpolation. One of the most important aspects of this technique is the choice of grid and collocation points. Theoretical results are developed which identify an optimal strategy for the distribution of collocation points for piecewise constant interpolation. The method, as outlined,...

This paper constructs adaptive sparse grid collocation method onto arbitrary order piecewise polynomial space. The is a popular technique for high dimensional problems, and the associated has been well studied in literature. contribution of this work introduction systematic framework high-order space that allowed to be discontinuous. We consider both Lagrange Hermite interpolation methods on ne...

In this paper, we present rational approximations based on Fourier series representation. For periodic piecewise analytic functions, the well-known Gibbs phenomenon hampers the convergence of the standard Fourier method. Here, for a given set of the Fourier coefficients from a periodic piecewise analytic function, we define Fourier–Padé–Galerkin and Fourier–Padé collocation methods by expressin...

Consider a model eigenvalue problem with a piecewise constant coefficient. We split the domain at the discontinuity of the coefficient function and define the multidomain variational formulation for the eigenproblem. The discrete multidomain variational formulations are defined for Legendre–Galerkin and Legendre-collocation methods. The spectral rate of convergence of the approximate eigensolut...

Gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity. The truncated spectral series has large oscillations near the jump, and the overshoot does not decay as the number of terms in the truncated series increases. There is therefore no convergence in the maximum norm, and convergence in smooth regions awa...

| Matrix decomposition algorithms employing fast Fourier transforms were developed recently by the authors to solve the systems of linear algebraic equations that arise when piecewise Hermite bicubic orthogonal spline collocation (OSC) is applied to certain separable elliptic boundary value problems on a rectangle. In this paper, these algorithms are interpreted as Fourier methods in analogy wi...

This paper reviews some developments in error estimation and mesh selection for collocation methods for ordinary differential equations. The basic idea of collocation has great generality and simplicity. Given some operator equation we look for an approximate solution in the form of a linear combination of some fixed basis functions. The coefficients in the linear combination are found by subst...

This article gives properties of the planar radiosity equation and methods for its numerical solution. Regularity properties of the radiosity solution are examined, including both the effects of corners and the effects of the visibility function. These are taken into account in the design of collocation methods with piecewise polynomial approximating functions. Numerical examples conclude the p...

This paper considers the numerical solution of boundary integral equations of the second kind for Laplace s equation u on connected regions D in R with boundary S The boundary S is allowed to be smooth or piecewise smooth and we let f K j K Ng be a triangulation of S The numerical method is collocation with approximations which are piecewise quadratic in the parametrization variables leading to...