Direct Error Bounds for Symmetric Rbf Collocation
نویسنده
چکیده
The standard error bounds for interpolation by kernels or radial basis functions are generalized to symmetric PDE collocation problems. This involves generalized Power Functions, and these can be explicitly calculated. Final error bounds then are a product of Power Function values and the norm of the solution in the native space of the kernel. Since the latter can also be estimated well, the whole procedure yields error bounds of the symmetric collocation solution on all points of the domain. Some numerical examples are given for illustration, including a greedy method for choosing good collocation points. On the theoretical side, a short proof of optimal convergence rates for problems with smooth solutions is provided.
منابع مشابه
New RBF collocation methods and kernel RBF with applications
Abstract. A few novel radial basis function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity principle, the boundary particle method is introduced for general inhomogeneous problems without using inner nodes. For domain-type ...
متن کاملRelationship between boundary integral equation and radial basis function
This paper aims to survey our recent work relating to the radial basis function (RBF) from some new views of points. In the first part, we established the RBF on numerical integration analysis based on an intrinsic relationship between the Green's boundary integral representation and RBF. It is found that the kernel function of integral equation is important to create efficient RBF. The fundame...
متن کاملRBF-Chebychev direct method for solving variational problems
This paper establishes a direct method for solving variational problems via a set of Radial basis functions (RBFs) with Gauss-Chebyshev collocation centers. The method consist of reducing a variational problem into a mathematical programming problem. The authors use some optimization techniques to solve the reduced problem. Accuracy and stability of the multiquadric, Gaussian and inverse multiq...
متن کاملA Symmetric Collocation Method with Fast Evaluation
Symmetric collocation, which can be used to numerically solve linear partial differential equations, is a natural generalization of the well-established scattered data interpolation method known as radial basis function (rbf) interpolation. As with rbf interpolation, a major shortcoming of symmetric collocation is the high cost, in terms of floating point operations, of evaluating the obtained ...
متن کاملNew Insights in Boundary-only and Domain-type RBF Methods
This paper has made some significant advances in the boundary-only and domain-type RBF techniques. The proposed boundary knot method (BKM) is different from the standard boundary element method in a number of important aspects. Namely, it is truly meshless, exponential convergence, integration-free (of course, no singular integration), boundary-only for general problems, and leads to symmetric ...
متن کامل