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Register a new userApproximation orders for interpolation by surface splines to rough functions
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In this paper we consider the approximation of functions by radial basis function interpolants. There is a plethora of results about the asymptotic behaviour of the error between appropriately smooth functions and their interpolants, as the interpolation points fill out a bounded domain in R^d. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function - the native space. In many cases, the native space contains functions possessing a certain amount of smoothness. We address the question of what can be said about these error estimates when the function being interpolated fails to have the required smoothness. These are the rough functions of the title. We limit our discussion to surface splines, as an exemplar of a wider class of radial basis functions, because we feel our techniques are most easily seen and understood in this setting.
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In this paper, we demonstrate the construction of generalized Rough Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on edge or derivative measurements. We prove the (quasi)-optimal localization and approximation properties of the obtained bases, and justify the theoretical results with numerical experiments.
We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we construct a new type of stochastic modified equation, which is a perturbation of the Wong--Zakai approximation of the rough differential equation. For a symplectic method applied to a rough Hamiltonian system, the associated stochastic modified equation is proved to have a Hamiltonian formulation. Second, the pathwise convergence order of the truncated modified equation to the numerical method is obtained by techniques in the rough path theory. Third, if increments of noises are simulated by truncated random variables, we show that the one-step error can be made exponentially small with respect to the time step size. Numerical experiments verify our theoretical results.
The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function -- the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough.
We propose an optimal approximation formula for analytic functions that are defined on a complex region containing the real interval $(-1,1)$ and possibly have algebraic singularities at the endpoints of the interval. As a space of such functions,we consider a Hardy space with the weight given by $w_{mu}(z) = (1-z^{2})^{mu/2}$ for $mu > 0$, and formulate the optimality of an approximation formula for the functions in the space. Then, we propose an optimal approximation formula for the space for any $mu > 0$ as opposed to existing results with the restriction $0 < mu < mu_{ast}$ for a certain constant $mu_{ast}$. We also provide the results of numerical experiments to show the performance of the proposed formula.
A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis.
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