Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Abstract

We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2 −nN ) in the H 1 norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection coefficients. Due to the fast convergence rate, very good approximations are found at low levels and with low Coiflet degrees, hence the size of corresponding linear systems is small. Numerical experiments confirm these claims.

Screening wells by multi-scale grids for multi-stage Markov Chain Monte Carlo simulation

“…We run three types of MCMC simulations until we obtain 400 accepted proposals. Shifted errors for accepted proposals by forward simulation, equation (6), are plotted in Figure 3. We see shifted errors converge to zero after obtaining a few accepted proposals and oscillate there afterwards, indicating chain is converged to steady state.…”

“…where a k and b k are known values. These properties make their definite integrals easy to evaluate, and they are highly demanded in the field of data compression and numerical computation, particularly for numerical solution of partial differential equations [6]. Example of data compression can be found in seismic signals which may be acoustics, geological, x-rays or tomographic images, biomedical signals, ultra wide-band wireless communication, etc [5].…”