A unified approach for the solution of the Fokker-Planck equation

This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.Comment: 19 page

Neural networks and separation of Cosmic Microwave Background and astrophysical signals in sky maps

The Independent Component Analysis (ICA) algorithm is implemented as a neural network for separating signals of different origin in astrophysical sky maps. Due to its self-organizing capability, it works without prior assumptions on the signals, neither on their frequency scaling, nor on the signal maps themselves; instead, it learns directly from the input data how to separate the physical components, making use of their statistical independence. To test the capabilities of this approach, we apply the ICA algorithm on sky patches, taken from simulations and observations, at the microwave frequencies, that are going to be deeply explored in a few years on the whole sky, by the Microwave Anisotropy Probe (MAP) and by the {\sc Planck} Surveyor Satellite. The maps are at the frequencies of the Low Frequency Instrument (LFI) aboard the {\sc Planck} satellite (30, 44, 70 and 100 GHz), and contain simulated astrophysical radio sources, Cosmic Microwave Background (CMB) radiation, and Galactic diffuse emissions from thermal dust and synchrotron. We show that the ICA algorithm is able to recover each signal, with precision going from 10% for the Galactic components to percent for CMB; radio sources are almost completely recovered down to a flux limit corresponding to $0.7\sigma_{CMB}$, where $\sigma_{CMB}$ is the rms level of CMB fluctuations. The signal recovering possesses equal quality on all the scales larger then the pixel size. In addition, we show that the frequency scalings of the input signals can be partially inferred from the ICA outputs, at the percent precision for the dominant components, radio sources and CMB.Comment: 15 pages; 6 jpg and 1 ps figures. Final version to be published in MNRA

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of on are proportional to where Λ( j ) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f ( x )= x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergencePeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43417/1/10915_2005_Article_9010.pd

The Cauchy singular integral operator and Clifford wavelets

We give an elementary, self-contained real-variable proof of the L 2-boundedness of the Cauchy singular integral operator on a Lipschitz surface. The main new feature is the role played by a system of Clifford algebra valued wavelets adapted to the geometry of the surface