Numerical Issues When Using Wavelets

International audienceWavelets and related multiscale representations pervade all areas of signal processing. The recent inclusion of wavelet algorithms in JPEG 2000 – the new still-picture compression standard– testifies to this lasting and significant impact. The reason of the success of the wavelets is due to the fact that wavelet basis represents well a large class of signals, and therefore allows us to detect roughly isotropic elements occurring at all spatial scales and locations. As the noise in the physical sciences is often not Gaussian, the modeling, in the wavelet space, of many kind of noise (Poisson noise, combination of Gaussian and Poisson noise, long-memory 1/f noise, non-stationary noise, ...) has also been a key step for the use of wavelets in scientific, medical, or industrial applications [1]. Extensive wavelet packages exist now, commercial (see for example [2]) or non commercial (see for example [3, 4]), which allows any researcher, doctor, or engineer to analyze his data using wavelets

Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations

We present and analyze a novel wavelet-Fourier technique for the numerical treatment of multidimensional advection-diffusion-reaction equations based on the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin technique with the compressed sensing approach, the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the PDE. In this paper, we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multi-dimensional setting. Our theoretical analysis relies on new estimates for the local a-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme is shown by numerical illustrations in the one-, two-, and three-dimensional case

Intertwining wavelets or Multiresolution analysis on graphs through random forests

We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov intertwining relation to provide a subgraph structure and a filter bank leading to a wavelet basis of the set of functions. Our construction involves two parameters q and q'. The first one controls the mean number of kept vertices in the downsampling, while the second one is a tuning parameter between space localization and frequency localization. We provide an explicit reconstruction formula, bounds on the reconstruction operator norm and on the error in the intertwining relation, and a Jackson-like inequality. These bounds lead to recommend a way to choose the parameters q and q'. We illustrate the method by numerical experiments.Comment: 39 pages, 12 figure

Review of modern numerical methods for a simple vanilla option pricing problem

Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better

Adaptive Wavelet Collocation Method for Simulation of Time Dependent Maxwell's Equations

This paper investigates an adaptive wavelet collocation time domain method for the numerical solution of Maxwell's equations. In this method a computational grid is dynamically adapted at each time step by using the wavelet decomposition of the field at that time instant. In the regions where the fields are highly localized, the method assigns more grid points; and in the regions where the fields are sparse, there will be less grid points. On the adapted grid, update schemes with high spatial order and explicit time stepping are formulated. The method has high compression rate, which substantially reduces the computational cost allowing efficient use of computational resources. This adaptive wavelet collocation method is especially suitable for simulation of guided-wave optical devices