Related papers: Wavelets and Fast Numerical Algorithms

Microlocal analysis provides deep insight into singularity structures and is often crucial for solving inverse problems, predominately, in imaging sciences. Of particular importance is the analysis of wavefront sets and the correct…

We develop fixed-point algorithms for the approximation of structured matrices with rank penalties. In particular we use these fixed-point algorithms for making approximations by sums of exponentials, or frequency estimation. For the basic…

The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions…

We present a method of using classical wavelet based multiresolution analysis to separate scales in model and observations during data assimilation with the ensemble Kalman filter. In many applications, the underlying physics of a phenomena…

We present a high-order spacetime numerical method for discretizing and solving linear initial-boundary value problems using wavelet-based techniques with user-prescribed error estimates. The spacetime wavelet discretization yields a system…

Despite the remarkable success of deep learning in pattern recognition, deep network models face the problem of training a large number of parameters. In this paper, we propose and evaluate a novel multi-path wavelet neural network…

We consider an application of variational-wavelet approach to nonlinear collective models of beam/plasma physics: Vlasov/Boltzmann-like reduction from general BBGKY hierachy. We obtain fast convergent multiresolution representations for…

The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be…

Many materials, processes, and structures in science and engineering have important features at multiple scales of time and/or space; examples include biological tissues, active matter, oceans, networks, and images. Explicitly extracting,…

Bayesian approaches are one of the primary methodologies to tackle an inverse problem in high dimensions. Such an inverse problem arises in hydrology to infer the permeability field given flow data in a porous media. It is common practice…

Deep learning models extract, before a final classification layer, features or patterns which are key for their unprecedented advantageous performance. However, the process of complex nonlinear feature extraction is not well understood, a…

In this paper, we consider the sparse regularization of manifold-valued data with respect to an interpolatory wavelet/multiscale transform. We propose and study variational models for this task and provide results on their well-posedness.…

The non-stationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasi-stationary spells. Given that standard theoretical and data analysis approaches usually rely on the assumption…

Kernel methods are a highly effective and widely used collection of modern machine learning algorithms. A fundamental limitation of virtually all such methods are computations involving the kernel matrix that naively scale quadratically…

This paper constructs the first quantum algorithm for wavelet packet transforms with a "parabolic scaling" tree structure, sometimes called wave atom transforms. Classically, wave atoms are used to construct sparse representations of…

Efficiently capturing the long-range patterns in sequential data sources salient to a given task -- such as classification and generative modeling -- poses a fundamental challenge. Popular approaches in the space tradeoff between the memory…

A set of semi-analytical techniques based on Fourier analysis is used to solve wave scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are newly developed conformal mapping…

Samplets are data adapted multiresolution analyses of localized discrete signed measures. They can be constructed on scattered data sites in arbitrary dimension such that they exhibit vanishing moments with respect to any prescribed set of…

We obtain a characterization of all wavelets leading to analytic wavelet transforms (WT). The characterization is obtained as a by-product of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude-only…

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived…