Related papers: A fast sparse spectral method for nonlinear integr…
We present a generic scheme to construct corrected trapezoidal rules with spectral accuracy for integral operators with weakly singular kernels in arbitrary dimensions. We assume that the kernel factorization of the form,…
A concise overview of the spectral theory of integral-functional operators is provided. In the context of analysis, a technique is described for deriving solutions to equations involving operators in a closed form. A constructive theorem…
In the spectral Petrov-Galerkin methods, the trial and test functions are required to satisfy particular boundary conditions. By a suitable linear combination of orthogonal polynomials, a basis, that is called the modal basis, is obtained.…
We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go…
Recovery of a sparse signal from a nonlinear system arises in many practical applications including compressive sensing, image reconstruction and machine learning. In this paper, a fast block nonlinear Bregman-Kaczmarz method with averaging…
The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive…
This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of the nonlinear Abel integral equations is studied completely. An…
Combination frequencies are not solutions of the perturbed stellar structure equations. In dense power spectra from a light curve of a given multi-periodic pulsating star, they can compromise the mode identification in an asteroseismic…
We establish a general form of explicit, input-dependent, measure-valued warpings for learning nonstationary kernels. While stationary kernels are ubiquitous and simple to use, they struggle to adapt to functions that vary in smoothness…
We look for spectral type differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…
Spectral analysis of operator-functions which are the symbols of the abstract integrodifferential equations of the Gurtin-Pipkin is provided. These equations represent abstract wave equations disturbed by terms involving Volterra operators.…
A nearly optimal explicitly-sparse representation for oscillatory kernels is presented in this work by developing a curvelet based method. Multilevel curvelet-like functions are constructed as the transform of the original nodal basis. Then…
We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the riveting properties of the single-species fast spectral method (Gamba et al. SIAM J. Sci. Comput., 39 pp. B658--B674…
We find all spectral type differential equations satisfied by the symmetric generalized ultraspherical polynomials which are orthogonal on the interval [-1,1] with respect to the classical symmetric weight function for the Jacobi…
Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the…
In this short paper, we aim at developing algorithms for sparse Volterra system identification when the system to be identified has infinite impulse response. Assuming that the impulse response is represented as a sum of exponentials and…
We propose non-stationary spectral kernels for Gaussian process regression. We propose to model the spectral density of a non-stationary kernel function as a mixture of input-dependent Gaussian process frequency density surfaces. We solve…
We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced…
We discuss the application of multistep collocation methods to Volterra integral equations which contain a weakly singular kernel $(t-\tau)^{\alpha-1}$ with $0 <\alpha <1.$ Convergence orders of the methods are determined and their…
In this paper, we investigate the abstract non-scalar Volterra difference equations. We employ the Poisson like transforms to connect the solutions of the abstract non-scalar Volterra integro-differential equations and the abstract…