Related papers: Understanding the ultraspherical spectral method
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
Ultra-fast and multi-dimensional spectroscopy gives a powerful looking glass into the dynamics of molecular systems. In particular two-dimensional electronic spectroscopy (2DES) provides a probe of coherence and the flow of energy within…
We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of…
Pseudospectral schemes are a class of numerical methods capable of solving smooth problems with high accuracy thanks to their exponential convergence to the true solution. When applied to discontinuous problems, such as fluid shocks and…
Adjusting for an unmeasured confounder is generally an intractable problem, but in the spatial setting it may be possible under certain conditions. In this paper, we derive necessary conditions on the coherence between the treatment…
Accurate camera calibration is a precondition for many computer vision applications. Calibration errors, such as wrong model assumptions or imprecise parameter estimation, can deteriorate a system's overall performance, making the reliable…
In high-dimensional Bayesian statistics, various methods have been developed, including prior distributions that induce parameter sparsity to handle many parameters. Yet, these approaches often overlook the rich spectral structure of the…
While considerable advance has been made to account for statistical uncertainties in astronomical analyses, systematic instrumental uncertainties have been generally ignored. This can be crucial to a proper interpretation of analysis…
We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We…
Quantum-inspired superresolution methods surpass the Rayleigh limit in imaging, or the analogous Fourier limit in spectroscopy. This is achieved by carefully extracting the information carried in the emitted optical field by engineered…
We present a novel spectral method for the Allen-Cahn equation on spheres, eliminating the reliance on conventional quadrature exactness conditions. By replacing these conditions with a restricted isometry relation derived from…
Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The…
Sampling a signal below the Shannon-Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also well-known that recovering spectral information from a signal using a parametric method can be ill-posed…
The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the…
In this work we study a DPG method for an ultra-weak variational formulation of a reaction-diffusion problem. We improve existing a priori convergence results by sharpening an approximation result for the numerical flux. By duality…
In an ultrasonic array system, increasing the aperture size to achieve a high resolution requires more transmit and receive channels, thus making it essential to have an analysis technique that can reconstruct the shape and physical…
In the paper we offer a functional-discrete method for solving the Cauchy problem for the first order ordinary differential equations (ODEs). This method (FD-method) is in some sense similar to the Adomian Decomposition Method. But it is…
Computational models in chemistry rely on a number of approximations. The effect of such approximations on observables derived from them is often unpredictable. Therefore, it is challenging to quantify the uncertainty of a computational…
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: first, we prove that…