We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.
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Factor models have been widely used to summarize the variability of high-dimensional data through a set of factors with much lower dimensionality. Gaussian linear factor models have been particularly popular due to their interpretability and ease of computation. However, in practice, data often violate the multivariate Gaussian assumption. To characterize higher-order dependence and nonlinearity, models that include factors as predictors in flexible multivariate regression are popular, with GP-LVMs using Gaussian process (GP) priors for the regression function and VAEs using deep neural networks. Unfortunately, such approaches lack identifiability and interpretability and tend to produce brittle and non-reproducible results. To address these problems by simplifying the nonparametric factor model while maintaining flexibility, we propose the NIFTY framework, which parsimoniously transforms uniform latent variables using one-dimensional nonlinear mappings and then applies a linear generative model. The induced multivariate distribution falls into a flexible class while maintaining simple computation and interpretation. We prove that this model is identifiable and empirically study NIFTY using simulated data, observing good performance in density estimation and data visualization. We then apply NIFTY to bird song data in an environmental monitoring application.
We introduce a novel algorithm that converges to level-set convex viscosity solutions of high-dimensional Hamilton-Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton-Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. A main contribution of our work is a new monotone scheme for approximating the direction of the gradient, which allows for monotone discretizations of pure partial derivatives in the direction of, and orthogonal to, the gradient. We provide a convergence analysis of the algorithm on both regular Cartesian grids and unstructured point clouds in any dimension and present numerical experiments that demonstrate the effectiveness of the algorithm in approximating solutions of the affine flow in two dimensions and the Tukey depth measure of high-dimensional datasets such as MNIST and FashionMNIST.
We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.
Learning nonparametric systems of Ordinary Differential Equations (ODEs) dot x = f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator and provide experimental comparisons with the state-of-the-art.
Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer frame (RIII CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. In this work, we aim at quantitatively assessing the impact and the range of validity of the RIII CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII CRD in realistic 1D atmospheric models. We focus on the chromospheric Ca i line at 4227 A and on the photospheric Sr i line at 4607 A.
This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schr\"odinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with some simple mesh-independent numerical fluxes we designed. We establish error estimates in the energy norm that require careful selection of a test function for the auxiliary equation involving the time derivative of the displacement variable. A critical part of the convergence analysis is to establish the L2 error bounds for the time derivative of the approximation error in the displacement variable by using the equation that determines its mean value. Using a specially chosen test function, we show that one can create a linear system for the time evolution of the unknowns even when dealing with nonlinear properties in the original problem. Extensive numerical experiments are provided to demonstrate the optimal convergence of the scheme in the L2 norm with our choices of the numerical flux.
We extend our formulation of Merge and Minimalism in terms of Hopf algebras to an algebraic model of a syntactic-semantic interface. We show that methods adopted in the formulation of renormalization (extraction of meaningful physical values) in theoretical physics are relevant to describe the extraction of meaning from syntactic expressions. We show how this formulation relates to computational models of semantics and we answer some recent controversies about implications for generative linguistics of the current functioning of large language models.
We present a space-time ultra-weak discontinuous Galerkin discretization of the linear Schr\"odinger equation with variable potential. The proposed method is well-posed and quasi-optimal in mesh-dependent norms for very general discrete spaces. Optimal $h$-convergence error estimates are derived for the method when test and trial spaces are chosen either as piecewise polynomials, or as a novel quasi-Trefftz polynomial space. The latter allows for a substantial reduction of the number of degrees of freedom and admits piecewise-smooth potentials. Several numerical experiments validate the accuracy and advantages of the proposed method.
Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.
This document describes an algorithm to scale a complex vector by the reciprocal of a complex value. The algorithm computes the reciprocal of the complex value and then scales the vector by the reciprocal. Some scaling may be necessary due to this 2-step strategy, and the proposed algorithm takes scaling into account. This algorithm is supposed to be faster than the naive approach of dividing each entry of the vector by the complex value, without losing much accuracy. It also serves as a single strategy for scaling vectors by the reciprocal of a complex value, which improves the software maintainability.
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