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We study time integration schemes for $\dot H^1$-solutions to the energy-(sub)critical semilinear wave equation on $\mathbb{R}^3$. We show first-order convergence in $L^2$ for the Lie splitting and convergence order $3/2$ for a corrected…
A family of implicit-in-time mixed finite element schemes is presented for the numerical approximation of the acoustic wave equation. The mixed space discretization is based on the displacement form of the wave equation and the…
We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes…
We test the analytical expressions for the first two eigenvalues of the harmonic oscillator with a Gaussian perturbation proposed recently. Our numerical eigenvalues show that those expressions are valid in an interval of the coupling…
In this paper, we consider an inference problem for the first order autoregressive process driven by a long memory stationary Gaussian process. Suppose that the covariance function of the noise can be expressed as $\abs{k}^{2H-2}$ times a…
We calculate certain estimates for the solution of the characteristic problem of the wave equation reduced to first order, in terms of the free data prescribed on two transverse surfaces, one of which is characteristic. Estimates of such…
We present randomized algorithms for sampling the standard Gaussian distribution restricted to a convex set and for estimating the Gaussian measure of a convex set, in the general membership oracle model. The complexity of integration is…
In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby modified equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the…
This paper derives non-asymptotic error bounds for nonlinear stochastic approximation algorithms in the Wasserstein-$p$ distance. To obtain explicit finite-sample guarantees for the last iterate, we develop a coupling argument that compares…
Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established,…
We advocate a Bayesian approach to optimal quantum frequency estimation - an important issue for future quantum enhanced atomic clock operation. The approach provides a clear insight into the interplay between decoherence and the extent of…
When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number $k$ increases, known as `pollution effect' which is directly related to the phase difference between the exact and numerical…
We study the use of Gaussian process emulators to approximate the parameter-to-observation map or the negative log-likelihood in Bayesian inverse problems. We prove error bounds on the Hellinger distance between the true posterior…
The $C^{1,1}$ estimate of the Dirichlet problem for degenerate $k$-Hessian equations with non-homogenous boundary conditions is an open problem, if the right hand side function $f$ is only assumed to satisfy $f^{1/(k-1)} \in C^{1,1}$. In…
In this paper we explore the influence of the shape parameter in the gaussian function on error estimates and present a set of criteria for its optimal choice.
Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian…
We study the filtering and smoothing problem for continuous-time linear Gaussian systems. While classical approaches such as the Kalman-Bucy filter and the Rauch-Tung-Striebel (RTS) smoother provide recursive formulas for the conditional…
In this paper we investigate the homogenization problem with a non-homogeneous Dirichlet condition. Our aim is to give error estimates with boundary data in $H^{1/2}(\partial\Omega)$. The tools used are those of the unfolding method in…
We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that…
It is often convenient to use Gaussian blur in studying image quality or in data augmentation pipelines for training convoluional neural networks. Because of their convenience, Guassians are sometimes used as first order approximations of…