Energy decay and stability of a perfectly matched layer for the wave equation

In [25,26], a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and L^2 stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations. Numerical results validate the theory

On Wave Splitting, Source Separation and Echo Removal with Absorbing Boundary Conditions

Starting from classical absorbing boundary con- ditions (ABC), we propose a method for the sep- aration of time-dependent wave fields given mea- surements of the total wave field. The method is local in space and time, deterministic, and makes no prior assumptions on the frequency spectrum and the location of sources or physical bound- aries. By using increasingly higher order ABC, the method can be made arbitrarily accurate and is, in that sense, exact. Numerical examples il- lustrate the usefulness for source separation and echo removal

Recent results on the singular local limit for nonlocal conservation laws

We provide an informal overview of recent developments concerning the singular local limit of nonlocal conservation laws. In particular, we discuss some counterexamples to convergence and we highlight the role of numerical viscosity in the numerical investigation of the nonlocal-to-local limit. We also state some open questions and describe recent related progress

Siegels’s lemma is sharp for almost all linear systems

The well-known Siegel Lemma gives an upper bound $cU^{m/(n−m)}$ for the size of the smallest non-zero integral solution of a linear system of $m \ge 1$ equations in $n > m$ unknowns whose coefficients are integers of absolute value at most $U \ge 1$; here $c = c(m, n) \ge 1$. In this paper we show that a better upper bound $U^{m/(n−m)}/B$ is relatively rare for large $B \ge 1$; for example there are $\theta = \theta(m,n) > 0$ and $c′ = c′(m,n)$ such that this happens for at most $c′U^{mn}/B^\theta$ out of the roughly $(2U)^{mn}$ possible such systems

An adaptive finite element method for high-frequency scattering problems with variable coefficients

We introduce a new method for the numerical approximation of time-harmonic acoustic scattering problems stemming from material inhomogeneities. The method works for any frequency $\omega$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: \emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; \emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; \emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in \emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $\omega$, as shown in the numerical experiments. We also present a new algorithm for computing the Fourier transform in \emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes

Smooth approximation is not a selection principle for the transport equation with rough vector field

In this paper we analyse the selection problem for weak solutions of the transport equation with rough vector field. We answer in the negative the question whether solutions of the equation with a regularized vector field converge to a unique limit, which would be the selected solution of the limit problem. To this aim, we give a new example of a vector field which admits infinitely many flows. Then we construct a smooth approximating sequence of the vector field for which the corresponding solutions have subsequences converging to different solutions of the limit equation

How to solve inverse scattering problems without knowing the source term: a three-step strategy

The solution of inverse scattering problems always presupposed knowledge of the incident wavefield and require repeated computations of the forward problem, for which knowing the source term is crucial. Here we present a three-step strategy to solve inverse scattering problems when the time signature of the source is unknown. The proposed strategy combines three recent techniques: (i) wave splitting to retrieve the incident and the scattered wavefields, (ii) time-reversed absorbing conditions (TRAC) for redatuming the data inside the computational domain, (iii) adaptive eigenspace inversion (AEI) to solve the inverse problem. Numerical results illustrate step-by-step the feasibility of the proposed strategy

Approximating solution spaces as a product of polygons

Solution spaces are regions of good designs in a potentially high-dimensional design space. Good designs satisfy by definition all requirements that are imposed on them as mathematical constraints. In previous work, the complete solution space was approximated by a hyper-rectangle, i.e., the Cartesian product of permissible intervals for design variables. These intervals serve as independent target regions for distributed and separated design work. For a better approximation, i.e., a larger resulting solution space, this article proposes to compute the Cartesian product of two-dimensional regions, so-called 2d-spaces, that are enclosed by polygons. 2d-spaces serve as target regions for pairs of variables and are independent of other 2d-spaces. A numerical algorithm for non-linear problems is presented that is based on iterative Monte-Carlo sampling

Age dependency in the transmission dynamics of the liver fluke, Opisthorchis viverrini and the effectiveness of interventions

We introduce a population-based model of the transmission dynamics of the liver fluke Opisthorchis viverrini, that allows the mean worm burden in humans to depend on the host age. We parameterise the model using data on intensity of infection in humans and prevalence data for cats, dogs, fish and snails from two island communities in Lao People’s Democratic Republic. We evaluate the steady state solution using a fixed point iteration and estimate the basic reproductive number. We optimise the coverage level of MDA in an adapted model of five age groups to compare varying coverages across age groups. Our results suggest that although adults have the strongest contribution to transmission and campaigns should target adults, if such targeting is operationally infeasible, achieving moderate coverage levels in all age groups can still have a substantial impact on reducing worm burden

Parallel Controllability Methods For the Helmholtz Equation

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures