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We consider the Cauchy problem for the Helmholtz equation with a domain in R^d, d>2 with N cylindrical outlets to infinity with bounded inclusions in R^{d-1}. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya proposed an alternating iterative method for solving Cauchy problems associated with elliptic,self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Mpinganzima et al. for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in R^2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters, the Robin-Dirichlet alternating iterative procedure is convergent.

We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.

We study approaches for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs).In this context, the empirical measure is contained within a natural convex set and can be approximated using convex optimization methods. Such an approximation gives under certain conditions rise to a coreset of data points. A key quantity that controls how large such a coreset has to be is the size of the largest ball around the empirical measure that is contained within the empirical convex set. The bulk of our work is concerned with deriving high probability lower bounds on the size of such a ball under various conditions. We complement this derivation of the lower bound by developing techniques that allow us to apply the compression approach to concrete inference problems such as kernel ridge regression. We conclude with a construction of an infinite dimensional RKHS for which the compression is poor, highlighting some of the difficulties one faces when trying to move to infinite dimensional RKHSs.

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.

We provide a decision theoretic analysis of bandit experiments. The setting corresponds to a dynamic programming problem, but solving this directly is typically infeasible. Working within the framework of diffusion asymptotics, we define suitable notions of asymptotic Bayes and minimax risk for bandit experiments. For normally distributed rewards, the minimal Bayes risk can be characterized as the solution to a nonlinear second-order partial differential equation (PDE). Using a limit of experiments approach, we show that this PDE characterization also holds asymptotically under both parametric and non-parametric distribution of the rewards. The approach further describes the state variables it is asymptotically sufficient to restrict attention to, and therefore suggests a practical strategy for dimension reduction. The upshot is that we can approximate the dynamic programming problem defining the bandit experiment with a PDE which can be efficiently solved using sparse matrix routines. We derive the optimal Bayes and minimax policies from the numerical solutions to these equations. The proposed policies substantially dominate existing methods such as Thompson sampling. The framework also allows for substantial generalizations to the bandit problem such as time discounting and pure exploration motives.

We introduce a novel methodology for particle filtering in dynamical systems where the evolution of the signal of interest is described by a SDE and observations are collected instantaneously at prescribed time instants. The new approach includes the discretisation of the SDE and the design of efficient particle filters for the resulting discrete-time state-space model. The discretisation scheme converges with weak order 1 and it is devised to create a sequential dependence structure along the coordinates of the discrete-time state vector. We introduce a class of space-sequential particle filters that exploits this structure to improve performance when the system dimension is large. This is numerically illustrated by a set of computer simulations for a stochastic Lorenz 96 system with additive noise. The new space-sequential particle filters attain approximately constant estimation errors as the dimension of the Lorenz 96 system is increased, with a computational cost that increases polynomially, rather than exponentially, with the system dimension. Besides the new numerical scheme and particle filters, we provide in this paper a general framework for discrete-time filtering in continuous-time dynamical systems described by a SDE and instantaneous observations. Provided that the SDE is discretised using a weakly-convergent scheme, we prove that the marginal posterior laws of the resulting discrete-time state-space model converge to the posterior marginal posterior laws of the original continuous-time state-space model under a suitably defined metric. This result is general and not restricted to the numerical scheme or particle filters specifically studied in this manuscript.

We study a class of enriched unfitted finite element or generalized finite element methods (GFEM) to solve a larger class of interface problems, that is, 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. The major challenge of GFEM development is to construct enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. The linear stable generalized finite element method (SGFEM) was recently developed using one enrichment function. We generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. So is the optimal order convergence at all nodes. To prove the efficiency of the SGFEM, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of $\delta$-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases $\delta$-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.

The Monge-Amp\`ere equation is a fully nonlinear partial differential equation (PDE) of fundamental importance in analysis, geometry and in the applied sciences. In this paper we solve the Dirichlet problem associated with the Monge-Amp\`ere equation using neural networks and we show that an ansatz using deep input convex neural networks can be used to find the unique convex solution. As part of our analysis we study the effect of singularities, discontinuities and noise in the source function, we consider nontrivial domains, and we investigate how the method performs in higher dimensions. We also compare this method to an alternative approach in which standard feed-forward networks are used together with a loss function which penalizes lack of convexity.