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Iterative refinement (IR) is a popular scheme for solving a linear system of equations based on gradually improving the accuracy of an initial approximation. Originally developed to improve upon the accuracy of Gaussian elimination, interest in IR has been revived because of its suitability for execution on fast low-precision hardware such as analog devices and graphics processing units. IR generally converges when the error associated with the solution method is small, but is known to diverge when this error is large. We propose and analyze a novel enhancement to the IR algorithm by adding a line search optimization step that guarantees the algorithm will not diverge. Numerical experiments verify our theoretical results and illustrate the effectiveness of our proposed scheme.

Invariant finite-difference schemes for the one-dimensional shallow water equations in the presence of a magnetic field for various bottom topographies are constructed. Based on the results of the group classification recently carried out by the authors, finite-difference analogues of the conservation laws of the original differential model are obtained. Some typical problems are considered numerically, for which a comparison is made between the cases of a magnetic field presence and when it is absent (the standard shallow water model). The invariance of difference schemes in Lagrangian coordinates and the energy preservation on the obtained numerical solutions are also discussed.

We revisit the Pseudo-Bayesian approach to the problem of estimating density matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has been shown to offer a powerful paradign for quantum tomography with attractive theoretical and empirical results. However, the computation of (Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional distribution, pose significant challenges that hampers their usages in practical settings. To overcome this problem, we present an efficient adaptive MCMC sampling method for the Pseudo-Bayesian estimator. We show in simulations that our approach is substantially faster than the previous implementation by at least two orders of magnitude which is significant for practical quantum tomography.

The solutions of scalar ordinary differential equations become more complex as their coefficients increase in magnitude. As a consequence, when a standard solver is applied to such an equation, its running time grows with the magnitudes of the equation's coefficients. It is well known, however, that scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions whose cost to represent via standard techniques is largely independent of the magnitude of the equation's coefficients. This observation is the basis of most methods for the asymptotic approximation of the solutions of ordinary differential equations, including the WKB method. Here, we introduce two numerical algorithms for constructing phase functions for scalar ordinary differential equations inspired by the classical Levin method for the calculation of oscillatory integrals. In the case of a large class of scalar ordinary differential equations with slowly-varying coefficients, their running times are independent of the magnitude of the equation's coefficients. The results of extensive numerical experiments demonstrating the properties of our algorithms are presented.

The recently introduced Genetic Column Generation (GenCol) algorithm has been numerically observed to efficiently and accurately compute high-dimensional optimal transport plans for general multi-marginal problems, but theoretical results on the algorithm have hitherto been lacking. The algorithm solves the OT linear program on a dynamically updated low-dimensional submanifold consisting of sparse plans. The submanifold dimension exceeds the sparse support of optimal plans only by a fixed factor $\beta$. Here we prove that for $\beta \geq 2$ and in the two-marginal case, GenCol always converges to an exact solution, for arbitrary costs and marginals. The proof relies on the concept of c-cyclical monotonicity. As an offshoot, GenCol rigorously reduces the data complexity of numerically solving two-marginal OT problems from $O(\ell^2)$ to $O(\ell)$ without any loss in accuracy, where $\ell$ is the number of discretization points for a single marginal. At the end of the paper we also present some insights into the convergence behavior in the multi-marginal case.

We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.

This paper presents a numerical method for the simulation of elastic solid materials coupled to fluid inclusions. The application is motivated by the modeling of vascularized tissues and by problems in medical imaging which target the estimation of effective (i.e., macroscale) material properties, taking into account the influence of microscale dynamics, such as fluid flow in the microvasculature. The method is based on the recently proposed Reduced Lagrange Multipliers framework. In particular, the interface between solid and fluid domains is not resolved within the computational mesh for the elastic material but discretized independently, imposing the coupling condition via non-matching Lagrange multipliers. Exploiting the multiscale properties of the problem, the resulting Lagrange multipliers space is reduced to a lower-dimensional characteristic set. We present the details of the stability analysis of the resulting method considering a non-standard boundary condition that enforces a local deformation on the solid-fluid boundary. The method is validated with several numerical examples.

The spectral clustering algorithm is often used as a binary clustering method for unclassified data by applying the principal component analysis. To study theoretical properties of the algorithm, the assumption of homoscedasticity is often supposed in existing studies. However, this assumption is restrictive and often unrealistic in practice. Therefore, in this paper, we consider the allometric extension model, that is, the directions of the first eigenvectors of two covariance matrices and the direction of the difference of two mean vectors coincide, and we provide a non-asymptotic bound of the error probability of the spectral clustering algorithm for the allometric extension model. As a byproduct of the result, we obtain the consistency of the clustering method in high-dimensional settings.

The efficacy of numerical methods like integral estimates via Gaussian quadrature formulas depends on the localization of the zeros of the associated family of orthogonal polynomials. In this regard, following the renewed interest in quadrature formulas on the unit circle, and $R_{II}$-type polynomials, which include the complementary Romanovski-Routh polynomials, in this work we present a collection of properties of their zeros. Our results include extreme bounds, convexity, and density, alongside the connection of such polynomials to classical orthogonal polynomials via asymptotic formulas.

We develop a novel method of virtual sources to formulate boundary integral equations for exterior wave propagation problems. However, by contrast to classical boundary integral formulations, we displace the singularity of the Green's function by a small distance $h>0$. As a result, the discretization can be performed on the actual physical boundary with continuous kernels so that any naive quadrature scheme can be used to approximate integral operators. Using on-surface radiation conditions, we combine single- and double-layer potential representations of the solution to arrive at a well-conditioned system upon discretization. The virtual displacement parameter $h$ controls the conditioning of the discrete system. We provide mathematical guidance to choose $h$, in terms of the wavelength and mesh refinements, in order to strike a balance between accuracy and stability. Proof-of-concept implementations are presented, including piecewise linear and isogeometric element formulations in two- and three-dimensional settings. We observe exceptionally well-behaved spectra, and solve the corresponding systems using matrix-free GMRES iterations. The results are compared to analytical solutions for canonical problems. We conclude that the proposed method leads to accurate solutions and good stability for a wide range of wavelengths and mesh refinements.