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We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem constrained by a convection-dominated problem. We prove global optimal convergence rates using an inf-sup condition, with the diffusion parameter $\varepsilon$ and regularization parameter $\beta$ explicitly tracked. We then propose a multilevel preconditioner based on downwind ordering to solve the discretized system. The preconditioner only requires two approximate solves of single convection-dominated equations using multigrid methods. Moreover, for the strongly convection-dominated case, only two sweeps of block Gauss-Seidel iterations are needed. We also derive a simple bound indicating the role played by the multigrid preconditioner. Numerical results are shown to support our findings.

Estimating the sharing of genetic effects across different conditions is important to many statistical analyses of genomic data. The patterns of sharing arising from these data are often highly heterogeneous. To flexibly model these heterogeneous sharing patterns, Urbut et al. (2019) proposed the multivariate adaptive shrinkage (MASH) method to jointly analyze genetic effects across multiple conditions. However, multivariate analyses using MASH (as well as other multivariate analyses) require good estimates of the sharing patterns, and estimating these patterns efficiently and accurately remains challenging. Here we describe new empirical Bayes methods that provide improvements in speed and accuracy over existing methods. The two key ideas are: (1) adaptive regularization to improve accuracy in settings with many conditions; (2) improving the speed of the model fitting algorithms by exploiting analytical results on covariance estimation. In simulations, we show that the new methods provide better model fits, better out-of-sample performance, and improved power and accuracy in detecting the true underlying signals. In an analysis of eQTLs in 49 human tissues, our new analysis pipeline achieves better model fits and better out-of-sample performance than the existing MASH analysis pipeline. We have implemented the new methods, which we call ``Ultimate Deconvolution'', in an R package, udr, available on GitHub.

We consider the problem of estimating the error when solving a system of differential algebraic equations. Richardson extrapolation is a classical technique that can be used to judge when computational errors are irrelevant and estimate the discretization error. We have simulated molecular dynamics with constraints using the GROMACS library and found that the output is not always amenable to Richardson extrapolation. We derive and illustrate Richardson extrapolation using a variety of numerical experiments. We identify two necessary conditions that are not always satisfied by the GROMACS library.

We study the problem of parametric estimation for continuously observed stochastic differential equation driven by fractional Brownian motion. Under some assumptions on drift and diffusion coefficients, we construct maximum likelihood estimator and establish its the asymptotic normality and moment convergence of the drift parameter when a small dispersion coefficient vanishes.

We demonstrate the feasibility of a scheme to obtain approximate weak solutions to the (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.

Reliably measuring the collinearity of bivariate data is crucial in statistics, particularly for time-series analysis or ongoing studies in which incoming observations can significantly impact current collinearity estimates. Leveraging identities from Welford's online algorithm for sample variance, we develop a rigorous theoretical framework for analyzing the maximal change to the Pearson correlation coefficient and its p-value that can be induced by additional data. Further, we show that the resulting optimization problems yield elegant closed-form solutions that can be accurately computed by linear- and constant-time algorithms. Our work not only creates new theoretical avenues for robust correlation measures, but also has broad practical implications for disciplines that span econometrics, operations research, clinical trials, climatology, differential privacy, and bioinformatics. Software implementations of our algorithms in Cython-wrapped C are made available at //github.com/marc-harary/sensitivity for reproducibility, practical deployment, and future theoretical development.

We consider a convex constrained Gaussian sequence model and characterize necessary and sufficient conditions for the least squares estimator (LSE) to be optimal in a minimax sense. For a closed convex set $K\subset \mathbb{R}^n$ we observe $Y=\mu+\xi$ for $\xi\sim N(0,\sigma^2\mathbb{I}_n)$ and $\mu\in K$ and aim to estimate $\mu$. We characterize the worst case risk of the LSE in multiple ways by analyzing the behavior of the local Gaussian width on $K$. We demonstrate that optimality is equivalent to a Lipschitz property of the local Gaussian width mapping. We also provide theoretical algorithms that search for the worst case risk. We then provide examples showing optimality or suboptimality of the LSE on various sets, including $\ell_p$ balls for $p\in[1,2]$, pyramids, solids of revolution, and multivariate isotonic regression, among others.

We set up, at the abstract Hilbert space setting, the general question on when an inverse linear problem induced by an operator of Friedrichs type admits solutions belonging to (the closure of) the Krylov subspace associated to such operator. Such Krylov solvability of abstract Friedrichs systems allows to predict when, for concrete differential inverse problems, truncation algorithms can or cannot reproduce the exact solutions in terms of approximants from the Krylov subspace.

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme, obtain a global reliability bound, and investigate local efficiency estimates.

We consider the application of the generalized Convolution Quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments.