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We propose the Terminating-Random Experiments (T-Rex) selector, a fast variable selection method for high-dimensional data. The T-Rex selector controls a user-defined target false discovery rate (FDR) while maximizing the number of selected variables. This is achieved by fusing the solutions of multiple early terminated random experiments. The experiments are conducted on a combination of the original predictors and multiple sets of randomly generated dummy predictors. A finite sample proof based on martingale theory for the FDR control property is provided. Numerical simulations confirm that the FDR is controlled at the target level while allowing for a high power. We prove under mild conditions that the dummies can be sampled from any univariate probability distribution with finite expectation and variance. The computational complexity of the proposed method is linear in the number of variables. The T-Rex selector outperforms state-of-the-art methods for FDR control on a simulated genome-wide association study (GWAS), while its sequential computation time is more than two orders of magnitude lower than that of the strongest benchmark methods. The open source R package TRexSelector containing the implementation of the T-Rex selector is available on CRAN.

We study the deviation inequality for a sum of high-dimensional random matrices and operators with dependence and arbitrary heavy tails. There is an increase in the importance of the problem of estimating high-dimensional matrices, and dependence and heavy-tail properties of data are among the most critical topics currently. In this paper, we derive a dimension-free upper bound on the deviation, that is, the bound does not depend explicitly on the dimension of matrices, but depends on their effective rank. Our result is a generalization of several existing studies on the deviation of the sum of matrices. Our proof is based on two techniques: (i) a variational approximation of the dual of moment generating functions, and (ii) robustification through truncation of eigenvalues of matrices. We show that our results are applicable to several problems such as covariance matrix estimation, hidden Markov models, and overparameterized linear regression models.

In the present paper, we examine a Crouzeix-Raviart approximation for non-linear partial differential equations having a $(p,\delta)$-structure for some $p\in (1,\infty)$ and $\delta\ge 0$. We establish a priori error estimates, which are optimal for all $p\in (1,\infty)$ and $\delta\ge 0$, medius error estimates, i.e., best-approximation results, and a primal-dual a posteriori error estimate, which is both reliable and efficient. The theoretical findings are supported by numerical experiments.

Causal discovery from observational and interventional data is challenging due to limited data and non-identifiability: factors that introduce uncertainty in estimating the underlying structural causal model (SCM). Selecting experiments (interventions) based on the uncertainty arising from both factors can expedite the identification of the SCM. Existing methods in experimental design for causal discovery from limited data either rely on linear assumptions for the SCM or select only the intervention target. This work incorporates recent advances in Bayesian causal discovery into the Bayesian optimal experimental design framework, allowing for active causal discovery of large, nonlinear SCMs while selecting both the interventional target and the value. We demonstrate the performance of the proposed method on synthetic graphs (Erdos-R\`enyi, Scale Free) for both linear and nonlinear SCMs as well as on the \emph{in-silico} single-cell gene regulatory network dataset, DREAM.

Modified Patankar-Runge-Kutta (MPRK) methods preserve the positivity as well as conservativity of a production-destruction system (PDS) of ordinary differential equations for all time step sizes. As a result, higher order MPRK schemes do not belong to the class of general linear methods, i.e. the iterates are generated by a nonlinear map $\mathbf g$ even when the PDS is linear. Moreover, due to the conservativity of the method, the map $\mathbf g$ possesses non-hyperbolic fixed points. Recently, a new theorem for the investigation of stability properties of non-hyperbolic fixed points of a nonlinear iteration map was developed. We apply this theorem to understand the stability properties of a family of second order MPRK methods when applied to a nonlinear PDS of ordinary differential equations. It is shown that the fixed points are stable for all time step sizes and members of the MPRK family. Finally, experiments are presented to numerically support the theoretical claims.

In this paper, we develop the constrained energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with mixed boundary conditions (Dirichlet and Neumann) for the elasticity equations in high contrast media. By a special treatment of mixed boundary conditions separately, and combining the construction of the relaxed and constraint version of the CEM-GMsFEM, we discover that the method offers some advantages such as the independence of the target region's contrast from precision, while the sizes of oversampling domains have a significant impact on numerical accuracy. Moreover, to our best knowledge, this is the first proof of the convergence of the CEM-GMsFEM with mixed boundary conditions for the elasticity equations given. Some numerical experiments are provided to demonstrate the method's performance.

Simulation of stochastic partial differential equations (SPDE) on a general domain requires a discretization of the noise. In this paper, the noise is discretized by a piecewise linear interpolation. The error caused by this is analyzed in the context of a fully discrete finite element approximation of a semilinear stochastic reaction-advection-diffusion equation on a convex polygon. The noise is Gaussian, white in time and correlated in space. It is modeled as a standard cylindrical Wiener process on the reproducing kernel Hilbert space associated to the covariance kernel. The noise is assumed to extend to a larger polygon than the SPDE domain to allow for sampling by the circulant embedding method. The interpolation error is analyzed under mild assumptions on the kernel. The main tools used are Hilbert--Schmidt bounds of multiplication operators onto negative order Sobolev spaces and an error bound for the finite element interpolant in fractional Sobolev norms. Examples with covariance kernels encountered in applications are illustrated in numerical simulations using the FEniCS finite element software. Conclusions from the analysis include that interpolation of noise with Mat\'ern kernels does not cause an additional error, that there exist kernels where the interpolation error dominates and that generation of noise on a coarser mesh than that of the SPDE discretization does not always result in a loss of accuracy.

In this paper, we present a multiscale framework for solving the Helmholtz equation in heterogeneous media without scale separation and in the high frequency regime where the wavenumber $k$ can be large. The main innovation is that our methods achieve a nearly exponential rate of convergence with respect to the computational degrees of freedom, using a coarse grid of mesh size $O(1/k)$ without suffering from the well-known pollution effect. The key idea is a non-overlapped domain decomposition and its associated coarse-fine scale decomposition of the solution space that adapts to the media property and wavenumber; this decomposition is inspired by the multiscale finite element method (MsFEM). We show that the coarse part is of \textit{low complexity} in the sense that it can be approximated with a nearly exponential rate of convergence via local basis functions, due to the compactness of a restriction operator that maps Helmholtz-harmonic functions to their interpolation residues on edges, while the fine part is \textit{local} such that it can be computed efficiently using the local information of the right hand side. The combination of the two parts yields the overall nearly exponential rate of convergence of our multiscale method. Our method draws many connections to multiscale methods in the literature, which we will comment in detail. We demonstrate the effectiveness of our methods theoretically and numerically; an exponential rate of convergence is consistently observed and confirmed. In addition, we observe the robustness of our methods regarding the high contrast in the media numerically. We specifically focus on 2D problems in our exposition since the geometry of non-overlapped domain decomposition is simplest to explain in such cases; generalizations to 3D will be outlined at the end.

Generalization in Reinforcement Learning (RL) aims to learn an agent during training that generalizes to the target environment. This paper studies RL generalization from a theoretical aspect: how much can we expect pre-training over training environments to be helpful? When the interaction with the target environment is not allowed, we certify that the best we can obtain is a near-optimal policy in an average sense, and we design an algorithm that achieves this goal. Furthermore, when the agent is allowed to interact with the target environment, we give a surprising result showing that asymptotically, the improvement from pre-training is at most a constant factor. On the other hand, in the non-asymptotic regime, we design an efficient algorithm and prove a distribution-based regret bound in the target environment that is independent of the state-action space.

Many functions have approximately-known upper and/or lower bounds, potentially aiding the modeling of such functions. In this paper, we introduce Gaussian process models for functions where such bounds are (approximately) known. More specifically, we propose the first use of such bounds to improve Gaussian process (GP) posterior sampling and Bayesian optimization (BO). That is, we transform a GP model satisfying the given bounds, and then sample and weight functions from its posterior. To further exploit these bounds in BO settings, we present bounded entropy search (BES) to select the point gaining the most information about the underlying function, estimated by the GP samples, while satisfying the output constraints. We characterize the sample variance bounds and show that the decision made by BES is explainable. Our proposed approach is conceptually straightforward and can be used as a plug in extension to existing methods for GP posterior sampling and Bayesian optimization.