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This paper presents a learnable solver tailored to iteratively solve sparse linear systems from discretized partial differential equations (PDEs). Unlike traditional approaches relying on specialized expertise, our solver streamlines the algorithm design process for a class of PDEs through training, which requires only training data of coefficient distributions. The proposed method is anchored by three core principles: (1) a multilevel hierarchy to promote rapid convergence, (2) adherence to linearity concerning the right-hand-side of equations, and (3) weights sharing across different levels to facilitate adaptability to various problem sizes. Built on these foundational principles and considering the similar computation pattern of the convolutional neural network (CNN) as multigrid components, we introduce a network adept at solving linear systems from PDEs with heterogeneous coefficients, discretized on structured grids. Notably, our proposed solver possesses the ability to generalize over right-hand-side terms, PDE coefficients, and grid sizes, thereby ensuring its training is purely offline. To evaluate its effectiveness, we train the solver on convection-diffusion equations featuring heterogeneous diffusion coefficients. The solver exhibits swift convergence to high accuracy over a range of grid sizes, extending from $31 \times 31$ to $4095 \times 4095$. Remarkably, our method outperforms the classical Geometric Multigrid (GMG) solver, demonstrating a speedup of approximately 3 to 8 times. Furthermore, our numerical investigation into the solver's capacity to generalize to untrained coefficient distributions reveals promising outcomes.

We propose and analyze a new asymptotic preserving (AP) finite volume scheme for the multidimensional compressible barotropic Euler equations to simulate low Mach number flows. The proposed scheme uses a stabilized upwind numerical flux, with the stabilization term being proportional to the stiff pressure gradient, and we prove its conditional energy stability and consistency. Utilizing the concept of dissipative measure-valued (DMV) solutions, we rigorously illustrate the AP properties of the scheme for well-prepared initial data. In particular, we prove that the numerical solutions will converge weakly to a DMV solution of the compressible Euler equations as the mesh parameter vanishes, while the Mach number is fixed. The DMV solutions then converge to a classical solution of the incompressible Euler system as the Mach number goes to zero. Conversely, we show that if the mesh parameter is kept fixed, we obtain an energy stable and consistent finite-volume scheme approximating the incompressible Euler equations as the Mach number goes to zero. The numerical solutions generated by this scheme then converge weakly to a DMV solution of the incompressible Euler system as the mesh parameter vanishes. Invoking the weak-strong uniqueness principle, we conclude that the DMV solution and classical solution of the incompressible Euler system coincide, proving the AP property of the scheme. We also present an extensive numerical case study in order to illustrate the theoretical convergences, wherein we utilize the techniques of K-convergence.

This paper analyzes the estimation of econometric models by penalizing the sum of squares of the residuals with a factor that makes the model estimates approximate those that would be obtained when considering the possible simple regressions between the dependent variable of the econometric model and each of its independent variables. It is shown that the ridge estimator is a particular case of the penalized estimator obtained, which, upon analysis of its main characteristics, presents better properties than the ridge especially in reference to the individual boostrap inference of the coefficients of the model and the numerical stability of the estimates obtained. This improvement is due to the fact that instead of shrinking the estimator towards zero, the estimator shrinks towards the estimates of the coefficients of the simple regressions discussed above.

This work introduces the Matrix Minimum Covariance Determinant (MMCD) method, a novel robust location and covariance estimation procedure designed for data that are naturally represented in the form of a matrix. Unlike standard robust multivariate estimators, which would only be applicable after a vectorization of the matrix-variate samples leading to high-dimensional datasets, the MMCD estimators account for the matrix-variate data structure and consistently estimate the mean matrix, as well as the rowwise and columnwise covariance matrices in the class of matrix-variate elliptical distributions. Additionally, we show that the MMCD estimators are matrix affine equivariant and achieve a higher breakdown point than the maximal achievable one by any multivariate, affine equivariant location/covariance estimator when applied to the vectorized data. An efficient algorithm with convergence guarantees is proposed and implemented. As a result, robust Mahalanobis distances based on MMCD estimators offer a reliable tool for outlier detection. Additionally, we extend the concept of Shapley values for outlier explanation to the matrix-variate setting, enabling the decomposition of the squared Mahalanobis distances into contributions of the rows, columns, or individual cells of matrix-valued observations. Notably, both the theoretical guarantees and simulations show that the MMCD estimators outperform robust estimators based on vectorized observations, offering better computational efficiency and improved robustness. Moreover, real-world data examples demonstrate the practical relevance of the MMCD estimators and the resulting robust Shapley values.

Randomized iterative methods, such as the Kaczmarz method and its variants, have gained growing attention due to their simplicity and efficiency in solving large-scale linear systems. Meanwhile, absolute value equations (AVE) have attracted increasing interest due to their connection with the linear complementarity problem. In this paper, we investigate the application of randomized iterative methods to generalized AVE (GAVE). Our approach differs from most existing works in that we tackle GAVE with non-square coefficient matrices. We establish more comprehensive sufficient and necessary conditions for characterizing the solvability of GAVE and propose precise error bound conditions. Furthermore, we introduce a flexible and efficient randomized iterative algorithmic framework for solving GAVE, which employs sampling matrices drawn from user-specified distributions. This framework is capable of encompassing many well-known methods, including the Picard iteration method and the randomized Kaczmarz method. Leveraging our findings on solvability and error bounds, we establish both almost sure convergence and linear convergence rates for this versatile algorithmic framework. Finally, we present numerical examples to illustrate the advantages of the new algorithms.

The high dimensional nature of genomics data complicates feature selection, in particular in low sample size studies - not uncommon in clinical prediction settings. It is widely recognized that complementary data on the features, `co-data', may improve results. Examples are prior feature groups or p-values from a related study. Such co-data are ubiquitous in genomics settings due to the availability of public repositories. Yet, the uptake of learning methods that structurally use such co-data is limited. We review guided adaptive shrinkage methods: a class of regression-based learners that use co-data to adapt the shrinkage parameters, crucial for the performance of those learners. We discuss technical aspects, but also the applicability in terms of types of co-data that can be handled. This class of methods is contrasted with several others. In particular, group-adaptive shrinkage is compared with the better-known sparse group-lasso by evaluating feature selection. Finally, we demonstrate the versatility of the guided shrinkage methodology by showing how to `do-it-yourself': we integrate implementations of a co-data learner and the spike-and-slab prior for the purpose of improving feature selection in genetics studies.

An additive Runge-Kutta method is used for the time stepping, which integrates the linear stiff terms by an explicit singly diagonally implicit Runge-Kutta (ESDIRK) method and the nonlinear terms by an explicit Runge-Kutta (ERK) method. In each time step, the implicit solve is performed by the recently developed Hierarchical Poincar\'e-Steklov (HPS) method. This is a fast direct solver for elliptic equations that decomposes the space domain into a hierarchical tree of subdomains and builds spectral collocation solvers locally on the subdomains. These ideas are naturally combined in the presented method since the singly diagonal coefficient in ESDIRK and a fixed time-step ensures that the coefficient matrix in the implicit solve of HPS remains the same for all time stages. This means that the precomputed inverse can be efficiently reused, leading to a scheme with complexity (in two dimensions) $\mathcal{O}(N^{1.5})$ for the precomputation where the solution operator to the elliptic problems is built, and then $\mathcal{O}(N \log N)$ for the solve in each time step. The stability of the method is proved for first order in time and any order in space, and numerical evidence substantiates a claim of stability for a much broader class of time discretization methods. Numerical experiments supporting the accuracy of efficiency of the method in one and two dimensions are presented.

We introduce Semi-Implicit Lagrangian Voronoi Approximation (SILVA), a novel numerical method for the solution of the incompressible Euler and Navier-Stokes equations, which combines the efficiency of semi-implicit time marching schemes with the robustness of time-dependent Voronoi tessellations. In SILVA, the numerical solution is stored at particles, which move with the fluid velocity and also play the role of the generators of the computational mesh. The Voronoi mesh is rapidly regenerated at each time step, allowing large deformations with topology changes. As opposed to the reconnection-based Arbitrary-Lagrangian-Eulerian schemes, we need no remapping stage. A semi-implicit scheme is devised in the context of moving Voronoi meshes to project the velocity field onto a divergence-free manifold. We validate SILVA by illustrative benchmarks, including viscous, inviscid, and multi-phase flows. Compared to its closest competitor, the Incompressible Smoothed Particle Hydrodynamics (ISPH) method, SILVA offers a sparser stiffness matrix and facilitates the implementation of no-slip and free-slip boundary conditions.

It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.

In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier-spectral method is applied for approximation in the velocity space whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in [Remark 4.4, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].