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A comprehensive mathematical model of the multiphysics flow of blood and Cerebrospinal Fluid (CSF) in the brain can be expressed as the coupling of a poromechanics system and Stokes' equations: the first describes fluids filtration through the cerebral tissue and the tissue's elastic response, while the latter models the flow of the CSF in the brain ventricles. This model describes the functioning of the brain's waste clearance mechanism, which has been recently discovered to play an essential role in the progress of neurodegenerative diseases. To model the interactions between different scales in the porous medium, we propose a physically consistent coupling between Multi-compartment Poroelasticity (MPE) equations and Stokes' equations. In this work, we introduce a numerical scheme for the discretization of such coupled MPE-Stokes system, employing a high-order discontinuous Galerkin method on polytopal grids to efficiently account for the geometric complexity of the domain. We analyze the stability and convergence of the space semidiscretized formulation, we prove a-priori error estimates, and we present a temporal discretization based on a combination of Newmark's $\beta$-method for the elastic wave equation and the $\theta$-method for the other equations of the model. Numerical simulations carried out on test cases with manufactured solutions validate the theoretical error estimates. We also present numerical results on a two-dimensional slice of a patient-specific brain geometry reconstructed from diagnostic images, to test in practice the advantages of the proposed approach.

We consider several basic questions on distributed routing in directed graphs with multiple additive costs, or metrics, and multiple constraints. Distributed routing in this sense is used in several protocols, such as IS-IS and OSPF. A practical approach to the multi-constraint routing problem is to, first, combine the metrics into a single `composite' metric, and then apply one-to-all shortest path algorithms, e.g. Dijkstra, in order to find shortest path trees. We show that, in general, even if a feasible path exists and is known for every source and destination pair, it is impossible to guarantee a distributed routing under several constraints. We also study the question of choosing the optimal `composite' metric. We show that under certain mathematical assumptions we can efficiently find a convex combination of several metrics that maximizes the number of discovered feasible paths. Sometimes it can be done analytically, and is in general possible using what we call a 'smart iterative approach'. We illustrate these findings by extensive experiments on several typical network topologies.

In high-temperature plasma physics, a strong magnetic field is usually used to confine charged particles. Therefore, for studying the classical mathematical models of the physical problems it needs to consider the effect of external magnetic fields. One of the important model equations in plasma is the Vlasov-Poisson equation with an external magnetic field. This equation usually has multi-scale characteristics and rich physical properties, thus it is very important and meaningful to construct numerical methods that can maintain the physical properties inherited by the original systems over long time. This paper extends the corresponding theory in Cartesian coordinates to general orthogonal curvilinear coordinates, and proves that a Poisson-bracket structure can still be obtained after applying the corresponding finite element discretization. However, the Hamiltonian systems in the new coordinate systems generally cannot be decomposed into sub-systems that can be solved accurately, so it is impossible to use the splitting methods to construct the corresponding geometric integrators. Therefore, this paper proposes a semi-implicit method for strong magnetic fields and analyzes the asymptotic stability of this method.

Extremile (Daouia, Gijbels and Stupfler,2019) is a novel and coherent measure of risk, determined by weighted expectations rather than tail probabilities. It finds application in risk management, and, in contrast to quantiles, it fulfills the axioms of consistency, taking into account the severity of tail losses. However, existing studies (Daouia, Gijbels and Stupfler,2019,2022) on extremile involve unknown distribution functions, making it challenging to obtain a root n-consistent estimator for unknown parameters in linear extremile regression. This article introduces a new definition of linear extremile regression and its estimation method, where the estimator is root n-consistent. Additionally, while the analysis of unlabeled data for extremes presents a significant challenge and is currently a topic of great interest in machine learning for various classification problems, we have developed a semi-supervised framework for the proposed extremile regression using unlabeled data. This framework can also enhance estimation accuracy under model misspecification. Both simulations and real data analyses have been conducted to illustrate the finite sample performance of the proposed methods.

We propose MNPCA, a novel non-linear generalization of (2D)$^2${PCA}, a classical linear method for the simultaneous dimension reduction of both rows and columns of a set of matrix-valued data. MNPCA is based on optimizing over separate non-linear mappings on the left and right singular spaces of the observations, essentially amounting to the decoupling of the two sides of the matrices. We develop a comprehensive theoretical framework for MNPCA by viewing it as an eigenproblem in reproducing kernel Hilbert spaces. We study the resulting estimators on both population and sample levels, deriving their convergence rates and formulating a coordinate representation to allow the method to be used in practice. Simulations and a real data example demonstrate MNPCA's good performance over its competitors.

This work is concerned with the uniform accuracy of implicit-explicit backward differentiation formulas for general linear hyperbolic relaxation systems satisfying the structural stability condition proposed previously by the third author. We prove the uniform stability and accuracy of a class of IMEX-BDF schemes discretized spatially by a Fourier spectral method. The result reveals that the accuracy of the fully discretized schemes is independent of the relaxation time in all regimes. It is verified by numerical experiments on several applications to traffic flows, rarefied gas dynamics and kinetic theory.

Controlling spurious oscillations is crucial for designing reliable numerical schemes for hyperbolic conservation laws. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique in [Lu, Liu, and Shu, SIAM J. Numer. Anal., 59:1299-1324, 2021]. The OEDG method incorporates an OE procedure after each Runge-Kutta stage, devised by alternately evolving conventional semidiscrete DG scheme and a damping equation. A novel damping operator is carefully designed to possess scale-invariant and evolution-invariant properties. We rigorously prove optimal error estimates of the fully discrete OEDG method for linear scalar conservation laws. This might be the first generic fully-discrete error estimates for nonlinear DG schemes with automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems across various scales and wave speeds, without problem-specific parameters. It obviates the need for characteristic decomposition in hyperbolic systems. It retains key properties of conventional DG method, such as conservation, optimal convergence rates, and superconvergence. Moreover, it remains stable under normal CFL condition. The OE procedure is non-intrusive, facilitating integration into existing DG codes as an independent module. Its implementation is easy and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of damping operator as a modal filter and establishes close relations between the damping and spectral viscosity techniques. Extensive numerical results confirm the theoretical analysis and validate the effectiveness and advantages of the OEDG method.

Bayesian variable selection methods are powerful techniques for fitting and inferring on sparse high-dimensional linear regression models. However, many are computationally intensive or require restrictive prior distributions on model parameters. In this paper, we proposed a computationally efficient and powerful Bayesian approach for sparse high-dimensional linear regression. Minimal prior assumptions on the parameters are required through the use of plug-in empirical Bayes estimates of hyperparameters. Efficient maximum a posteriori (MAP) estimation is completed through a Parameter-Expanded Expectation-Conditional-Maximization (PX-ECM) algorithm. The PX-ECM results in a robust computationally efficient coordinate-wise optimization which -- when updating the coefficient for a particular predictor -- adjusts for the impact of other predictor variables. The completion of the E-step uses an approach motivated by the popular two-group approach to multiple testing. The result is a PaRtitiOned empirical Bayes Ecm (PROBE) algorithm applied to sparse high-dimensional linear regression, which can be completed using one-at-a-time or all-at-once type optimization. We compare the empirical properties of PROBE to comparable approaches with numerous simulation studies and analyses of cancer cell drug responses. The proposed approach is implemented in the R package probe.

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs -- which are dominated by snapshot generation -- of model reduction techniques for nonlinear advection-dominated problems.

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem, possibly with a local or a boundary control, for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits excellent performances and robustness with respect to the parameters of interest.