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We study the performance of stochastic first-order methods for finding saddle points of convex-concave functions. A notorious challenge faced by such methods is that the gradients can grow arbitrarily large during optimization, which may result in instability and divergence. In this paper, we propose a simple and effective regularization technique that stabilizes the iterates and yields meaningful performance guarantees even if the domain and the gradient noise scales linearly with the size of the iterates (and is thus potentially unbounded). Besides providing a set of general results, we also apply our algorithm to a specific problem in reinforcement learning, where it leads to performance guarantees for finding near-optimal policies in an average-reward MDP without prior knowledge of the bias span.

Many science and engineering applications demand partial differential equations (PDE) evaluations that are traditionally computed with resource-intensive numerical solvers. Neural operator models provide an efficient alternative by learning the governing physical laws directly from data in a class of PDEs with different parameters, but constrained in a fixed boundary (domain). Many applications, such as design and manufacturing, would benefit from neural operators with flexible domains when studied at scale. Here we present a diffeomorphism neural operator learning framework towards developing domain-flexible models for physical systems with various and complex domains. Specifically, a neural operator trained in a shared domain mapped from various domains of fields by diffeomorphism is proposed, which transformed the problem of learning function mappings in varying domains (spaces) into the problem of learning operators on a shared diffeomorphic domain. Meanwhile, an index is provided to evaluate the generalization of diffeomorphism neural operators in different domains by the domain diffeomorphism similarity. Experiments on statics scenarios (Darcy flow, mechanics) and dynamic scenarios (pipe flow, airfoil flow) demonstrate the advantages of our approach for neural operator learning under various domains, where harmonic and volume parameterization are used as the diffeomorphism for 2D and 3D domains. Our diffeomorphism neural operator approach enables strong learning capability and robust generalization across varying domains and parameters.

In this work, we present a high-order finite volume framework for the numerical simulation of shallow water flows. The method is designed to accurately capture complex dynamics inherent in shallow water systems, particularly suited for applications such as tsunami simulations. The arbitrarily high-order framework ensures precise representation of flow behaviors, crucial for simulating phenomena characterized by rapid changes and fine-scale features. Thanks to an {\it ad-hoc} reformulation in terms of production-destruction terms, the time integration ensures positivity preservation without any time-step restrictions, a vital attribute for physical consistency, especially in scenarios where negative water depth reconstructions could lead to unrealistic results. In order to introduce the preservation of general steady equilibria dictated by the underlying balance law, the high-order reconstruction and numerical flux are blended in a convex fashion with a well-balanced approximation, which is able to provide exact preservation of both static and moving equilibria. Through numerical experiments, we demonstrate the effectiveness and robustness of the proposed approach in capturing the intricate dynamics of shallow water flows, while preserving key physical properties essential for flood simulations.

Motivated by a recent work on a preconditioned MINRES for flipped linear systems in imaging, in this note we extend the scope of that research for including more precise boundary conditions such as reflective and anti-reflective ones. We prove spectral results for the matrix-sequences associated to the original problem, which justify the use of the MINRES in the current setting. The theoretical spectral analysis is supported by a wide variety of numerical experiments, concerning the visualization of the spectra of the original matrices in various ways. We also report numerical tests regarding the convergence speed and regularization features of the associated GMRES and MINRES methods. Conclusions and open problems end the present study.

Error estimates of cubic interpolated pseudo-particle scheme (CIP scheme) for the one-dimensional advection equation with periodic boundary conditions are presented. The CIP scheme is a semi-Lagrangian method involving the piecewise cubic Hermite interpolation. Although it is numerically known that the space-time accuracy of the scheme is third order, its rigorous proof remains an open problem. In this paper, denoting the spatial and temporal mesh sizes by $ h $ and $ \Delta t $ respectively, we prove an error estimate $ O(\Delta t^3 + \frac{h^4}{\Delta t}) $ in $ L^2 $ norm theoretically, which justifies the above-mentioned prediction if $ h = O(\Delta t) $. The proof is based on properties of the interpolation operator; the most important one is that it behaves as the $ L^2 $ projection for the second-order derivatives. We remark that the same strategy perfectly works as well to address an error estimate for the semi-Lagrangian method with the cubic spline interpolation.

Boundary value problems involving elliptic PDEs such as the Laplace and the Helmholtz equations are ubiquitous in mathematical physics and engineering. Many such problems can be alternatively formulated as integral equations that are mathematically more tractable. However, an integral-equation formulation poses a significant computational challenge: solving large dense linear systems that arise upon discretization. In cases where iterative methods converge rapidly, existing methods that draw on fast summation schemes such as the Fast Multipole Method are highly efficient and well-established. More recently, linear complexity direct solvers that sidestep convergence issues by directly computing an invertible factorization have been developed. However, storage and computation costs are high, which limits their ability to solve large-scale problems in practice. In this work, we introduce a distributed-memory parallel algorithm based on an existing direct solver named ``strong recursive skeletonization factorization.'' Specifically, we apply low-rank compression to certain off-diagonal matrix blocks in a way that minimizes computation and data movement. Compared to iterative algorithms, our method is particularly suitable for problems involving ill-conditioned matrices or multiple right-hand sides. Large-scale numerical experiments are presented to show the performance of our Julia implementation.

We present an algorithm for the exact computer-aided construction of the Voronoi cells of lattices with known symmetry group. Our algorithm scales better than linearly with the total number of faces and is applicable to dimensions beyond 12, which previous methods could not achieve. The new algorithm is applied to the Coxeter-Todd lattice $K_{12}$ as well as to a family of lattices obtained from laminating $K_{12}$. By optimizing this family, we obtain a new best 13-dimensional lattice quantizer (among the lattices with published exact quantizer constants).

In a recent work (Dick et al, arXiv:2310.06187), we considered a linear stochastic elasticity equation with random Lam\'e parameters which are parameterized by a countably infinite number of terms in separate expansions. We estimated the expected values over the infinite dimensional parametric space of linear functionals ${\mathcal L}$ acting on the continuous solution $\vu$ of the elasticity equation. This was achieved by truncating the expansions of the random parameters, then using a high-order quasi-Monte Carlo (QMC) method to approximate the high dimensional integral combined with the conforming Galerkin finite element method (FEM) to approximate the displacement over the physical domain $\Omega.$ In this work, as a further development of aforementioned article, we focus on the case of a nearly incompressible linear stochastic elasticity equation. To serve this purpose, in the presence of stochastic inhomogeneous (variable Lam\'e parameters) nearly compressible material, we develop a new locking-free symmetric nonconforming Galerkin FEM that handles the inhomogeneity. In the case of nearly incompressible material, one known important advantage of nonconforming approximations is that they yield optimal order convergence rates that are uniform in the Poisson coefficient. Proving the convergence of the nonconforming FEM leads to another challenge that is summed up in showing the needed regularity properties of $\vu$. For the error estimates from the high-order QMC method, which is needed to estimate the expected value over the infinite dimensional parametric space of ${\mathcal L}\vu,$ we %rely on (Dick et al. 2022). We are required here to show certain regularity properties of $\vu$ with respect to the random coefficients. Some numerical results are delivered at the end.

In this article we aim to obtain the Fisher Riemann geodesics for nonparametric families of probability densities as a weak limit of the parametric case with increasing number of parameters.