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This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavorably with the number of evaluation points, which can result in computational inefficiency. To address this issue, we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple, we focus only on one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to more complicated two- and three-dimensional problems with appropriate modifications.

We discuss the asymptotic-preserving properties of a hybridizable discontinuous Galerkin method for the Westervelt model of ultrasound waves. More precisely, we show that the proposed method is robust with respect to small values of the sound diffusivity damping parameter~$\delta$ by deriving low- and high-order energy stability estimates, and \emph{a priori} error bounds that are independent of~$\delta$. Such bounds are then used to show that, when~$\delta \rightarrow 0^+$, the method remains stable and the discrete acoustic velocity potential~$\psi_h^{(\delta)}$ converges to~$\psi_h^{(0)}$, where the latter is the singular vanishing dissipation limit. Moreover, we prove optimal convergence for the approximation of the acoustic particle velocity variable~$\bv = \nabla \psi$. The established theoretical results are illustrated with some numerical experiments.

Physics-informed neural networks (PINNs) have emerged as powerful tools for solving a wide range of partial differential equations (PDEs). However, despite their user-friendly interface and broad applicability, PINNs encounter challenges in accurately resolving PDEs, especially when dealing with singular cases that may lead to unsatisfactory local minima. To address these challenges and improve solution accuracy, we propose an innovative approach called Annealed Adaptive Importance Sampling (AAIS) for computing the discretized PDE residuals of the cost functions, inspired by the Expectation Maximization algorithm used in finite mixtures to mimic target density. Our objective is to approximate discretized PDE residuals by strategically sampling additional points in regions with elevated residuals, thus enhancing the effectiveness and accuracy of PINNs. Implemented together with a straightforward resampling strategy within PINNs, our AAIS algorithm demonstrates significant improvements in efficiency across a range of tested PDEs, even with limited training datasets. Moreover, our proposed AAIS-PINN method shows promising capabilities in solving high-dimensional singular PDEs. The adaptive sampling framework introduced here can be integrated into various PINN frameworks.

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and5zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-B{\'e}nard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modelling, and precipitation forecasting).

Stability of the BDF methods of order up to five for parabolic equations can be established by the energy technique via Nevanlinna--Odeh multipliers. The nonexistence of Nevanlinna--Odeh multipliers makes the six-step BDF method special; however, the energy technique was recently extended by the authors in [Akrivis et al., SIAM J. Numer. Anal. \textbf{59} (2021) 2449--2472] and covers all six stable BDF methods. The seven-step BDF method is unstable for parabolic equations, since it is not even zero-stable. In this work, we construct and analyze a stable linear combination of two non zero-stable schemes, the seven-step BDF method and its shifted counterpart, referred to as WSBDF7 method. The stability regions of the WSBDF$q, q\leqslant 7$, with a weight $\vartheta\geqslant1$, increase as $\vartheta$ increases, are larger than the stability regions of the classical BDF$q,$ corresponding to $\vartheta=1$. We determine novel and suitable multipliers for the WSBDF7 method and establish stability for parabolic equations by the energy technique. The proposed approach is applicable for mean curvature flow, gradient flows, fractional equations and nonlinear equations.

We study the weak convergence behaviour of the Leimkuhler--Matthews method, a non-Markovian Euler-type scheme with the same computational cost as the Euler scheme, for the approximation of the stationary distribution of a one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The particular class under study is known as mean-field (overdamped) Langevin equations (MFL). We provide weak and strong error results for the scheme in both finite and infinite time. We work under a strong convexity assumption. Based on a careful analysis of the variation processes and the Kolmogorov backward equation for the particle system associated with the MV-SDE, we show that the method attains a higher-order approximation accuracy in the long-time limit (of weak order convergence rate $3/2$) than the standard Euler method (of weak order $1$). While we use an interacting particle system (IPS) to approximate the MV-SDE, we show the convergence rate is independent of the dimension of the IPS and this includes establishing uniform-in-time decay estimates for moments of the IPS, the Kolmogorov backward equation and their derivatives. The theoretical findings are supported by numerical tests.

In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate for the $L^2$-norm of the space-time discretization of SHE by a semi-implicit Euler scheme with respect to time and a TPFA scheme with respect to space and the variational solution of SHE. The only regularity assumptions additionally needed is spatial regularity of the initial datum and smoothness of the diffusive term.

The numerical solution of problems in nonlinear magnetostatics is typically based on a variational formulation in terms of magnetic potentials, the discretization by finite elements, and iterative solvers like the Newton method. The vector potential approach aims at minimizing a certain energy functional and, in three dimensions, requires the use of edge elements and appropriate gauging conditions. The scalar potential approach, on the other hand, seeks to maximize the negative coenergy and can be realized by standard Lagrange finite elements, thus reducing the number of degrees of freedom and simplifying the implementation. The number of Newton iterations required to solve the governing nonlinear system, however, has been observed to be usually higher than for the vector potential formulation. In this paper, we propose a method that combines the advantages of both approaches, i.e., it requires as few Newton iterations as the vector potential formulation while involving the magnetic scalar potential as the primary unknown. We discuss the variational background of the method, its well-posedness, and its efficient implementation. Numerical examples are presented for illustration of the accuracy and the gain in efficiency compared to other approaches.

We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can also overcome the non-physical issues that arise from many traditional numerical schemes. Due to their preservation of physical nature, these schemes can be straightforwardly used as backbones for further developing randomized and quantum algorithms in simulating Lindblad equations. In this work, we systematically study these methods and perform a detailed error analysis, which is validated through numerical examples.

We establish a connection between problems studied in rigidity theory and matroids arising from linear algebraic constructions like tensor products and symmetric products. A special case of this correspondence identifies the problem of giving a description of the correctable erasure patterns in a maximally recoverable tensor code with the problem of describing bipartite rigid graphs or low-rank completable matrix patterns. Additionally, we relate dependencies among symmetric products of generic vectors to graph rigidity and symmetric matrix completion. With an eye toward applications to computer science, we study the dependency of these matroids on the characteristic by giving new combinatorial descriptions in several cases, including the first description of the correctable patterns in an (m, n, a=2, b=2) maximally recoverable tensor code.