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High-dimensional parabolic partial differential equations (PDEs) often involve large-scale Hessian matrices, which are computationally expensive for deep learning methods relying on automatic differentiation to compute derivatives. This work aims to address this issue. In the proposed method, the PDE is reformulated into a martingale formulation, which allows the computation of loss functions to be derivative-free and parallelized in time-space domain. Then, the martingale formulation is enforced using a Galerkin method via adversarial learning techniques, which eliminate the need of computing conditional expectations in the margtingale property. This method is further extended to solve Hamilton-Jacobi-Bellman (HJB) equations and the associated Stochastic optimal control problems, enabling the simultaneous solution of the value function and optimal feedback control in a derivative-free manner. Numerical results demonstrate the effectiveness and efficiency of the proposed method, capable of solving HJB equations accurately with dimensionality up to 10,000.

We present a novel technique for imposing non-linear entropy conservative and entropy stable wall boundary conditions for the resistive magnetohydrodynamic equations in the presence of an adiabatic wall or a wall with a prescribed heat entropy flow, addressing three scenarios: electrically insulating walls, thin walls with finite conductivity, and perfectly conducting walls. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts, and simultaneous-approximation-term operators. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions is coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators) on high-order unstructured grids are used to demonstrate the new procedure's accuracy, robustness, and efficacy for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, nodal and modal discontinuous Galerkin, and flux reconstruction schemes.

This study focuses on addressing the challenge of solving the reduced biquaternion equality constrained least squares (RBLSE) problem. We develop algebraic techniques to derive both complex and real solutions for the RBLSE problem by utilizing the complex and real forms of reduced biquaternion matrices. Additionally, we conduct a perturbation analysis for the RBLSE problem and establish an upper bound for the relative forward error of these solutions. Numerical examples are presented to illustrate the effectiveness of the proposed approaches and to verify the accuracy of the established upper bound for the relative forward errors.

We propose and analyse a boundary-preserving numerical scheme for the weak approximations of some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter splitting followed by exact simulation and exact integration in time. We prove weak convergence of optimal order 1/4 for globally Lipschitz continuous test functions of the scheme by proving strong convergence towards a strong solution driven by a different noise process. Boundary-preservation is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the effectiveness of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing methods for SPDEs.

The phenomenon of finite time blow-up in hydrodynamic partial differential equations is central in analysis and mathematical physics. While numerical studies have guided theoretical breakthroughs, it is challenging to determine if the observed computational results are genuine or mere numerical artifacts. Here we identify numerical signatures of blow-up. Our study is based on the complexified Euler equations in two dimensions, where instant blow-up is expected. Via a geometrically consistent spatiotemporal discretization, we perform several numerical experiments and verify their computational stability. We then identify a signature of blow-up based on the growth rates of the supremum norm of the vorticity with increasing spatial resolution. The study aims to be a guide for cross-checking the validity for future numerical experiments of suspected blow-up in equations where the analysis is not yet resolved.

We consider a fully discretized numerical scheme for parabolic stochastic partial differential equations with multiplicative noise. Our abstract framework can be applied to formulate a non-iterative domain decomposition approach. Such methods can help to parallelize the code and therefore lead to a more efficient implementation. The domain decomposition is integrated through the Douglas-Rachford splitting scheme, where one split operator acts on one part of the domain. For an efficient space discretization of the underlying equation, we chose the discontinuous Galerkin method as this suits the parallelization strategy well. For this fully discretized scheme, we provide a strong space-time convergence result. We conclude the manuscript with numerical experiments validating our theoretical findings.

The dynamics of magnetization in ferromagnetic materials are modeled by the Landau-Lifshitz equation, which presents significant challenges due to its inherent nonlinearity and non-convex constraint. These complexities necessitate efficient numerical methods for micromagnetics simulations. The Gauss-Seidel Projection Method (GSPM), first introduced in 2001, is among the most efficient techniques currently available. However, existing GSPMs are limited to first-order accuracy. This paper introduces two novel second-order accurate GSPMs based on a combination of the biharmonic equation and the second-order backward differentiation formula, achieving computational complexity comparable to that of solving the scalar biharmonic equation implicitly. The first proposed method achieves unconditional stability through Gauss-Seidel updates, while the second method exhibits conditional stability with a Courant-Friedrichs-Lewy constant of 0.25. Through consistency analysis and numerical experiments, we demonstrate the efficacy and reliability of these methods. Notably, the first method displays unconditional stability in micromagnetics simulations, even when the stray field is updated only once per time step.

We present a rigorous convergence analysis of a new method for density-based topology optimization: Sigmoidal Mirror descent with a Projected Latent variable. SiMPL, pronounced like "simple," provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, SiMPL delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm.

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

We propose and analyze a space--time finite element method for Westervelt's quasilinear model of ultrasound waves in second-order formulation. The method combines conforming finite element spatial discretizations with a discontinuous-continuous Galerkin time stepping. Its analysis is challenged by the fact that standard Galerkin testing approaches for wave problems do not allow for bounding the discrete energy at all times. By means of redesigned energy arguments for a linearized problem combined with Banach's fixed-point argument, we show the well-posedness of the scheme, a priori error estimates, and robustness with respect to the strong damping parameter $\delta$. Moreover, the scheme preserves the asymptotic preserving property of the continuous problem; more precisely, we prove that the discrete solutions corresponding to $\delta>0$ converge, in the singular vanishing dissipation limit, to the solution of the discrete inviscid problem. We use several numerical experiments in $(2 + 1)$-dimensions to validate our theoretical results.