We present an innovative interpretation of Kalman filter (KF) combining the ideas of Schwarz domain decomposition (DD) and parallel in time (PinT) approaches. Thereafter we call it DD-KF. In contrast to standard DD approaches which are already incorporated in KF and other state estimation models, implementing a straightforward data parallelism inside the loop over time, DD-KF ab-initio partitions the whole model, including filter equations and dynamic model along both space and time directions/steps. As a consequence, we get local KFs reproducing the original filter at smaller dimensions on local domains. Also, sub problems could be solved in parallel. In order to enforce the matching of local solutions on overlapping regions, and then to achieve the same global solution of KF, local KFs are slightly modified by adding a correction term keeping track of contributions of adjacent subdomains to overlapping regions. Such a correction term balances localization errors along overlapping regions, acting as a regularization constraint on local solutions. Furthermore, such a localization excludes remote observations from each analyzed location improving the conditioning of the error covariance matrices. As dynamic model we consider shallow water equations which can be regarded a consistent tool to get a proof of concept of the reliability assessment of DD-KF in monitoring and forecasting of weather systems and ocean currents.

Deep learning techniques for solving elliptic interface problems have gained significant attentions. In this paper, we introduce a hybrid residual and weak form (HRW) loss aimed at mitigating the challenge of model training. HRW utilizes the functions residual loss and Ritz method in an adversary-system, which enhances the probability of jumping out of the local optimum even when the loss landscape comprises multiple soft constraints (regularization terms), thus improving model’s capability and robustness. For the problem with interface conditions, unlike existing methods that use the domain decomposition, we design a Pre-activated ResNet of ResNet (PRoR) network structure employing a single network to feed both coordinates and corresponding subdomain indicators, thus reduces the number of parameters. The effectiveness and improvements of the PRoR with HRW are verified on two-dimensional interface problems with regular or irregular interfaces. We then apply the PRoR with HRW to solve the size-modified Poisson-Boltzmann equation, an improved dielectric continuum model for predicting the electrostatic potentials in an ionic solvent by considering the steric effects. Our findings demonstrate that the PRoR with HRW accurately approximates solvation free-energies of three proteins with irregular interfaces, showing the competitive results compared to the ones obtained using the finite element method.

In this paper, we first reinvestigate the convergence of the vanilla SGD method in the sense of $L^2$ under more general learning rates conditions and a more general convex assumption, which relieves the conditions on learning rates and does not need the problem to be strongly convex. Then, by taking advantage of the Lyapunov function technique, we present the convergence of the momentum SGD and Nesterov accelerated SGD methods for the convex and non-convex problem under $L$-smooth assumption that extends the bounded gradient limitation to a certain extent. The convergence of time averaged SGD was also analyzed.

We present a new category of physics-informed neural networks called physics informed variational embedding generative adversarial network (PI-VEGAN), that effectively tackles the forward, inverse, and mixed problems of stochastic differential equations. In these scenarios, the governing equations are known, but only a limited number of sensor measurements of the system parameters are available. We integrate the governing physical laws into PI-VEGAN with automatic differentiation, while introducing a variational encoder for approximating the latent variables of the actual distribution of the measurements. These latent variables are integrated into the generator to facilitate accurate learning of the characteristics of the stochastic partial equations. Our model consists of three components, namely the encoder, generator, and discriminator, each of which is updated alternatively employing the stochastic gradient descent algorithm. We evaluate the effectiveness of PI-VEGAN in addressing forward, inverse, and mixed problems that require the concurrent calculation of system parameters and solutions. Numerical results demonstrate that the proposed method achieves satisfactory stability and accuracy in comparison with the previous physics-informed generative adversarial network (PI-WGAN).

The problem of finding fixed points of nonexpansive mappings on Hadamard manifolds is considered in this paper. To solve this kind of problem, we propose a modified Riemannian Mann algorithm with inertial effect. Under the assumption of existence of fixed points of the nonexpansive mapping, the global convergence of the proposed algorithm is established. To show the efficiency of the proposed algorithm, numerical comparisons with some existing algorithms are reported.

In recent years, numerical solutions of the inverse eigenvalue problems with multiple eigenvalues have attracted the attention of some researchers, and there have been a few algorithms with quadratic convergence. We propose here an extended two-step method for solving the inverse eigenvalue problems with multiple eigenvalues. Under appropriate assumptions, the convergence analysis of the extended method is presented and the cubic root-convergence rate is proved. Numerical experiments are provided to confirm the theoretical results and comparisons with the inexact Cayley transform method are made. Our extended method and convergence result in the present paper may enrich the results of numerical solutions of the inverse eigenvalue problems with multiple eigenvalues.

We develop a class of conservative integrators for the regularized logarithmic Schrödinger equation (RLogSE) using the quadratization technique and symplectic Runge-Kutta schemes. To preserve the highly nonlinear energy functional, the regularized equation is first transformed into an equivalent system that admits two quadratic invariants by adopting the invariant energy quadratization approach. The reformulation is then discretized using the Fourier pseudo-spectral method in the space direction, and integrated in the time direction by a class of diagonally implicit Runge-Kutta schemes that conserve both quadratic invariants to round-off errors. For comparison purposes, a class of multi-symplectic integrators are developed for RLogSE to conserve the multi-symplectic conservation law and global mass conservation law in the discrete level. Numerical experiments illustrate the convergence, efficiency, and conservative properties of the proposed methods.

Although there are many effective methods for removing impulse noise in image restoration, there is still much room for improvement. In this paper, we propose a new two-phase method for solving such a problem, which combines the nuclear norm and the total variation regularization with box constraint. The popular alternating direction method of multipliers and the proximal alternating direction method of multipliers are employed to solve this problem. Compared with other algorithms, the obtained algorithm has an explicit solution at each step. Numerical experiments demonstrate that the proposed method performs better than the state-of-the-art methods in terms of both subjective and objective evaluations.

Krylov subspace methods are widely used for solving sparse linear algebraic equations, but they rely heavily on preconditioners, and it is difficult to find an effective preconditioner that is efficient and stable for all problems. In this paper, a novel projection strategy including the orthogonal and the oblique projection is proposed to improve the preconditioner, which can enhance the efficiency and stability of iteration. The proposed strategy can be considered as a minimization process, where the orthogonal projection minimizes the energy norm of error and the oblique projection minimizes the 2-norm of the residual, meanwhile they can be regarded as approaches to correct the approximation by solving low-rank inverse of the matrices. The strategy is a wide-ranging approach and provides a way to transform the constant preconditioner into a variable one. The paper discusses in detail the projection strategy for sparse approximate inverse (SPAI) preconditioner applied to flexible GMRES and conducts the numerical test for problems from different applications. The results show that the proposed projection strategy can significantly improve the solving process, especially for some non-converging and slowly convergence systems.

In this paper, we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations (BSDEs). By the nonlinear Feynman-Kac formula, we reformulate the BSDE into a pair of reference ordinary differential equations (ODEs), which can be directly discretized by many standard ODE solvers, yielding the corresponding numerical schemes for BSDEs. In particular, by applying strong stability preserving (SSP) time discretizations to the reference ODEs, we can propose new SSP multistep schemes for BSDEs. Theoretical analyses are rigorously performed to prove the consistency, stability and convergency of the proposed SSP multistep schemes. Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.

In the present work we aim to simulate shallow water flows over movable bottom with suspended and bedload transport. In order to numerically approximate such a system, we proceed step by step. We start by considering shallow water equations with non-constant density of the mixture water-sediment. Then, the Exner equation is included to take into account bedload sediment transport. Finally, source terms for friction, erosion and deposition processes are considered. Indeed, observe that the sediment particle could go in suspension into the water or being deposited on the bottom. For the numerical scheme, we rely on well-balanced Lagrange-projection methods. In particular, since sediment transport is generally a slow process, we aim to develop semi-implicit schemes in order to obtain fast simulations. The Lagrange-projection splitting is well-suited for such a purpose as it entails a decomposition of the (fast) acoustic waves and the (slow) material waves of the model. Hence, in subsonic regimes, an implicit approximation of the acoustic equations allows us to neglect the corresponding CFL condition and to obtain fast numerical schemes with large time step.