In this paper, a new multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter for high-order local discontinuous Galerkin (LDG) method is designed for solving Navier-Stokes equations on triangular meshes. This MR-WENO limiter is a new extension of the finite volume MR-WENO schemes. Such new limiter uses information of the LDG solution essentially only within the troubled cell itself, to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the highest degree of the LDG method. As an example, a third-order LDG method with associated same order MR-WENO limiter has been developed in this paper, which could maintain the original order of accuracy in smooth regions and could simultaneously suppress spurious oscillations near strong shocks or contact discontinuities. The linear weights of such new MR-WENO limiter can be any positive numbers on condition that their summation is one. This is the first time that a series of different degree polynomials within the troubled cell are applied in a WENO-type fashion to modify the freedom of degrees of the LDG solutions in the troubled cell. This MR-WENO limiter is very simple to construct, and can be easily implemented to arbitrary high-order accuracy and in higher dimensions on unstructured meshes. Such spatial reconstruction methodology improves the robustness in the numerical simulation on the same compact spatial stencil of the original LDG methods on triangular meshes. Some classical viscous examples are given to show the good performance of this third-order LDG method with associated MR-WENO limiter.

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach is the requirement for manual hyperparameter tuning, making it impractical in the absence of validation data or prior knowledge of the solution. Our investigations of the loss landscapes and backpropagated gradients in the presence of physics reveal that existing methods produce non-convex loss landscapes that are hard to navigate. Our findings demonstrate that high-order PDEs contaminate backpropagated gradients and hinder convergence. To address these challenges, we introduce a novel method that bypasses the calculation of high-order derivative operators and mitigates the contamination of backpropagated gradients. Consequently, we reduce the dimension of the search space and make learning PDEs with non-smooth solutions feasible. Our method also provides a mechanism to focus on complex regions of the domain. Besides, we present a dual unconstrained formulation based on Lagrange multiplier method to enforce equality constraints on the model’s prediction, with adaptive and independent learning rates inspired by adaptive subgradient methods. We apply our approach to solve various linear and non-linear PDEs.

For steady Euler equations in complex boundary domains, high-order shock-capturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations, and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary. This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.

New superconvergent structures are proposed and analyzed for the finite volume element (FVE) method over tensorial meshes in general dimension $d$ (for $d≥2$); we call these orthogonal superconvergent structures. In this framework, one has the freedom to choose the superconvergent points of tensorial $k$-order FVE schemes (for $k≥3$). This flexibility contrasts with the superconvergent points (such as Gauss points and Lobatto points) for current FE schemes and FVE schemes, which are fixed. The orthogonality condition and the modified M-decomposition (MMD) technique that are developed over tensorial meshes help in the construction of proper superclose functions for the FVE solutions and in ensuring the new superconvergence properties of the FVE schemes. Numerical experiments are provided to validate our theoretical results.

The ensemble Kalman inversion (EKI), inspired by the well-known ensemble Kalman filter, is a derivative-free and parallelizable method for solving inverse problems. The method is appealing for applications in a variety of fields due to its low computational cost and simple implementation. In this paper, we propose an adaptive ensemble Kalman inversion with statistical linearization (AEKI-SL) method for solving inverse problems from a hierarchical Bayesian perspective. Specifically, by adaptively updating the unknown with an EKI and updating the hyper-parameter in the prior model, the method can improve the accuracy of the solutions to the inverse problem. To avoid semi-convergence, we employ Morozov’s discrepancy principle as a stopping criterion. Furthermore, we extend the method to simultaneous estimation of noise levels in order to reduce the randomness of artificially ensemble noise levels. The convergence of the hyper-parameter in prior model is investigated theoretically. Numerical experiments show that our proposed methods outperform the traditional EKI and EKI with statistical linearization (EKI-SL) methods.

It is found that the solution remapping technique proposed in [Numer. Math. Theor. Meth. Appl., 2020, 13(4)] and [J. Sci. Comput., 2021, 87(3): 1-26] does not work out for the Navier-Stokes equations with a high Reynolds number. The shape deformations usually reach several boundary layer mesh sizes for viscous flow, which far exceed one-layer mesh that the original method can tolerate. The direct application to Navier-Stokes equations can result in the unphysical pressures in remapped solutions, even though the conservative variables are within the reasonable range. In this work, a new solution remapping technique with lower bound preservation is proposed to construct initial values for the new shapes, and the global minimum density and pressure of the current shape which serve as lower bounds of the corresponding variables are used to constrain the remapped solutions. The solution distribution provided by the present method is proven to be acceptable as an initial value for the new shape. Several numerical experiments show that the present technique can substantially accelerate the flow convergence for large deformation problems with 70%-80% CPU time reduction in the viscous airfoil drag minimization.

Fast and accurate MRI reconstruction is a key concern in modern clinical practice. Recently, numerous Deep-Learning methods have been proposed for MRI reconstruction, however, they usually fail to reconstruct sharp details from the subsampled $k$-space data. To solve this problem, we propose a lightweight and accurate Edge Attention MRI Reconstruction Network (EAMRI) to reconstruct images with edge guidance. Specifically, we design an efficient Edge Prediction Network to directly predict accurate edges from the blurred image. Meanwhile, we propose a novel Edge Attention Module (EAM) to guide the image reconstruction utilizing the extracted edge priors, as inspired by the popular self-attention mechanism. EAM first projects the input image and edges into ${\rm Q_{image}},$ ${\rm K_{edge}},$ and ${\rm V_{image}},$ respectively. Then EAM pairs the ${\rm Q_{image}}$ with ${\rm K_{edge}}$ along the channel dimension, such that 1) it can search globally for the high-frequency image features that are activated by the edge priors; 2) the overall computation burdens are largely reduced compared with the traditional spatial-wise attention. With the help of EAM, the predicted edge priors can effectively guide the model to reconstruct high-quality MR images with accurate edges. Extensive experiments show that our proposed EAMRI outperforms other methods with fewer parameters and can recover more accurate edges.

We develop a new moving-water equilibria preserving partial relaxation (PR) scheme for the two-dimensional (2-D) Saint-Venant system of shallow water equations. The new scheme is a 2-D generalization of the one-dimensional (1-D) PR scheme recently proposed in [X. Liu, X. Chen, S. Jin, A. Kurganov, and H. Yu, SIAM J. Sci. Comput., 42 (2020), pp. A2206–A2229]. Our scheme is based on the PR approximation, which is designed in two steps. First, the geometric source terms are incorporated into the discharge fluxes, which results in a hyperbolic system with global fluxes. Second, the discharge equations are relaxed so that the nonlinearity is moved into the stiff right-hand side of the four added auxiliary equation. The obtained PR system is then numerically integrated using a semi-discrete hybrid upwind/central-upwind finite-volume method combined with an efficient semi-implicit ODE solver. The new 2-D PR scheme inherits the main advantages of the 1-D PR scheme: (i) no special treatment of the geometric source terms is required, (ii) no nonlinear (cubic) equations should be solved to obtain the point values of the water depth out of the reconstructed equilibrium variables. The performance of the proposed PR scheme is illustrated on a number of numerical examples, in which we demonstrate that the PR scheme not only capable of exactly preserving quasi 1-D moving-water steady states and accurately capturing their small perturbations, but can also handle genuinely 2-D steady states and their small perturbations in a non-oscillatory manner.

In this paper, we study three families of $C^m (m=0,1,2)$ finite element methods for one dimensional fourth-order equations. They include $C^0$ and $C^1$ Galerkin methods and a $C^2-C^0$ Petrov-Galerkin method. Existence, uniqueness and optimal error estimates of the numerical solution are established. A unified approach is proposed to study the superconvergence property of these methods. We prove that, for $k$th-order elements, the $C^0$ and $C^1$ finite element solutions and their derivative are superconvergent with rate $h^{2k−2} (k≥3)$ at all mesh nodes; while the solution of the $C^2-C^0$ Petrov-Galerkin method and its first- and second-order derivatives are superconvergent with rate $h^{2k−4} (k≥5)$ at all mesh nodes. Furthermore, interior superconvergence points for the $l$-${\rm th} (0≤l≤m+1)$ derivate approximations are also discovered, which are identified as roots of special Jacobi polynomials, Lobatto points, and Gauss points. As a by-product, we prove that the $C^m$ finite element solution is superconvergent towards a particular Jacobi projection of the exact solution in the $H^l (0≤ l ≤ m+1)$ norms. All theoretical findings are confirmed by numerical experiments.

This note provides a correction of a missing weight constant in the MscaleDNN formula and some comments on the performance of the corrected algorithm.