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In this paper, we propose a semigroup method for solving high-dimensional elliptic partial differential equations (PDEs) and the associated eigenvalue problems based on neural networks. For the PDE problems, we reformulate the original equations as variational problems with the help of semigroup operators and then solve the variational problems with neural network (NN) parameterization. The main advantages are that no mixed second-order derivative computation is needed during the stochastic gradient descent training and that the boundary conditions are taken into account automatically by the semigroup operator. Unlike popular methods like PINN \cite{raissi2019physics} and Deep Ritz \cite{weinan2018deep} where the Dirichlet boundary condition is enforced solely through penalty functions and thus changes the true solution, the proposed method is able to address the boundary conditions without penalty functions and it gives the correct true solution even when penalty functions are added, thanks to the semigroup operator. For eigenvalue problems, a primal-dual method is proposed, efficiently resolving the constraint with a simple scalar dual variable and resulting in a faster algorithm compared with the BSDE solver \cite{han2020solving} in certain problems such as the eigenvalue problem associated with the linear Schr\"odinger operator. Numerical results are provided to demonstrate the performance of the proposed methods.

In this work, we determine the full expression for the global truncation error of hyperbolic partial differential equations (PDEs). In particular, we use theoretical analysis and symbolic algebra to find exact expressions for the coefficients of the generic global truncation error. Our analysis is valid for any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method, belonging to the Method of Lines. Furthermore, we discuss the practical implications of this analysis. If we employ a stable numerical scheme and the orders of accuracy of the global solution error and the global truncation error agree, we make the following asymptotic observations: (a) the order of convergence at constant ratio of $\Delta t$ to $\Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. An implication of (a) is that it is impractical to invest in a time-stepping method of order higher than the spatial discretization. In addition to (b), we demonstrate that under certain circumstances, the error can even monotonically increase with refinement only in space or only in time, and explain why this phenomenon occurs. To verify our theoretical findings, we conduct convergence studies of linear and non-linear advection equations using finite difference and finite volume spatial discretizations, and predictor-corrector and multistep time-stepping methods. Finally, we study the effect of slope limiters and monotonicity-preserving strategies on the order of accuracy.

Spin-Transfer Torque Magnetic RAM (STT-MRAM) is known as the most promising replacement for SRAM technology in large Last-Level Caches (LLCs). Despite its high-density, non-volatility, near-zero leakage power, and immunity to radiation as the major advantages, STT-MRAM-based cache suffers from high error rates mainly due to retention failure, read disturbance, and write failure. Existing studies are limited to estimating the rate of only one or two of these error types for STT-MRAM cache. However, the overall vulnerability of STT-MRAM caches, which its estimation is a must to design cost-efficient reliable caches, has not been offered in any of previous studies. In this paper, we propose a system-level framework for reliability exploration and characterization of errors behavior in STT-MRAM caches. To this end, we formulate the cache vulnerability considering the inter-correlation of the error types including all three errors as well as the dependency of error rates to workloads behavior and Process Variations (PVs). Our analysis reveals that STT-MRAM cache vulnerability is highly workload-dependent and varies by orders of magnitude in different cache access patterns. Our analytical study also shows that this vulnerability divergence significantly increases by process variations in STT-MRAM cells. To evaluate the framework, we implement the error types in the gem5 full-system simulator, and the experimental results show that the total error rate in a shared LLC varies by 32.0x for different workloads. A further 6.5x vulnerability variation is observed when considering PVs in the STT-MRAM cells. In addition, the contribution of each error type in total LLC vulnerability highly varies in different cache access patterns and moreover, error rates are differently affected by PVs.

The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nedelec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that a) kh/p is sufficient small and b) p/\ln k is bounded from below.

Optimal experimental design (OED) plays an important role in the problem of identifying uncertainty with limited experimental data. In many applications, we seek to minimize the uncertainty of a predicted quantity of interest (QoI) based on the solution of the inverse problem, rather than the inversion model parameter itself. In these scenarios, we develop an efficient method for goal-oriented optimal experimental design (GOOED) for large-scale Bayesian linear inverse problem that finds sensor locations to maximize the expected information gain (EIG) for a predicted QoI. By deriving a new formula to compute the EIG, exploiting low-rank structures of two appropriate operators, we are able to employ an online-offline decomposition scheme and a swapping greedy algorithm to maximize the EIG at a cost measured in model solutions that is independent of the problem dimensions. We provide detailed error analysis of the approximated EIG, and demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm for a contaminant transport inverse problem with infinite-dimensional parameter field.

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using Galerkin finite element method in space and finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem, and present a thorough error analysis for the reconstructed potential. Our derived \textsl{a priori} error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis.

In their article "Coupling at a distance HDG and BEM", Cockburn, Sayas and Solano proposed an iterative coupling of the hybridizable discontinuous Galerkin method (HDG) and the boundary element method (BEM) to solve an exterior Dirichlet problem. The novelty of the numerical scheme consisted of using a computational domain for the HDG discretization whose boundary did not coincide with the coupling interface. In their article, the authors provided extensive numerical evidence for convergence, but the proof of convergence and the error analysis remained elusive at that time. In this article we fill the gap by proving the convergence of a relaxation of the algorithm and providing a priori error estimates for the numerical solution.

Inclusion of a term $-\gamma \nabla \nabla \cdot u$, forcing $\nabla \cdot u$ to be pointwise small, is an effective tool for improving mass conservation in discretizations of incompressible flows. However, the added grad-div term couples all velocity components, decreases sparsity and increases the condition number in the linear systems that must be solved every time step. To address these three issues various sparse grad-div regularizations and a modular grad-div method have been developed. We develop and analyze herein a synthesis of a fully decoupled, parallel sparse grad-div method of Guermond and Minev with the modular grad-div method. Let $G^{\ast }=-diag(\partial _{x}^{2},\partial _{y}^{2},\partial _{z}^{2})$ denote the diagonal of $% G=-\nabla \nabla \cdot $, and $\alpha \geq 0$\ an adjustable parameter. The 2-step method considered is \begin{eqnarray} 1 &:&\frac{\widetilde{u}^{n+1}-u^{n}}{k}+u^{n}\cdot \nabla \widetilde{u}^{n+1}+\nabla p^{n+1}-\nu \Delta \widetilde{u}^{n+1}=f\text{ & }\nabla \cdot \widetilde{u}^{n+1}=0,\\ 2 &:&\left[ \frac{1}{k}I+(\gamma +\alpha )G^{\ast }\right] u^{n+1}=\frac{1}{k }\widetilde{u}^{n+1}+\left[ (\gamma +\alpha )G^{\ast }-\gamma G\right] u^{n}. \end{eqnarray} The analysis also establishes that the method controls the persistent size of $\Vert \nabla \cdot u \Vert$ in general and controls the transients in $\Vert\nabla \cdot u\Vert$ for a cold start when $\alpha >0.5\gamma $. Consistent numerical tests are presented.

In this paper we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems. We discuss the convergence properties of the method in the continuous case first and then apply the arguments to the finite difference discretization case. In both cases, we prove that the Schwarz alternating method is convergent if its counterpart for an elliptic equation is convergent. Meanwhile, the convergence rate of the method for the elliptic equation under the maximum norm also gives a uniform upper bound (with respect to the regularization parameter $\alpha$) of the convergence rate of the method for the optimal control problem under the maximum norm of proper error merit functions in the continuous case or vectors in the discrete case. Our numerical results corroborate our theoretical results and show that with $\alpha$ decreasing to zero, the method will converge faster. We also give some exposition of this phenomenon.

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair $\{A,L\}$, where we propose three approaches to compute approximate generalized singular values and vectors. We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of $\{A,L\}$. The results show that, under some mild conditions, the semiorthogonality of Lanczos type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We also establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.