Relaxation methods such as Jacobi or Gauss-Seidel are often applied as smoothers in algebraic multigrid. Incomplete factorizations can also be employed, however, direct triangular solves are comparatively slow on GPUs. Previous work by Antz et al. \cite{Anzt2015} proposed an iterative approach for solving such sparse triangular systems. However, when using the stationary Jacobi iteration, if the upper or lower triangular factor is highly non-normal, the iterations will diverge. An ILUT smoother is introduced for classical Ruge-St\"uben C-AMG that applies Ruiz scaling to mitigate the non-normality of the upper triangular factor. Our approach facilitates the use of Jacobi iteration in place of the inherently sequential triangular solve. Because the scaling is applied to the upper triangular factor as opposed to the global matrix, it can be done locally on an MPI-rank for a diagonal block of the global matrix. A performance model is provided along with numerical results for matrices extracted from the PeleLM \cite{PeleLM} pressure continuity solver.
相關內容
We introduce and analyze various Regularized Combined Field Integral Equations (CFIER) formulations of time-harmonic Navier equations in media with piece-wise constant material properties. These formulations can be derived systematically starting from suitable coercive approximations of Dirichlet-to-Neumann operators (DtN), and we present a periodic pseudodifferential calculus framework within which the well posedness of CIER formulations can be established. We also use the DtN approximations to derive and analyze Optimized Schwarz (OS) methods for the solution of elastodynamics transmission problems. The pseudodifferential calculus we develop in this paper relies on careful singularity splittings of the kernels of Navier boundary integral operators which is also the basis of high-order Nystr\"om quadratures for their discretizations. Based on these high-order discretizations we investigate the rate of convergence of iterative solvers applied to CFIER and OS formulations of scattering and transmission problems. We present a variety of numerical results that illustrate that the CFIER methodology leads to important computational savings over the classical CFIE one, whenever iterative solvers are used for the solution of the ensuing discretized boundary integral equations. Finally, we show that the OS methods are competitive in the high-frequency high-contrast regime.
Given a set $P$ of $n$ points in the plane, we consider the problem of computing the number of points of $P$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matou\v{s}ek's results, we can build a data structure of $O(n)$ space so that each query can be answered in $O(\sqrt{n})$ time. Our techniques lead to improvements for several other classical problems, such as batched range searching, counting/reporting intersecting pairs of unit circles, distance selection, discrete 2-center, etc. For example, given a set of $n$ unit disks and a set of $n$ points in the plane, the batched range searching problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $O(n^{4/3}\log n)$ time while our new algorithm runs in $O(n^{4/3})$ time.
In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that considering a time discretization with a positive step size $h$ an error bound of size $h$ can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size $k$ an error bound of size $O(k/h)$ can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under similar assumptions to those of the time discrete case, that the error of the fully discrete case is in fact $O(h+k)$ which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behaviour $1/h$ from the bound $O(k/h)$ have not been observed.
We study online convex optimization with switching costs, a practically important but also extremely challenging problem due to the lack of complete offline information. By tapping into the power of machine learning (ML) based optimizers, ML-augmented online algorithms (also referred to as expert calibration in this paper) have been emerging as state of the art, with provable worst-case performance guarantees. Nonetheless, by using the standard practice of training an ML model as a standalone optimizer and plugging it into an ML-augmented algorithm, the average cost performance can be even worse than purely using ML predictions. In order to address the "how to learn" challenge, we propose EC-L2O (expert-calibrated learning to optimize), which trains an ML-based optimizer by explicitly taking into account the downstream expert calibrator. To accomplish this, we propose a new differentiable expert calibrator that generalizes regularized online balanced descent and offers a provably better competitive ratio than pure ML predictions when the prediction error is large. For training, our loss function is a weighted sum of two different losses -- one minimizing the average ML prediction error for better robustness, and the other one minimizing the post-calibration average cost. We also provide theoretical analysis for EC-L2O, highlighting that expert calibration can be even beneficial for the average cost performance and that the high-percentile tail ratio of the cost achieved by EC-L2O to that of the offline optimal oracle (i.e., tail cost ratio) can be bounded. Finally, we test EC-L2O by running simulations for sustainable datacenter demand response. Our results demonstrate that EC-L2O can empirically achieve a lower average cost as well as a lower competitive ratio than the existing baseline algorithms.
This paper studies the application of reconfigurable intelligent surface (RIS) to cooperative non-orthogonal multiple access (C-NOMA) networks with simultaneous wireless information and power transfer (SWIPT). We aim for maximizing the rate of the strong user with guaranteed weak user's quality of service (QoS) by jointly optimizing power splitting factors, beamforming coefficients, and RIS reflection coefficients in two transmission phases. The formulated problem is difficult to solve due to its complex and non-convex constraints. To tackle this challenging problem, we first use alternating optimization (AO) framework to transform it into three subproblems, and then use the penalty-based arithmetic-geometric mean approximation (PBAGM) algorithm and the successive convex approximation (SCA)-based method to solve them. Numerical results verify the superiority of the proposed algorithm over the baseline schemes.
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). This article is to propose a Deep Learning Galerkin Method (DGM) for the closed-loop geothermal system, which is a new coupled multi-physics PDEs and mainly consists of a framework of underground heat exchange pipelines to extract the geothermal heat from the geothermal reservoir. This method is a natural combination of Galerkin Method and machine learning with the solution approximated by a neural network instead of a linear combination of basis functions. We train the neural network by randomly sampling the spatiotemporal points and minimize loss function to satisfy the differential operators, initial condition, boundary and interface conditions. Moreover, the approximate ability of the neural network is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in L^2 norm under certain conditions. Finally, some numerical examples are carried out to demonstrate the approximation ability of the neural networks intuitively.
Multigrid is a powerful solver for large-scale linear systems arising from discretized partial differential equations. The convergence theory of multigrid methods for symmetric positive definite problems has been well developed over the past decades, while, for nonsymmetric problems, such theory is still not mature. As a foundation for multigrid analysis, two-grid convergence theory plays an important role in motivating multigrid algorithms. Regarding two-grid methods for nonsymmetric problems, most previous works focus on the spectral radius of iteration matrix or rely on convergence measures that are typically difficult to compute in practice. Moreover, the existing results are confined to two-grid methods with exact solution of the coarse-grid system. In this paper, we analyze the convergence of a two-grid method for nonsymmetric positive definite problems (e.g., linear systems arising from the discretizations of convection-diffusion equations). In the case of exact coarse solver, we establish an elegant identity for characterizing two-grid convergence factor, which is measured by a smoother-induced norm. The identity can be conveniently used to derive a class of optimal restriction operators and analyze how the convergence factor is influenced by restriction. More generally, we present some convergence estimates for an inexact variant of the two-grid method, in which both linear and nonlinear coarse solvers are considered.
We study a class of enriched unfitted finite element or generalized finite element methods (GFEM) to solve a larger class of interface problems, that is, 1D elliptic interface problems with discontinuous solutions, including those having implicit or Robin-type interface jump conditions. The major challenge of GFEM development is to construct enrichment functions that capture the imposed discontinuity of the solution while keeping the condition number from fast growth. The linear stable generalized finite element method (SGFEM) was recently developed using one enrichment function. We generalized it to an arbitrary degree using two simple discontinuous one-sided enrichment functions. Optimal order convergence in the $L^2$ and broken $H^1$-norms are established. So is the optimal order convergence at all nodes. To prove the efficiency of the SGFEM, the enriched linear, quadratic, and cubic elements are applied to a multi-layer wall model for drug-eluting stents in which zero-flux jump conditions and implicit concentration interface conditions are both present.
A new numerical method for mean field games (MFGs) is proposed. The target MFGs are derived from optimal control problems for multidimensional systems with advection terms, which are difficult to solve numerically with existing methods. For such MFGs, linearization using the Cole-Hopf transformation and iterative computation using fictitious play are introduced. This leads to an implementation-friendly algorithm that iteratively solves explicit schemes. The convergence properties of the proposed scheme are mathematically proved by tracking the error of the variable through iterations. Numerical calculations show that the proposed method works stably for both one- and two-dimensional control problems.
Convolutional neural networks (CNNs) have shown dramatic improvements in single image super-resolution (SISR) by using large-scale external samples. Despite their remarkable performance based on the external dataset, they cannot exploit internal information within a specific image. Another problem is that they are applicable only to the specific condition of data that they are supervised. For instance, the low-resolution (LR) image should be a "bicubic" downsampled noise-free image from a high-resolution (HR) one. To address both issues, zero-shot super-resolution (ZSSR) has been proposed for flexible internal learning. However, they require thousands of gradient updates, i.e., long inference time. In this paper, we present Meta-Transfer Learning for Zero-Shot Super-Resolution (MZSR), which leverages ZSSR. Precisely, it is based on finding a generic initial parameter that is suitable for internal learning. Thus, we can exploit both external and internal information, where one single gradient update can yield quite considerable results. (See Figure 1). With our method, the network can quickly adapt to a given image condition. In this respect, our method can be applied to a large spectrum of image conditions within a fast adaptation process.
小貼士
登錄享
相關主題
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191