In this paper we propose an algorithm for solving the large-scale discrete ill-conditioned linear problems arising from the discretization of linear or nonlinear inverse problems. The algorithm combines some existing regularization techniques and regularization parameter choice rules with a randomized singular value decomposition (SVD) so that only much smaller-scale systems are needed to solve, instead of the original large-scale regularized system. The algorithm can directly apply to some existing regularization methods such as the Tikhonov and truncated SVD methods, with some popular regularization parameter choice rules such as the L-curve, GCV function, quasi-optimality and discrepancy principle. The error of the approximate regularized solution is analyzed and the efficiency of the method is well demonstrated by the numerical examples.

Abstract. Coarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichletto-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings.

Abstract. Since the cost of communication (moving data) greatly exceeds the cost of doing arithmetic on current and future computing platforms, we are motivated to devise algorithms that communicate as little as possible, even if they do slightly more arithmetic, and as long as they still get the right answer. This paper is about getting the right answer for such an algorithm.It discusses CALU, a communication avoiding LU factorization algorithm based on a new pivoting strategy, that we refer to as ca-pivoting. The reason to consider CALU is that it does an optimal amount of communication, and asymptotically less than Gaussian elimination with partial pivoting (GEPP), and so will be much faster on platforms where communication is expensive, as shown in previous work.We show that the Schur complement obtained after each step of performing CALU on a matrix A is the same as the Schur complement obtained after performing GEPP on a larger matrix whose entries are the same as the entries of A (sometimes slightly perturbed) and zeros. Hence we expect that CALU will behave as GEPP and it will be also very stable in practice. In addition, extensive experiments on random matrices and a set of special matrices show that CALU is stable in practice. The upper bound on the growth factor of CALU is worse than of GEPP. However, we present examples showing that neither GEPP nor CALU is uniformly more stable than the other.Key words. LU factorization, communication optimal algorithm, numerical stability AMS subject classifications. 65F50, 65F05, 68R101. Introduction. In this paper we discuss CALU, a communication avoiding LU factorization algorithm. The main part of the paper focuses on showing that CALU is stable in practice. We also show that CALU minimizes communication. For this, we use lower bounds on communication for dense LU factorization that were introduced in [5]. These bounds were obtained by showing through reduction that lower bounds on dense matrix multiplication [15,16] represent lower bounds for dense LU factorization as well. These bounds show that a sequential algorithm that computes the LU factorization of a dense n× n matrix transfers between slow and fast memory at least Ω(n 3 /W 1/2 ) number of words and Ω(n 3 /W 3/2 ) number of messages, where W denotes the fast memory size and we assume a message consists of at most W words in consecutive memory locations. On a parallel machine with P processors, if we consider that the local memory size used on each processor is on the order of n 2 /P , so a lower bound on the number of words is Ω(n 2 / √ P ) and a lower bound on the number of messages is Ω( √ P ). Here we consider square matrices, but later we consider the more general case of an m × n matrix.Gaussian elimination with partial pivoting (GEPP) is one of the most stable algorithms for solving a linear system through LU factorization. At each step of the algorithm, the maximum element in each column of L is permuted in diagonal position and used as a pivot. Efficient implementations of this algorithm exist ...

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