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Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such as SVD and CPQR, are either prohibitively expensive or cannot provide sufficiently accurate results. A solution is to use randomized low-rank matrix approximation methods such as randomized SVD , and randomized LU decomposition on extremely large data sets. In this paper, we focus on the randomized LU decomposition method. First, we employ a reorthogonalization procedure to perform the power iteration of the existing randomized LU algorithm to compensate for the rounding errors caused by the power method. Then we propose a novel randomized LU algorithm, called PowerLU, for the fixed low-rank approximation problem. PowerLU allows for an arbitrary number of passes of the input matrix, $v \geq 2$. Recall that the existing randomized LU decomposition only allows an even number of passes. We prove the theoretical relationship between PowerLU and the existing randomized LU. Numerical experiments show that our proposed PowerLU is generally faster than the existing randomized LU decomposition, while remaining accurate. We also propose a version of PowerLU, called PowerLU_FP, for the fixed precision low-rank matrix approximation problem. PowerLU_FP is based on an efficient blocked adaptive rank determination Algorithm 4.1 proposed in this paper. We present numerical experiments that show that PowerLU_FP can achieve almost the same accuracy and is faster than the randomized blocked QB algorithm by Martinsson and Voronin. We finally propose a single-pass algorithm based on LU factorization. Tests show that the accuracy of our single-pass algorithm is comparable with the existing single-pass algorithms., Comment: 28 pages, 9 figures

In this article, we consider the general problem of checking the correctness of matrix multiplication. Given three $n \times n$ matrices $A$, $B$, and $C$, the goal is to verify that $A \times B=C$ without carrying out the computationally costly operations of matrix multiplication and comparing the product $A \times B$ with $C$, term by term. This is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors. Here we extend Freivalds' algorithm to a Gaussian Variant of Freivalds' Algorithm (GVFA) by projecting the product $A \times B$ as well as $C$ onto a Gaussian random vector and then comparing the resulting vectors. The computational complexity of GVFA is consistent with that of Freivalds' algorithm, which is $O(n^{2})$. However, unlike Freivalds' algorithm, whose probability of a false positive is $2^{-k}$, where $k$ is the number of iterations. Our theoretical analysis shows that when $A \times B \neq C$, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic. When we introduce round-off error and floating point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives. Moreover, by iterating GVFA $k$ times, the probability of a false positive decreases as $p^k$, where $p$ is a very small value depending on the nature of the fault on the result matrix and the arithmetic system's floating-point precision. Unlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA. Thus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures. In particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations.

We describe two efficient methods of estimating the fluid permeability of standard models of porous media by using the statistics of continuous Brownian motion paths that initiate outside a sample and terminate on contacting the porous sample. The first method associates the "penetration depth" with a specific property of the Brownian paths, then uses the standard relation between penetration depth and permeability to calculate the latter. The second method uses Brownian paths to calculate an effective capacitance for the sample, then relates the capacitance, via angle-averaging theorems to the translational hydrodynamic friction of the sample. Finally, a result of Felderhof is used to relate the latter quantity to the permeability of the sample. We find that the penetration depth method is highly accurate in predicting permeability of porous material.

A discrete random walk method on grids was proposed and used to solve the linearized Poisson-Boltzmann equation (LPBE) \cite{Rammile}. Here, we present a new and efficient grid-free random walk method. Based on a modified `` Walk On Spheres" (WOS) algorithm \cite{Elepov-Mihailov1973} for the LPBE, this Monte Carlo algorithm uses a survival probability distribution function for the random walker in a continuous and free diffusion region. The new simulation method is illustrated by computing four analytically solvable problems. In all cases, excellent agreement is observed., Comment: 12 pages, 5 figures