Provably Efficient Algorithms for Numerical Tensor Algebra

This article aims to fill in the gap of the second-order accurate schemes for the time-fractional subdiffusion equation with unconditional stability. Two fully discrete schemes are first proposed for the time-fractional subdiffusion equation with space discretized by finite element and time discretized by the fractional linear multistep methods. These two methods are unconditionally stable with...

Nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension. The separated representation, introduced previously, allows many operations to be performed with scaling that is formally linear in the dimension. In this paper we further develop this representation by (i) discussing the variety of mechanisms that allow it to b...

Abstract. The performance of numerical algorithms, both regarding stability and complexity, can be understood in a unified way in terms of condition numbers. This requires to identify the appropriate geometric settings and to characterize condition in geometric ways. A probabilistic analysis of numerical algorithms can be reduced to a corresponding analysis of condition numbers, which leads to ...

-In this paper, the authors combine two algorithms for application to the recognition of unconstrained isolated handwritten numerals. The first algorithm employs a modified quadratic discriminant function utilizing direction sensitive spatial features of the numeral image. The second algorithm utilizes features derived from the profile of the character in a structural configuration to recognize...

Practical methods of differentiating a signal known only through its on-line samples are much needed, given the numerous areas in control theory and practice where differentiation is encountered. This communication presents theoretical as well as implementation details on several numerical differentiation algorithms which may be useful in the area of nonlinear estimation. In particular, these a...

We propose an efficient numerical algorithm for relative error model reduction based on balanced stochastic truncation. The method uses full-rank factors of the Gramians to be balanced versus each other and exploits the fact that for large-scale systems these Gramians are often of low numerical rank. We use the easy-to-parallelize sign function method as the major computational tool in determin...

We present cache-efficient algorithms for scientific computations using graphics processing units (GPUs). Our approach is based on mapping the nested loops in the numerical algorithms to the texture mapping hardware and efficiently utilizing GPU caches. This mapping exploits the inherent parallelism, pipelining and high memory bandwidth on GPUs. We further improve the performance of numerical a...

Automatic diierentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. This article discusses the application of automatic diierentiation to numerical integration algorithms for ordinary diierential equations (ODEs), in particular, the ramiications of the fact that AD is applied not only to the solution of such an algorith...