Efficient Algorithms for CUR and Interpolative Matrix Decompositions

The manuscript describes efficient algorithms for the computation of the CUR and ID decompositions. The methods used are based on simple modifications to the classical truncated pivoted QR decomposition, which means that highly optimized library codes can be utilized for implementation. For certain applications, further acceleration can be attained by incorporating techniques based on randomized projections. Numerical experiments demonstrate advantageous performance compared to existing techniques for computing CUR factorizations

Survey of Computational Methods for Inverse Problems

Inverse problems occur in a wide range of scientific applications, such as in the fields of signal processing, medical imaging, or geophysics. This work aims to present to the field practitioners, in an accessible and concise way, several established and newer cutting-edge computational methods used in the field of inverse problems—and when and how these techniques should be employed

Compression approaches for the regularized solutions of linear systems from large-scale inverse problems

We introduce and compare new compression approaches to obtain regularized solutions of large linear systems which are commonly encountered in large scale inverse problems. We first describe how to approximate matrix vector operations with a large matrix through a sparser matrix with fewer nonzero elements, by borrowing from ideas used in wavelet image compression. Next, we describe and compare approaches based on the use of the low rank singular value decomposition (SVD), which can result in further size reductions. We describe how to obtain the approximate low rank SVD of the original matrix using the sparser wavelet compressed matrix. Some analytical results concerning the various methods are presented and the results of the proposed techniques are illustrated using both synthetic data and a very large linear system from a seismic tomography application, where we obtain significant compression gains with our methods, while still resolving the main features of the solutions.European Research Council (Advanced Grant 226837)United States. Defense Advanced Research Projects Agency (Contract N66001-13-1-4050)National Science Foundation (U.S.) (Contracts 1320652 and 0748488

Introduction to Numerical Approaches for Forward and Inverse Heat Transfer Problems

We describe the formulation of forward and inverse problems in heat transfer and illustrate several techniques applicable to their solution

Testing the Annular Mode Autocorrelation Time Scale in Simple Atmospheric General Circulation Models

A new diagnostic for measuring the ability of atmospheric models to reproduce realistic low-frequency variability is introduced in the context of Held and Suarez’s 1994 proposal for comparing the dynamics of different general circulation models. A simple procedure to compute τ, the e-folding time scale of the annular mode autocorrelation function, is presented. This quantity concisely quantifies the strength of low-frequency variability in a model and is easy to compute in practice. The sensitivity of τ to model numerics is then studied for two dry primitive equation models driven with the Held–Suarez forcings: one pseudospectral and the other finite volume. For both models, τ is found to be unrealistically large when the horizontal resolutions are low, such as those that are often used in studies in which long integrations are needed to analyze model variability on low frequencies. More surprising is that it is found that, for the pseudospectral model, τ is particularly sensitive to vertical resolution, especially with a triangular truncation at wavenumber 42 (a very common resolution choice). At sufficiently high resolution, the annular mode autocorrelation time scale τ in both models appears to converge around values of 20–25 days, suggesting the existence of an intrinsic time scale at which the extratropical jet vacillates in the Held and Suarez system. The importance of τ for computing the correct response of a model to climate change is explicitly demonstrated by perturbing the pseudospectral model with simple torques. The amplitude of the model’s response to external forcing increases as τ increases, as suggested by the fluctuation–dissipation theorem

Randomized Matrix Decompositions Using R

Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality reduction, and data compression. Massive datasets, however, pose a computational challenge for traditional algorithms, placing significant constraints on both memory and processing power. Recently, the powerful concept of randomness has been introduced as a strategy to ease the computational load. The essential idea of probabilistic algorithms is to employ some amount of randomness in order to derive a smaller matrix from a high-dimensional data matrix. The smaller matrix is then used to compute the desired low-rank approximation. Such algorithms are shown to be computationally efficient for approximating matrices with low-rank structure. We present the R package rsvd, and provide a tutorial introduction to randomized matrix decompositions. Specifically, randomized routines for the singular value decomposition, (robust) principal component analysis, interpolative decomposition, and CUR decomposition are discussed. Several examples demonstrate the routines, and show the computational advantage over other methods implemented in R