Circulant matrix

About: Circulant matrix is a research topic. Over the lifetime, 3476 publications have been published within this topic receiving 50116 citations.

Fast and Exact Simulation of Stationary Gaussian Processes through Circulant Embedding of the Covariance Matrix

Abstract: Geostatistical simulations often require the generation of numerous realizations of a stationary Gaussian process over a regularly meshed sample grid $\Omega$. This paper shows that for many important correlation functions in geostatistics, realizations of the associated process over $m+1$ equispaced points on a line can be produced at the cost of an initial FFT of length $2m$ with each new realization requiring an additional FFT of the same length. In particular, the paper first notes that if an $(m+1)\times(m+1) $ Toeplitz correlation matrix $R$ can be embedded in a nonnegative definite $2M\times2M$ circulant matrix $S$, exact realizations of the normal multivariate $y \sim {\cal N}(0,R)$ can be generated via FFTs of length $2M$. Theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which $M = m$ is nonnegative definite. Extensions to simulations of stationary fields in higher dimensions are also provided and illustrated.

An Optimal Circulant Preconditioner for Toeplitz Systems

Abstract: Given a Toeplitz matrix A, we derive an optimal circulant preconditioner C in the sense of minimizing ${\|C - A\|}_F $. It is in general different from the one proposed earlier by Strang [“A proposal for Toeplitz matrix calculations,” Stud. Appl. Math., 74(1986), pp. 171–176], except in the case when A is itself circulant. The new preconditioner is easy to compute and in preliminary numerical experiments performs better than Strang's preconditioner in terms of reducing the condition number of $C^{ - 1} A$ and comparably in terms of clustering the spectrum around unity.

Simulation of Stationary Gaussian Processes in [0, 1] d

Abstract: A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1]d ⊂ℝd is described It is assumed that the process is stationary with respect to translations of ℝd, but the method does not require the process to be isotropic As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝd, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝd, is strictly positive The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer An approximate procedure is also proposed to cover cases where it is not feasible to apply

Graph Clustering Via a Discrete Uncoupling Process

Abstract: A discrete uncoupling process for finite spaces is introduced, called the Markov Cluster Process or the MCL process. The process is the engine for the graph clustering algorithm called the MCL algorithm. The MCL process takes a stochastic matrix as input, and then alternates expansion and inflation, each step defining a stochastic matrix in terms of the previous one. Expansion corresponds with taking the $k$th power of a stochastic matrix, where $k\in\N$. Inflation corresponds with a parametrized operator $\Gamma_r$, $r\geq 0$, that maps the set of (column) stochastic matrices onto itself. The image $\Gamma_r M$ is obtained by raising each entry in $M$ to the $r$th power and rescaling each column to have sum 1 again. In practice the process converges very fast towards a limit that is invariant under both matrix multiplication and inflation, with quadratic convergence around the limit points. The heuristic behind the process is its expected behavior for (Markov) graphs possessing cluster structure. The process is typically applied to the matrix of random walks on a given graph $G$, and the connected components of (the graph associated with) the process limit generically allow a clustering interpretation of $G$. The limit is in general extremely sparse and iterands are sparse in a weighted sense, implying that the MCL algorithm is very fast and highly scalable. Several mathematical properties of the MCL process are established. Most notably, the process (and algorithm) iterands posses structural properties generalizing the mapping from process limits onto clusterings. The inflation operator $\Gamma_r$ maps the class of matrices that are diagonally similar to a symmetric matrix onto itself. The phrase diagonally positive semi-definite (dpsd) is used for matrices that are diagonally similar to a positive semi-definite matrix. For $r\in\N$ and for $M$ a stochastic dpsd matrix, the image $\Gamma_r M$ is again dpsd. Determinantal inequalities satisfied by a dpsd matrix $M$ imply a natural ordering among the diagonal elements of $M$, generalizing the mapping of process limits onto clusterings. The spectrum of $\Gamma_{\infty} M$ is of the form $\{0^{n-k}, 1^k\}$, where $k$ is the number of endclasses of the ordering associated with $M$, and $n$ is the dimension of $M$. This attests to the uncoupling effect of the inflation operator.