Circulant Embedding Research Articles

Evaluating the impact of uncertainty in ground motion forecasts for post-earthquake impact modeling applications

The US Geological Survey’s (USGS) ShakeMap system provides a rapid characterization of strong ground shaking in areas directly affected by an earthquake. This study focuses on studying the aggregate effects of macroseismic shaking estimates from ShakeMap, expressed in terms of modified Mercalli intensity (MMI), when accounting for the uncertainty in forecasted ground motions. We use a Monte Carlo approach to generate numerous spatially correlated realizations of ground motions by utilizing a combination of circulant embedding and kriging techniques for efficiently handling the correlations. We then assessed the aggregate effects of shaking by looking at bin counts across these realizations. We demonstrate that the aggregate shaking regarding the mean macroseismic intensity estimates (from the ShakeMap output) is a biased representation of the aggregate shaking when shaking uncertainty is included. Incorporating shaking uncertainty can help to improve various downstream earthquake impact applications, such as the USGS Prompt Assessment of Global Earthquakes for Response (PAGER) overall earthquake fatality distribution or estimates of shaking-induced ground failure impacts from consequential earthquakes.

Thermally activated particle motion in biased correlated Gaussian disorder potentials.

Thermally activated particle motion in disorder potentials is controlled by the large-ΔV tail of the distribution of height ΔV of the potential barriers created by the disorder. We employ the optimal fluctuation method to evaluate this tail for correlated quenched Gaussian potentials in one dimension in the presence of a small bias of the potential. We focus on the mean escape time (MET) of overdamped particles averaged over the disorder. We show that the bias leads to a strong (exponential) reduction of the MET in the direction along the bias. The reduction depends both on the bias and on detailed properties of the covariance of the disorder, such as its derivatives and asymptotic behavior at large distances. We verify our theoretical predictions for the large-ΔV tail of the barrier height distribution, as well as earlier predictions of this tail for zero bias, by performing large-deviation simulations of the potential disorder. The simulations employ correlated random potential sampling based on the circulant embedding method and the Wang-Landau algorithm, which enable us to probe probability densities smaller than 10^{-1200}.

Initialization of the Circulant Embedding method to speed up the generation of Gaussian random fields

The Circulant Embedding Method (CEM) is a well known technique to generate stationary Gaussian Random Fields (GRF). The main idea is to embed the covariance matrix in a larger nested block circulant matrix, whose factorization can be rapidly computed thanks to the fast Fourier transform (FFT) algorithm. The CEM requires the extended matrix to be at least positive semidefinite which is proven to be the case if the enclosing domain is sufficiently large, as proven by Theorem 2.3 in [9] for cubic domains. In this paper, we generalize this theorem to the case of rectangular parallelepipeds. Then we propose a new initialization stage of the CEM algorithm that makes it possible to quickly jump to a domain size close to the one needed for the CEM algorithm to work. These domain size estimates are based on fitting functions. Examples of fitting functions are given for the Matérn family of covariances. These functions are inspired by our numerical simulations and by the theoretical work from [9]. The parameters estimation of the fitting functions is done numerically. Several numerical tests are performed to show the efficiency of the proposed algorithms, for both isotropic and anisotropic Matérn covariances.

Adapting conditional simulation using circulant embedding for irregularly spaced spatial data

Computing an ensemble of random fields using conditional simulation is an ideal method for retrieving accurate estimates of a field conditioned on available data and for quantifying the uncertainty of these realizations. Methods for generating random realizations, however, are computationally demanding, especially when the estimates are conditioned on numerous observed data and for large domains. In this article, a new, approximate conditional simulation approach is applied that builds on circulant embedding (CE), a fast method for simulating stationary Gaussian processes. The standard CE is restricted to simulating stationary Gaussian processes (possibly anisotropic) on regularly spaced grids. In this work, we explore two possible algorithms, namely, local Kriging and approximate grid embedding, that extend CE for irregularly spaced data points. We establish the accuracy of these methods to be suitable for practical inference and the speedup in computation allows for generating conditional fields close to an interactive time frame. The methods are motivated by the U.S. Geological Survey's software ShakeMap, which provides near real‐time maps of shaking intensity after the occurrence of a significant earthquake. An example for the 2019 event in Ridgecrest, California, is used to illustrate our method.

An efficient numerical approach for stochastic evolution PDEs driven by random diffusion coefficients and multiplicative noise

<abstract><p>In this paper, we investigate the stochastic evolution equations (SEEs) driven by a bounded $ \log $-Whittle-Mat$ \acute{{\mathrm{e}}} $rn (W-M) random diffusion coefficient field and $ Q $-Wiener multiplicative force noise. First, the well-posedness of the underlying equations is established by proving the existence, uniqueness, and stability of the mild solution. A sampling approach called approximation circulant embedding with padding is proposed to sample the random coefficient field. Then a spatio-temporal discretization method based on semi-implicit Euler-Maruyama scheme and finite element method is constructed and analyzed. An estimate for the strong convergence rate is derived. Numerical experiments are finally reported to confirm the theoretical result.</p></abstract>

Random Toeplitz matrices: The condition number under high stochastic dependence

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

Hurst exponent estimation of fractional surfaces for mammogram images analysis

The work presents a methodology to precise simulation and parameter estimation of multidimensional fractional Brownian motion (fBm). The simulation approach uses circulant embedding algorithm and solution of Poisson equation, while generalizing it to multiple dimensions. For estimation, a method using Wishart distribution and maximum likelihood is presented and verified on simulated data. Unlike approximate methods for generating multidimensional fBm and its Hurst exponent estimation, this approach shows unbiased results for all processes with short memory and majority of cases with long memory. The methodology is applied to mammography screening images to find significant differences between benign and cancerous breast lumps.

Nonparametric spectral methods for multivariate spatial and spatial–temporal data

We propose computationally efficient methods for estimating stationary multivariate spatial and spatial–temporal spectra from incomplete gridded data. The methods are iterative and rely on successive imputation of data and updating of model estimates. Imputations are done according to a periodic model on an expanded domain. The periodicity of the imputations is a key feature that reduces edge effects in the periodogram and is facilitated by efficient circulant embedding techniques. In addition, we describe efficient methods for decomposing the estimated cross spectral density function into a linear model of coregionalization plus a residual process. The methods are applied to two storm datasets, one of which is from Hurricane Florence, which struck the southeastern United States in September 2018. The application demonstrates how fitted models from different datasets can be compared, and how the methods are computationally feasible on datasets with more than 200,000 total observations.

Estimation of Macrodispersivity in Bounded Formations by Circulant Embedding and Analysis of Variance

AbstractWe present a new method to estimate the displacement covariance and macrodispersivity for solute transport in bounded formations. Here we use circulant embedding, which is based on fast Fourier transform and is much more efficient than eigen‐decomposition, for the factorization of random spatial fields. We compute the displacement covariances by the analysis of variance approach, and introduce an interpolation process to significantly reduce the number of forward simulations. Once the effect of each eigenvector on the displacement covariance is obtained, it is unnecessary to rerun the simulator for different spatial covariance functions or anisotropy ratios, which saves a large amount of computational effort. The proposed method is validated in various tests in two‐dimensional and three‐dimensional examples and accurately matches the results from the Monte Carlo simulation. It is found that the longitudinal dispersivity is less sensitive to the impervious boundaries, while the transverse and vertical dispersivities are greatly affected. The method is applied to the Borden site and provides a better explanation of the observed data after considering the effect of vertical boundaries. These results show that our method could serve as a promising tool for studying and predicting the characteristics of solute transport in heterogeneous media.

Estimation of Macrodispersivity in Bounded Formations by Circulant Embedding and Analysis of Variance

Estimation of Macrodispersivity in Bounded Formations by Circulant Embedding and Analysis of Variance

Efficient Bayesian shape-restricted function estimation with constrained Gaussian process priors

This article revisits the problem of Bayesian shape-restricted inference in the light of a recently developed approximate Gaussian process that admits an equivalent formulation of the shape constraints in terms of the basis coefficients. We propose a strategy to efficiently sample from the resulting constrained posterior by absorbing a smooth relaxation of the constraint in the likelihood and using circulant embedding techniques to sample from the unconstrained modified prior. We additionally pay careful attention to mitigate the computational complexity arising from updating hyperparameters within the covariance kernel of the Gaussian process. The developed algorithm is shown to be accurate and highly efficient in simulated and real data examples.

Unified Analysis of Periodization-Based Sampling Methods for Matérn Covariances

The periodization of a stationary Gaussian random field on a sufficiently large torus comprising the spatial domain of interest is the basis of various efficient computational methods, such as the classical circulant embedding technique using the fast Fourier transform for generating samples on uniform grids. For the family of Matérn covariances with smoothness index $\nu$ and correlation length $\lambda$, we analyze the nonsmooth periodization (corresponding to classical circulant embedding) and an alternative procedure using a smooth truncation of the covariance function. We solve two open problems: the first concerning the $\nu$-dependent asymptotic decay of eigenvalues of the resulting circulant in the nonsmooth case, the second concerning the required size in terms of $\nu$, $\lambda$ of the torus when using a smooth periodization. In doing this we arrive at a complete characterization of the performance of these two approaches. Both our theoretical estimates and the numerical tests provided here show substantial advantages of smooth truncation.

Development of hybrid low salinity water polymer flooding numerical reservoir simulator and smart proxy model for chemical enhanced oil recovery (CEOR)

A MATLAB based finite volume CEOR simulator was developed for modeling the recovery of oil by low salinity water polymer flooding (LSWPF) in a proposed geostatistical method of generating porous media. Firstly, this paper coupled ion exchange, wettability and fluid flow transport models to the proposed heterogeneous porous media to investigate LSWPF oil recovery performance. Secondly, for the first time, a new hybrid smart proxy model of particle swarm optimization artificial neural network (PSO-ANN) recovery factor (RF) correlation was derived for LSWPF oil recovery performance using the extracted weights and biases. The proposed porous media as well as stochastic multi-fractal random field and circulant embedding method (CEM) permeability field realizations did not exhibit artificial banding unlike turning bands method (TBM) realizations when compared. A fast convergence pre-conditioner solution technique of algebraic multigrid (AMG) linear solver combined with an acceleration method of biconjugate gradients stabilized method (BI-CGSTAB) was used to solve the fully implicit system of coupled linearized equations. The simulator was validated with PHREEQC geochemical package and a secondary LSWPF coreflood test with outstanding history match. Numerical 2D examples study in this paper revealed the synergetic effects of LSWPF and high salinity water polymer flooding (HSWPF) with higher oil RFs of 39% and 31%, respectively than the standalone LSWF of 23% and normal waterflood of 11% for 50 days of production. The optimal architecture of the PSO-ANN RF model is 7 inputs, 3 hidden neurons and 1 output. Also, the PSO-ANN RF model predictions were compared to the LSWPF numerical simulation results with the former remarkably having reduced computational time in seconds with training and blind testing errors <1% average absolute percentage error (AAPE) for various permeability field realizations investigated. Finally, the PSO-ANN RF model will serve as a fast-alternative reservoir management and planning tool to supplement the high-fidelity LSWPF CEOR simulator developed to minimize computational overheads, risks and uncertainties in oil recovery performance modeling.

Spectral density estimation for random fields via periodic embeddings

We introduce methods for estimating the spectral density of a random field on a [Formula: see text]-dimensional lattice from incomplete gridded data. Data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function. The imputations are convenient computationally, in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in [Formula: see text] time and [Formula: see text] memory. However, these so-called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates. In addition, we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias. The paper contains theoretical results on properties of the imputed-data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios. We present an application to a gridded satellite surface temperature dataset with missing values.

Fast random field generation with H-matrices

We use the H-matrix technology to compute the approximate square root of a covariance matrix in linear cost. This allows us to generate normal and log-normal random fields on general point sets with optimal cost. We derive rigorous error estimates which show convergence of the method. Our approach requires only mild assumptions on the covariance function and on the point set. Therefore, it might be also a nice alternative to the circulant embedding approach which applies only to regular grids and stationary covariance functions.

Fast and exact synthesis of some operator scaling Gaussian random fields

Abstract Operator scaling Gaussian random fields, as anisotropic generalizations of self-similar fields, know an increasing interest for theoretical studies in the literature. However, up to now, they were only defined through stochastic integrals, without explicit covariance functions. In this paper we exhibit explicit covariance functions, as anisotropic generalizations of fractional Brownian fields ones, and define corresponding Operator scaling Gaussian random fields. This allows us to propose a fast and exact method of simulation in dimension 2 based on the circulant embedding matrix method, following ideas of Stein [34] for fractional Brownian surfaces syntheses. This is a first piece of work to popularize these models in anisotropic spatial data modeling.

Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

In a previous paper (Graham et al. in J Comput Phys 230:3668–3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

Sparse Circulant Binary Embedding: An Asymptotic Analysis

Binary embedding refers to methods for embedding points in $\mathbb {R}^{d}$ into vertices in the Hamming cube of dimension ${\mathcal{O}}(d)$ , such that the normalized Hamming distance between the codes preserves a prespecified distance between vectors in the original space. A common approach to binary embedding is to use random projection , followed by one-bit quantization to produce binary codes. Of particular interest, in this letter, is sparse circulant binary embedding (SCBE), where a sparse random circulant matrix with random sampling at a rate of $1-s$ for $s\in (0, 1)$ is used for random projection. The SCBE has the space complexity ${\mathcal{O}}((1-s)d)$ , while unstructured random projection has the space complexity ${\mathcal{O}}(d^2)$ . We present an asymptotic analysis of SCBE, when $d$ approaches $\infty$ , showing that the performance of SCBE is comparable to that of binary embedding with unstructured random projection whilethe former has the space complexity ${\mathcal{O}}((1-s)d)$ and the time complexity ${\mathcal{O}}(d\log d)$ but the latter has both space and time complexities ${\mathcal{O}}(d^2)$ .

Efficient Modeling of Gravity Fields Caused by Sources with Arbitrary Geometry and Arbitrary Density Distribution

We present a brief review of gravity forward algorithms in Cartesian coordinate system, including both space-domain and Fourier-domain approaches, after which we introduce a truly general and efficient algorithm, namely the convolution-type Gauss fast Fourier transform (Conv-Gauss-FFT) algorithm, for 2D and 3D modeling of gravity potential and its derivatives due to sources with arbitrary geometry and arbitrary density distribution which are defined either by discrete or by continuous functions. The Conv-Gauss-FFT algorithm is based on the combined use of a hybrid rectangle-Gaussian grid and the fast Fourier transform (FFT) algorithm. Since the gravity forward problem in Cartesian coordinate system can be expressed as continuous convolution-type integrals, we first approximate the continuous convolution by a weighted sum of a series of shifted discrete convolutions, and then each shifted discrete convolution, which is essentially a Toeplitz system, is calculated efficiently and accurately by combining circulant embedding with the FFT algorithm. Synthetic and real model tests show that the Conv-Gauss-FFT algorithm can obtain high-precision forward results very efficiently for almost any practical model, and it works especially well for complex 3D models when gravity fields on large 3D regular grids are needed.

Fast and exact simulation of univariate and bivariate Gaussian random fields

Circulant embedding is a powerful algorithm for fast simulation of stationary Gaussian random fields on a rectangular grid in , which works perfectly for compactly supported covariance functions. Cut‐off circulant embedding techniques have been developed for univariate random fields for dimensions up to and rely on the modification of a covariance function outside the simulation window, such that the modified covariance function is compactly supported. In this paper, we propose extensions of the cut‐off approach for univariate and bivariate Gaussian random fields. In particular, we provide a method for simulating bivariate fields with a powered exponential model and the Matérn model for certain sets of parameters. Copyright © 2018 John Wiley &amp; Sons, Ltd.