Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Saibaba, Arvind K.; Ambikasaran, Sivaram; Li, J. Yue; Kitanidis, Peter K.; Darve, Eric

doi:10.2516/ogst/2012064

Cited by 30 publications

(46 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…General limitations of the tensor technique are that 1) it could be time consuming to compute a low-rank tensor decomposition; 2) it requires axes-parallel mesh; 3) some theoretical estimations exist for functions depending on |x − y| (although more general functions have a low-rank representation in practice). During the last few years, there has been great interest in numerical methods for representing and approximating large covariance matrices [44,54,56,51,1,2,43]. Low-rank tensors were previously applied to accelerated kriging and spatial design by orders of magnitude [51].…”

Section: Introductionmentioning

confidence: 99%

“…The H-matrix techniques [19,23,21] provide the efficient data sparse approximation for the differential and integral operators in R d , d = 1, 2, 3. H-matrices are very robust for approximating the covariance matrix [38,56,2,26], [1,4], its inverse [1], and its Cholesky decomposition [38,44,43], but can also be expensive, especially for large n in 3D. Namely, the complexity in 3D will be C k d−1 N log N , where N = n d , d = 3, k ≪ n is the rank and C is a large constant which scales exponentially in dimension d, see [23].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

Litvinenko

Keyes

Khoromskaia

et al. 2018

Computational Methods in Applied Mathematics

View full text Add to dashboard Cite

In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matérn-and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential O(n d ) to a linear scaling O(drn), where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axesparallel) grid and that the covariance function depends on a distance, x − y .

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

Litvinenko

Keyes

Khoromskaia

et al. 2018

Computational Methods in Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Hierarchical matrices have been described in detail [29,28,24,30,45]. Applications of the H-matrix technique to the covariance matrices can be found in [3,65,67,11,32,1,2,41]. There are many implementations exist.…”

Section: Hierarchical Approximation Of Covariance Matricesmentioning

confidence: 99%

“…The white blocks are empty. In the last few years, there has been great interest in numerical methods for representing and approximating large covariance matrices in the applied mathematics community [60,11,67,56,1,2,69,5].…”

mentioning

confidence: 99%

“…In particular, they approximated Matérn covariance matrices in the H-matrix format, suggested the pivoted H-Cholesky method and provided an a posteriori error estimate in the trace norm. In [67,65], the authors applied H-matrices to linear inverse problems in large-scale geostatistics. They started with a detailed explanation of the H-matrix technique, then reduced the cost of dense matrix-vector products, combined H-matrices with a matrix-free Krylov subspace and solved the system of equations that arise from the geostatistical approach.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Likelihood approximation with hierarchical matrices for large spatial datasets

Litvinenko

Sun

Genton

et al. 2019

Computational Statistics & Data Analysis

View full text Add to dashboard Cite

We consider measurements to estimate the unknown parameters (variance, smoothness, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, we approximate the discretized covariance function in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and storage O(kn log n), where the rank k is a small integer, and n is the number of locations. The H-matrix technique allows us to work with general covariance matrices efficiently, since H-matrices can approximate inhomogeneous covariance functions, with a fairly general mesh that is not necessarily axes-parallel, and neither the covariance matrix itself nor its inverse has to be sparse. We research how the H-matrix approximation error influences on the estimated parameters. We demonstrate our method with Monte Carlo simulations with known true values of parameters and an application to soil moisture data with unknown parameters. The C, C++ codes and data are freely available. becomes challenging, due to the computation of the inverse and log-determinant of the n-by-n covariance matrix C(θ). Indeed, this requires O(n 2 ) memory and O(n 3 ) computational steps. Hence, scalable methods that can process larger sample sizes are needed.Stationary covariance functions, discretized on a rectangular grid, have block Toeplitz structure. This structure can be further extended to a block circulant form and resolved with the Fast Fourier Transform (FFT) [79,16,25,74,18]. The computing cost, in this case, is O(n log n). However, this approach either does not work for data measured at irregularly spaced locations or requires expensive, non-trivial modifications.During the past two decades, a large amount of research has been devoted to tackling the aforementioned computational challenge of developing scalable methods: for example, low-rank tensor methods [52,56], covariance tapering [21,38,68], likelihood approximations in both the spatial [72,73] and spectral [19] domains, latent processes such as Gaussian predictive processes [6] and fixed-rank kriging [15], and Gaussian Markov random-field approximations [64,63,20]; see [77] for a review. A generalization of the Vecchia approach and a general Vecchia framework was introduced in [37, 78]. Each of these methods has its strengths and drawbacks. For instance, covariance tapering sometimes performs even worse than assuming independent blocks in the covariance [70]; low-rank approximations have their own limitations [71]; and Markov models depend on the observation locations, requiring irregular locations to be realigned on a much finer grid with missing values [76]. A matrix-free approach for solving the multi-parametric Gaussian maximum likelihood problem was developed in [4]. To further improve on these issues, other methods that have been recently developed include the nearest-neighbor Gaussian process models [17], low-rank update [66], multiresolution Gaussian process models [57], equivalent ...

show abstract

A Kalman filter powered by H2-matrices for quasi-continuous data assimilation problems

Ambikasaran

Darve

et al. 2014

Water Resour. Res.

View full text Add to dashboard Cite

Continuously tracking the movement of a fluid or a plume in the subsurface is a challenge that is often encountered in applications, such as tracking a plume of injected CO 2 or of a hazardous substance. Advances in monitoring techniques have made it possible to collect measurements at a high frequency while the plume moves, which has the potential advantage of providing continuous high-resolution images of fluid flow with the aid of data processing. However, the applicability of this approach is limited by the high computational cost associated with having to analyze large data sets within the time constraints imposed by real-time monitoring. Existing data assimilation methods have computational requirements that increase superlinearly with the size of the unknowns m. In this paper, we present the HiKF, a new Kalman filter (KF) variant powered by the hierarchical matrix approach that dramatically reduces the computational and storage cost of the standard KF from Oðm 2 Þ to OðmÞ, while producing practically the same results. The version of HiKF that is presented here takes advantage of the so-called random walk dynamical model, which is tailored to a class of data assimilation problems in which measurements are collected quasi-continuously. The proposed method has been applied to a realistic CO 2 injection model and compared with the ensemble Kalman filter (EnKF). Numerical results show that HiKF can provide estimates that are more accurate than EnKF and also demonstrate the usefulness of modeling the system dynamics as a random walk in this context.

show abstract

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Cited by 30 publications

References 52 publications

Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

Tucker Tensor Analysis of Matérn Functions in Spatial Statistics

Likelihood approximation with hierarchical matrices for large spatial datasets

A Kalman filter powered by H2-matrices for quasi-continuous data assimilation problems

Contact Info

Product

Resources

About