## 13 Citations

### A diffusion-based spatio-temporal extension of Gaussian Mat\'ern fields

- Computer Science, Mathematics
- 2020

This work develops a spatio-temporal extension of the Gaussian Mat ´ ern ﬁelds formulated as solutions to a stochastic partial differential equation that provides a sparse representation based on a sparse element approximation that is well suited for statistical inference and which is implemented in the R-INLA software.

### The reproducing kernel Hilbert spaces underlying linear SDE Estimation, Kalman filtering and their relation to optimal control

- Mathematics
- 2022

It is often said that control and estimation problems are in duality. Recently, in Aubin-Frankowski (2021a), we found new reproducing kernels in Linear-Quadratic optimal control by focusing on the…

### Finite Element Representations of Gaussian Processes: Balancing Numerical and Statistical Accuracy

- Computer ScienceSIAM/ASA J. Uncertain. Quantification
- 2022

The theory implies that, under certain smoothness assumptions, one can reduce the computation and memory cost without hindering the estimation accuracy by setting 𝑛 ≪ 𝓁 in the large 𝐁 asymptotics.

### The SPDE approach for spatio-temporal datasets with advection and diffusion

- Computer Science
- 2022

This work presents the advection-diﬀusion SPDE with ﬁrst order derivative in time to enlarge the SPDE family to the space-time context and proposes Computationally eﬃcient methods to estimate the parameters of the SPDe and to predict the spatio-temporalﬁeld by kriging.

### Realistic and Fast Modeling of Spatial Extremes over Large Geographical Domains

- Computer Science
- 2021

A more realistic Bayesian framework based on a novel Gaussian scale mixture model, where the Gaussian process component is defined by a stochastic partial differential equation that yields a sparse precision matrix, and the random scale component is modeled as a lowrank Pareto-tailed or Weibull-tailed spatial process determined by compactly supported basis functions.

### Covariance-based rational approximations of fractional SPDEs for computationally efficient Bayesian inference

- Computer Science
- 2022

A new method based on approximating the covariance operator L − 2 β of the Gaussianﬁeld u by a ﬁnite element method combined with a rational approximation of the fractional power results in a numerically stable GMRF approximation which can be combined with the integrated nested Laplace approximation (INLA) method for fast Bayesian inference.

### Efficient algorithms for Bayesian Inverse Problems with Whittle-Matérn Priors

- Mathematics, Computer ScienceArXiv
- 2022

An eﬃcient method for handling all admissible noninteger values of the exponent is derived and it is shown how to incorporate this prior representation into the inﬁnite-dimensional Bayesian formulation, and how to compute the maximum a posteriori estimate, and approximate the posterior variance.

### Numerical Approximation of Gaussian random fields on Closed Surfaces

- Mathematics, Computer ScienceArXiv
- 2022

The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs and consists of a sinc quadrature coupled with a standard surface ﬁnite element method.

### Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

- Computer Science, MathematicsTheory of Probability and Mathematical Statistics
- 2022

A new Boltzmann–Gibbs model which features local interactions in the energy functional and is embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics.

### Beyond Stationary Simulation; Modern Approaches to Stochastic Modelling

- Computer Science
- 2022

Three stochastic simulation methods using non-stationary covariance, multipoint simulation, and conditional GANs are presented: an SPDE method was used as a benchmark comparison and the application of machine learning techniques can be used to surpass previous simulation generation speeds and allows for greater parameterization.

## References

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- MathematicsBernoulli
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This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models…

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- Computer Science, Mathematics
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It is shown that, using an approximate stochastic weak solution to (linear) stochastically partial differential equations, some Gaussian fields in the Matérn class can provide an explicit link, for any triangulation of , between GFs and GMRFs, formulated as a basis function representation.

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- Computer Science
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The article studies non‐Gaussian extensions of a recently discovered link between certain Gaussian random fields, expressed as solutions to stochastic partial differential equations (SPDEs), and…

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- Computer Science, Mathematics
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The SPDE method is extended to a larger class of non-Gaussian random fields with Matern covariance functions, including certain Laplace Moving Average (LMA) models, and it is shown how the SPDE formulation can be used to obtain an efficient simulation method and an accurate parameter estimation technique for a LMA model.

### Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction

- MathematicsArXiv
- 2021

It is proved that the precision and covariance operators may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section, and a tapering strategy by thresholding results in optimally numerically sparse approximations.

### Matern Gaussian processes on Riemannian manifolds

- Computer Science, MathematicsNeurIPS
- 2020

This work proposes techniques for computing the kernels of these processes via spectral theory of the Laplace--Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods.