The rational SPDE approach for Gaussian random fields with general smoothness

A popular approach for modeling and inference in spatial statistics is to represent Gaussian random fields as solutions to stochastic partial differential equations (SPDEs) of the form $L^{\beta}u = \mathcal{W}$, where $\mathcal{W}$ is Gaussian white noise, $L$ is a second-order differential operator, and $\beta>0$ is a parameter that determines the smoothness of $u$. However, this approach has been limited to the case $2\beta\in\mathbb{N}$, which excludes several important models and makes it necessary to keep $\beta$ fixed during inference. We propose a new method, the rational SPDE approach, which in spatial dimension $d\in\mathbb{N}$ is applicable for any $\beta>d/4$, and thus remedies the mentioned limitation. The presented scheme combines a finite element discretization with a rational approximation of the function $x^{-\beta}$ to approximate $u$. For the resulting approximation, an explicit rate of convergence to $u$ in mean-square sense is derived. Furthermore, we show that our method has the same computational benefits as in the restricted case $2\beta\in\mathbb{N}$. Several numerical experiments and a statistical application are used to illustrate the accuracy of the method, and to show that it facilitates likelihood-based inference for all model parameters including $\beta$.Comment: 28 pages, 4 figure

Approximation Theory for Matrices

We review the theory of optimal polynomial and rational Chebyshev approximations, and Zolotarev's formula for the sign function over the range (\epsilon \leq |z| \leq1). We explain how rational approximations can be applied to large sparse matrices efficiently by making use of partial fraction expansions and multi-shift Krylov space solvers.Comment: 10 pages, 7 figure

Large Eddy Simulation of Turbulent Channel Flows by the Rational LES Model

The rational large eddy simulation (RLES) model is applied to turbulent channel flows. This approximate deconvolution model is based on a rational (subdiagonal Pade') approximation of the Fourier transform of the Gaussian filter and is proposed as an alternative to the gradient (also known as the nonlinear or tensor-diffusivity) model. We used a spectral element code to perform large eddy simulations of incompressible channel flows at Reynolds numbers based on the friction velocity and the channel half-width Re{sub tau} = 180 and Re{sub tau} = 395. We compared the RLES model with the gradient model. The RLES results showed a clear improvement over those corresponding to the gradient model, comparing well with the fine direct numerical simulation. For comparison, we also present results corresponding to a classical subgrid-scale eddy-viscosity model such as the standard Smagorinsky model.Comment: 31 pages including 15 figure

The Combinatorial World (of Auctions) According to GARP

Revealed preference techniques are used to test whether a data set is compatible with rational behaviour. They are also incorporated as constraints in mechanism design to encourage truthful behaviour in applications such as combinatorial auctions. In the auction setting, we present an efficient combinatorial algorithm to find a virtual valuation function with the optimal (additive) rationality guarantee. Moreover, we show that there exists such a valuation function that both is individually rational and is minimum (that is, it is component-wise dominated by any other individually rational, virtual valuation function that approximately fits the data). Similarly, given upper bound constraints on the valuation function, we show how to fit the maximum virtual valuation function with the optimal additive rationality guarantee. In practice, revealed preference bidding constraints are very demanding. We explain how approximate rationality can be used to create relaxed revealed preference constraints in an auction. We then show how combinatorial methods can be used to implement these relaxed constraints. Worst/best-case welfare guarantees that result from the use of such mechanisms can be quantified via the minimum/maximum virtual valuation function

An approach to rational approximation of power spectral densities on the unit circle

In this article, we propose a new approach to determining the best rational approximation of a given irrational power spectral density defined on the unit circle such that the approximant has McMillan degree less than or equal to some positive integer $n$. The main result is that we prove the existence of an optimal solution and that this solution can be found by standard methods of optimization

Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in L2{L}^2 of the Circle

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L^2-sense on the unit circle, to functions f, where f is a sum of a Cauchy transform of a complex measure \mu supported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of \mu.Comment: 28 page