Approximation of Bayesian inverse problems for PDEs

Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier–Stokes equation from Lagrangian and Eulerian observations, respectively

Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

Besov priors for Bayesian inverse problems

We consider the inverse problem of estimating a function $u$ from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem. This approach has a long history for inversion, dating back to 1970, and has, over the last decade, gained importance as a practical tool. However most of the existing theory has been developed for Gaussian prior measures. Recently Lassas, Saksman and Siltanen (Inv. Prob. Imag. 2009) showed how to construct Besov prior measures, based on wavelet expansions with random coefficients, and used these prior measures to study linear inverse problems. In this paper we build on this development of Besov priors to include the case of nonlinear measurements. In doing so a key technical tool, established here, is a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which, when matched to properties of the prior Besov measure, imply the well-definedness and well-posedness of the posterior measure. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.Comment: 18 page

Optical Spectral Variability of the Very-High-Energy Gamma-Ray Blazar 1ES 1011+496

We present results of five years of optical (UBVRI) observations of the very-high-energy gamma-ray blazar 1ES 1011+496 at the MDM Observatory. We calibrated UBVRI magnitudes of five comparison stars in the field of the object. Most of our observations were done during moderately faint states of 1ES 1011+496 with R > 15.0. The light curves exhibit moderate, closely correlated variability in all optical wavebands on time scales of a few days. A cross-correlation analysis between optical bands does not show significant evidence for time lags. We find a positive correlation (Pearson's r = 0.57; probability of non-correlation P(>r) ~ 4e-8) between the R-band magnitude and the B - R color index, indicating a bluer-when-brighter trend. Snap-shot optical spectral energy distributions (SEDs) exhibit a peak within the optical regime, typically between the V and B bands. We find a strong (r = 0.78; probability of non-correlation P (>r) ~ 1e-15) positive correlation between the peak flux and the peak frequency, best fit by a relation $\nu F_{\nu}^{\rm pk} \propto \nu_{\rm pk}^k$ with k = 2.05 +/- 0.17. Such a correlation is consistent with the optical (synchrotron) variability of 1ES 1011+496 being primarily driven by changes in the magnetic field.Comment: Accepted for publication in ApJ. 16 pages, including 7 figure

A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations

Many scientific and engineering problems require to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, choosing an appropriate prior distribution is an important task. In particular we consider problems where the function to infer is subject to sharp jumps which render the commonly used Gaussian measures unsuitable. On the other hand, the so-called total variation (TV) prior can only be defined in a finite dimensional setting, and does not lead to a well-defined posterior measure in function spaces. In this work we present a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect sharp jumps of the function, and the Gaussian distribution is used as a reference measure so that it results in a well-defined posterior measure in the function space. We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the TG prior. With numerical examples we demonstrate the performance of the TG prior and the efficiency of the proposed MCMC algorithm

Abelian Sandpile Model on the Honeycomb Lattice

We check the universality properties of the two-dimensional Abelian sandpile model by computing some of its properties on the honeycomb lattice. Exact expressions for unit height correlation functions in presence of boundaries and for different boundary conditions are derived. Also, we study the statistics of the boundaries of avalanche waves by using the theory of SLE and suggest that these curves are conformally invariant and described by SLE2.Comment: 24 pages, 5 figure

Electromagnetically induced spatial light modulation

We theoretically report that, utilizing electromagnetically induced transparency (EIT), the transverse spatial properties of weak probe fields can be fast modulated by using optical patterns (e.g. images) with desired intensity distributions in the coupling fields. Consequently, EIT systems can function as high-speed optically addressed spatial light modulators. To exemplify our proposal, we indicate the generation and manipulation of Laguerre-Gaussian beams based on either phase or amplitude modulation in hot vapor EIT systems.Comment: 8 pages, 3 figure

CLOCK 3111 T/C SNP Interacts with Emotional Eating Behavior for Weight-Loss in a Mediterranean Population

Objective: The goals of this research was (1) to analyze the role of emotional eating behavior on weight-loss progression during a 30-week weight-loss program in 1,272 individuals from a large Mediterranean population and (2) to test for interaction between CLOCK 3111 T/C SNP and emotional eating behavior on the effectiveness of the weight-loss program. Design and Methods: A total of 1,272 overweight and obese participants (BMI: 31±5 kg/m2), aged 20 to 65 years, attending outpatient weight-loss clinics were recruited for this analysis. Emotional eating behavior was assessed by the Emotional Eating Questionnaire (EEQ), a questionnaire validated for overweight and obese Spanish subjects. Anthropometric measures, dietary intake and weight-loss progression were assessed and analyzed throughout the 30-week program. Multivariate analysis and linear regression models were performed to test for gene-environment interaction. Results: Weight-loss progression during the 30-week program differed significantly according to the degree of emotional eating behavior. Participants classified as 'very emotional eaters' experienced more irregular (P = 0.007) weight-loss, with a lower rate of weight decline (−0.002 vs. −0.003, P = 11), lost significantly less weight than those C carriers with a low emotional score (<11) (P = 0.005). Conclusions: Emotional eating behavior associates with weight-loss pattern, progression and total weight-loss. Additionally, CLOCK 3111 T/C SNP interacts with emotional eating behavior to modulate total weight loss. These results suggest that the assessment of this locus and emotional eating behavior could improve the development of effective, long-tern weight-management interventions