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In a variety of applications it is important to extract information from a probability measure μ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest tofind a measure v, from within a simple class of measures, which approximates μ. This problem is studied in the case where the Kullback-Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where v is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms. [ABSTRACT FROM AUTHOR]

The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs. Abstractly, it can be viewed as determining a "good" n-dimensional space Hn to be used in approximating the elements of a compact set. F in a Hilbert space H. One by now popular computational approach is to find Hn through a greedy strategy. It is natural to compare the approximation performance of the Hn generated by this strategy with that of the Kolmogorov widths dn(F) since the latter gives the smallest error that can be achieved by subspaces of fixed dimension n. The first such comparisons, given in [A. Buffa et al., ESAIM Math. Model. Numer. Anal., 2011, to appear], show that the approximation error, σn(F) := dist(F, Hn), obtained by the greedy strategy satisfies σn(F) ≤ Cn2ndn(F). In this paper, various improvements of this result will be given. Among these, it is shown that whenever dn(F) ≤ Mn-α for all n > 0 and some M,α > 0, we also have σn(F) ≤ Cn2ndn for all n > 0, where Cα depends only on α. Similar results are derived for generalized exponential rates of the form Me-αn. The exact greedy algorithm is not always computationally feasible, and a commonly used computationally friendly variant can be formulated as a "weak greedy algorithm." The results of this paper are established for this version as well. [ABSTRACT FROM AUTHOR]

We consider the vanishing viscosity approximation of the traffic model, proposed by Lighthill, Whitham, and Richards, on a network composed by a single junction with n incoming and m outgoing roads. We prove that a solution of the parabolic approximation exists and, as the viscosity vanishes, the solution of the parabolic problem converges to a solution of the original problem. [ABSTRACT FROM AUTHOR]