The discontinuous Galerkin (DG) method is combined with the spectral deferred correction (SDC) time integration approach to solve the fluid dynamic equations. The moment limiter is generalized for nonuniform grids with hanging nodes that result from adaptive mesh refinement. The effect of characteristic, primitive, or conservative decomposition in the limiting stage is studied. In general, primitive variable decomposition is a better option, especially in two and three dimensions. The accuracy-preserving total variation diminishing (AP-TVD) marker for troubled-cell detection, which uses an averaged-derivative basis, is modified to use the Legendre polynomial basis. Given that the latest basis is generally used for DG, the new approach avoids transforming to the averaged-derivative basis, what results in a more efficient technique. Further, a new error estimator is proposed to determine where to refine or coarsen the grid. This estimator is compared against other estimator used in the literature and shows an improved performance. Canonical tests in one, two, and three dimensions are conducted to show the accuracy of the solver.

We give an alternative and unified derivation of the general framework developed in the last few years for analyzing nonstationary time series. A different approach for handling the resulting variational problem numerically is introduced. We further expand the framework by employing adaptive finite element algorithms and ideas from information theory to solve the problem of finding the most adequate model based on a maximum-entropy ansatz, thereby reducing the number of underlying probabilistic assumptions. In addition, we formulate and prove the result establishing the link between the optimal parametrizations of the direct and the inverse problems and compare the introduced algorithm to standard approaches like Gaussian mixture models, hidden Markov models, artificial neural networks and local kernel methods. Furthermore, based on the introduced general framework, we show how to create new data analysis methods for specific practical applications. We demonstrate the application of the framework to data samples from toy models as well as to real-world problems such as biomolecular dynamics, DNA sequence analysis and financial applications.

We present a method to approximate the law of positive random variables defined by their Laplace transforms. It is based on the study of the error in the Laplace domain and allows for many behaviors of the law, both at $0$ and infinity. In most cases, both the Kantorovich/Wasserstein error and the Kolmogorov–Smirnov error can be accurately computed. Two detailed examples illustrate our results.

We consider the problem of finding optimally stable polynomial approximations to the exponential for application to one-step integration of initial value ordinary and partial differential equations. The objective is to find the largest stable step size and corresponding method for a given problem when the spectrum of the initial value problem is known. The problem is expressed in terms of a general least deviation feasibility problem. Its solution is obtained by a new fast, accurate, and robust algorithm based on convex optimization techniques. Global convergence of the algorithm is proven in the case that the order of approximation is one and in the case that the spectrum encloses a starlike region. Examples demonstrate the effectiveness of the proposed algorithm even when these conditions are not satisfied.