Mario Bukal

Entropy-stable and entropy-dissipative approximations of a fourth-order quantum diffusion equation

Structure-preserving numerical schemes for a nonlinear parabolic fourth-order equation, modeling the electron transport in quantum semiconductors, with periodic boundary conditions are analyzed. First, a two-step backward differentiation formula (BDF) semi-discretization in time is investigated. The scheme preserves the nonnegativity of the solution, is entropy stable and dissipates a modified entropy functional. The existence of a weak semi-discrete solution and, in a particular case, its temporal second-order convergence to the continuous solution is proved. The proofs employ an algebraic relation which implies the G-stability of the two-step BDF. Second, an implicit Euler and

Extended information filter on matrix Lie groups

q

A multidimensional nonlinear sixth-order quantum diffusion equation☆

q-step BDF discrete variational derivative method are considered. This scheme, which exploits the variational structure of the equation, dissipates the discrete Fisher information (or energy). Numerical experiments show that the discrete (relative) entropies and Fisher information decay even exponentially fast to zero.

Direction-only tracking of moving objects on the unit sphere via probabilistic data association

Abstract This paper presents a systematic approach for component number reduction in mixtures of exponential families, putting a special emphasis on the von Mises mixtures. We propose to formulate the problem as an optimization problem utilizing a new class of computationally tractable composite distance measures as cost functions, namely the composite Renyi α -divergences, which include the composite Kullback–Leibler distance as a special case. Furthermore, we prove that the composite divergence bounds from above the corresponding intractable Renyi α -divergence between a pair of mixtures. As a solution to the optimization problem we synthesize that two existing suboptimal solution strategies, the generalized k-means and a pairwise merging approach, are actually minimization methods for the composite distance measures. Moreover, in the present paper the existing joining algorithm is also extended for comparison purposes. The algorithms are implemented and their reduction results are compared and discussed on two examples of von Mises mixtures: a synthetic mixture and a real-world mixture used in people trajectory shape analysis.

On the simultaneous homogenization and dimension reduction in elasticity and locality of \(\varGamma \)-closure

In this paper we propose a new state estimation algorithm called the extended information filter on Lie groups. The proposed filter is inspired by the extended Kalman filter on Lie groups and exhibits the advantages of the information filter with regard to multisensor update and decentralization, while keeping the accuracy of stochastic inference on Lie groups. We present the theoretical development and demonstrate its performance on multisensor rigid body attitude tracking by forming the state space on the SO(3)×R3 group, where the first and second component represent the orientation and angular rates, respectively. The performance of the filter is compared with respect to the accuracy of attitude tracking with parametrization based on Euler angles and with respect to execution time of the extended Kalman filter formulation on Lie groups. The results show that the filter achieves higher performance consistency and smaller error by tracking the state directly on the Lie group and that it keeps smaller computational complexity of the information form with respect to high number of measurements.

Derivation of homogenized Euler–Lagrange equations for von Kármán rods

Bayesian filters are often used in statistical inference and consist of recursively alternating between two steps: prediction and correction. Most commonly the Gaussian distribution is used within the Bayes filtering framework, but other distributions, which could model better the nature of the estimated phenomenon like the von Mises-Fisher distribution on the unit sphere, have also been subject of research interest. However, the von Mises-Fisher filter requires approximations since the prediction step does not yield an another von Mises-Fisher distribution. Furthermore, other advanced filtering methods require approximating a mixture of distributions with just a single component. In this paper we propose to use the score matching within the context of Bayesian assumed density filtering inlieu of the more common moment matching. Moment matching functions by assuming the type of the resulting distribution and then matching its moments with the prior distribution, which in the end minimizes the Kullback-Leibler divergence. Score matching also assumes the resulting distribution type, but finds optimal parameters by minimizing the relative Fisher information. In the paper we show that the score matching procedure results with identical performance, but with simpler equations that, unlike moment matching, do not require tedious numerical methods. In the end, we corroborate theoretical results by running the moment and score matching based filters for single and multiple object tracking on a large number of randomly generated trajectories on the unit sphere.