Wong–Zakai Approximation for Landau–Lifshitz–Gilbert Equation Driven by Geometric Rough Paths

We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

Regularization of inverse problems by an approximate matrix-function technique

In this work, we introduce and investigate a class of matrix-free regularization techniques for discrete linear ill-posed problems based on the approximate computation of a special matrix-function. In order to produce a regularized solution, the proposed strategy employs a regular approximation of the Heavyside step function computed into a small Krylov subspace. This particular feature allows our proposal to be independent from the structure of the underlying matrix. If on the one hand, the use of the Heavyside step function prevents the amplification of the noise by suitably filtering the responsible components of the spectrum of the discretization matrix, on the other hand, it permits the correct reconstruction of the signal inverting the remaining part of the spectrum. Numerical tests on a gallery of standard benchmark problems are included to prove the efficacy of our approach even for problems affected by a high level of noise.

Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

In this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

Regular Approximation of Weighted Linear Context-Free Tree Languages

We show how to train a weighted regular tree grammar such that it best approximates a weighted linear context-free tree grammar concerning the Kullback–Leibler divergence between both grammars. Furthermore, we construct a regular tree grammar that approximates the tree language induced by a context-free tree grammar.