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Stability of the Additive Splitting Methods for the Generalized Nonlinear Schrödinger Equation
No full textSplitting methods provide an efficient approach to solving evolutionary wave equations, especially in situations where dispersive and nonlinear effects on wave propagation can be separated, as in the generalized nonlinear Schrödinger equation (GNLSE). However, such methods are explicit and can lead to numerical instabilities. We study these instabilities in the context of the GNLSE. Results previously obtained for multiplicative splitting methods are extended to additive splittings. An estimate of the largest possible integration step is derived and tested. The results are important when many solutions of GNLSE are needed, e.g., in optimization problems or statistical calculations
First-order conditions for the optimal control of learning-informed nonsmooth PDEs
No full textIn this paper we study the optimal control of a class of semilinear elliptic partial differential equations which have nonlinear constituents that are only accessible by data and are approximated by nonsmooth ReLU neural networks. The optimal control problem is studied in detail. In particular, the existence and uniqueness of the state equation are shown, and continuity as well as directional differentiability properties of the corresponding control-to-state map are established. Based on approximation capabilities of the pertinent networks, we address fundamental questions regarding approximating properties of the learning-informed control-to-state map and the solution of the corresponding optimal control problem. Finally, several stationarity conditions are derived based on different notions of generalized differentiability
Optimal Control under Uncertainty with Joint Chance State Constraints: Almost-Everywhere Bounds, Variance Reduction, and Application to (Bi)linear Elliptic PDEs
No full textSelf-consistent thermal-opto-electronic model for the dynamics in high-power semiconductor lasers
Full text linkHigh-power broad-area diode lasers (BALs) generate significant heat, impacting performance. A 2+1 dimensional traveling wave (TW) model incorporates heating effects through an iterative coupling of electro-optical (EO) and heat-transport (HT) solvers. This method analyzes heat sources, temperature profiles, and thermally induced refractive index changes
Viscous flow past a translating body with oscillating boundary
No full textWe study an incompressible viscous flow around an obstacle with an oscillating boundary that moves by a translational periodic motion, and we show existence of strong time-periodic solutions for small data in different configurations: If the mean velocity of the body is zero, existence of time-periodic solutions is provided within a framework of Sobolev functions with isotropic pointwise decay. If the mean velocity is non-zero, this framework can be adapted, but the spatial behavior of the flow requires a setting of anisotropically weighted spaces. In the latter case, we also establish existence of solutions within an alternative framework of homogeneous Sobolev spaces. These results are based on the time-periodic maximal regularity of the associated linearizations, which is derived from suitable R-bounds for the Stokes and Oseen resolvent problems. The pointwise estimates are deduced from the associated time-periodic fundamental solutions
Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems
Full text linkWe investigate the convergence of spatial discretizations for reaction-diffusion systems with mass-action law satisfying a detailed balance condition. Considering systems on the d-dimensional torus, we construct appropriate space-discrete processes and show convergence not only on the level of solutions, but also on the level of the gradient systems governing the evolutions. As an important step, we prove chain rule inequalities for the reaction-diffusion systems as well as their discretizations, featuring a non-convex dissipation functional. The convergence is then obtained with variational methods by building on the recently introduced notion of gradient systems in continuity equation format
Joint distributions for total lengths of shortest-path trees in telecommunication networks
Full text linkShortest-path trees play an important role in the field of optimising fixed-access telecommunication networks with respect to costs and capacities. Distributional properties of the corresponding two half-trees originating from the root of such a tree are of special interest for engineers. In the present paper, we derive parametric approximation formulas for the marginal density functions of the total lengths of both half-trees. Besides, a parametric copula is used in order to combine the marginal distributions of these functionals to a bivariate joint distribution as, naturally, the total lengths of the half-trees are not independent random variables. Asymptotic results for infinitely sparse and infinitely dense networks are discussed as well
Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs
No full textWe consider complex stochastic systems in continuous time and space where the objects of interest are modelled via stochastic differential equations, in general high dimensional and with nonlinear coefficients. The extraction of quantifiable information from such systems has a long history and many aspects. We shall focus here on the perhaps most classical problems in this context: the filtering problem for nonlinear diffusions and the problem of parameter estimation, also for nonlinear and multidimensional diffusions. More specifically, we return to the question of robustness, first raised in the filtering community in the mid-1970s: will it be true that the conditional expectation of some observable of the signal process, given an observation (sample) path, depends continuously on the latter? Sadly, the answer here is no, as simple counterexamples show. Clearly, this is an unhappy state of affairs for users who effectively face an ill-posed situation: close observations may lead to vastly different predictions. A similar question can be asked in the context of (maximum likelihood) parameter estimation for diffusions. Some (apparently novel) counter examples show that, here again, the answer is no. Our contribution (Crisan et al., Ann Appl Probab 23(5):2139–2160, 2013); Diehl et al., A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs (2013, arXiv:1301.3799; Diehl et al., Pathwise stability of likelihood estimators for diffusions via rough paths (2013, arXiv:1311.1061) changed to yes, in other words: well-posedness is restored, provided one is willing or able to regard observations as rough paths in the sense of T. Lyons