Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
Instantaneous Gelation and Non-Existence of Weak Solutions for the Oort--Hulst--Safronov Coagulation Model
No full textThe possible occurrence of instantaneous gelation to the Oort–Hulst–Safronov (OHS) coagulation equation is investigated for a certain class of unbounded coagulation kernels. The existence of instantaneous gelation is confirmed by showing the non-existence of mass-conserving weak solutions. Finally, it is shown that for such kernels, there is no weak solution to the OHS coagulation equation at any time interval
Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium
Get PDFKernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium Pavel Dvurechensky , Jia-Jie Zhu Abstract The theoretical analysis of machine learning algorithms, such as deep generative modeling, motivates multiple recent works on the Mixed Nash Equilibrium (MNE) problem. Different from MNE, this paper formulates the Mixed Functional Nash Equilibrium (MFNE), which replaces one of the measure optimization problems with optimization over a class of dual functions, e.g., the reproducing kernel Hilbert space (RKHS) in the case of Mixed Kernel Nash Equilibrium (MKNE). We show that our MFNE and MKNE framework form the backbones that govern several existing machine learning algorithms, such as implicit generative models, distributionally robust optimization (DRO), and Wasserstein barycenters. To model the infinite-dimensional continuous- limit optimization dynamics, we propose the Interacting Wasserstein-Kernel Gradient Flow, which includes the RKHS flow that is much less common than the Wasserstein gradient flow but enjoys a much simpler convexity structure. Time-discretizing this gradient flow, we propose a primal-dual kernel mirror prox algorithm, which alternates between a dual step in the RKHS, and a primal step in the space of probability measures. We then provide the first unified convergence analysis of our algorithm for this class of MKNE problems, which establishes a convergence rate of O(1/N ) in the deterministic case and O(1/√N) in the stochastic case. As a case study, we apply our analysis to DRO, providing the first primal-dual convergence analysis for DRO with probability-metric constraints
Neutral delay differential equation Kerr cavity model
Get PDFA neutral delay differential equation (NDDE) model of a Kerr cavity with external coherent injection is developed that can be considered as a generalization of the Ikeda map with second and higher order dispersions being taken into account. It is shown that this model has solutions in the form of dissipative solitons both in the limit, where the model can be reduced to the Lugiato--Lefever equation (LLE), and beyond this limit, where the soliton is eventually destroyed by the Cherenkov radiation. Unlike the standard LLE the NDDE model is able to describe the overlap of multiple resonances associated with different cavity modes
Unrolled three-operator splitting for parameter-map learning in low dose X-ray CT reconstruction
Get PDFWe propose a method for fast and automatic estimation of spatially dependent regularization maps for total variation-based (TV) tomography reconstruction. The estimation is based on two distinct sub-networks, with the first sub-network estimating the regularization parameter-map from the input data while the second one unrolling T iterations of the Primal-Dual Three-Operator Splitting (PD3O) algorithm. The latter approximately solves the corresponding TV-minimization problem incorporating the previously estimated regularization parameter-map. The overall network is then trained end-to-end in a supervised learning fashion using pairs of clean-corrupted data but crucially without the need of having access to labels for the optimal regularization parameter-maps
High-probability bounds for stochastic optimization and variational inequalities: The case of unbounded variance
Get PDFDuring recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central α-th moment for α∈(1,2] in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization
Analysis and numerical approximation of energy-variational solutions to the Ericksen--Leslie equations
No full textWe define the concept of energy-variational solutions for the Ericksen-Leslie equations in three spatial dimensions. This solution concept is finer than dissipative solutions and satisfies the weak-strong uniqueness property. For a certain choice of the regularity weight, the existence of energy-variational solutions implies the existence of measure-valued solutions and for a different choice, we construct an energy-variational solution with the help of an implementable, structure-inheriting space-time discretization. Computational studies are performed in order to provide some evidence of the applicability of the proposed algorithm
Accurate evaluation of self-heterodyne laser linewidth measurements using Wiener filters
Get PDFSelf-heterodyne beat note measurements are widely used for the experimental characterization of the frequency noise power spectral density (FN-PSD) and the spectral linewidth of lasers. The measured data, however, must be corrected for the transfer function of the experimental setup in a post-processing routine. The standard approach disregards the detector noise and thereby induces reconstruction artifacts, i.e., spurious spikes, in the reconstructed FN-PSD. We introduce an improved post-processing routine based on a parametric Wiener filter that is free from reconstruction artifacts, provided a good estimate of the signal-to-noise ratio is supplied. Building on this potentially exact reconstruction, we develop a new method for intrinsic laser linewidth estimation that is aimed at deliberate suppression of unphysical reconstruction artifacts. Our method yields excellent results even in the presence of strong detector noise, where the intrinsic linewidth plateau is not even visible using the standard method. The approach is demonstrated for simulated time series from a stochastic laser model including 1/f-type noise
A drift-diffusion based electrothermal model for organic thin-film devices including electrical and thermal environment
Get PDFWe derive and investigate a stationary model for the electrothermal behavior of organic thin-film devices including their electrical and thermal environment. Whereas the electrodes are modeled by Ohm's law, the electronics of the organic device itself is described by a generalized van Roosbroeck system with temperature dependent mobilities and using Gauss--Fermi integrals for the statistical relation. The currents give rise to Joule heat which together with the heat generated by the generation/recombination of electrons and holes in the organic device occur as source terms in the heat flow equation that has to be considered on the whole domain. The crucial task is to establish that the quantities in the transfer conditions at the interfaces between electrodes and the organic semiconductor device have sufficient regularity. Therefore, we restrict the analytical treatment of the system to two spatial dimensions. We consider layered organic structures, where the physical parameters (total densities of transport states, LUMO and HOMO energies, disorder parameter, basic mobilities, activation energies, relative dielectric permittivity, heat conductivity) are piecewise constant. We prove the existence of weak solutions using Schauder's fixed point theorem and a regularity result for strongly coupled systems with nonsmooth data and mixed boundary conditions that is verified by Caccioppoli estimates and a Gehring-type lemma
Controlled Polyhedral Sweeping Processes: Existence, Stability, and Optimality Conditions
No full textThis paper is mainly devoted to the study of controlled sweeping processes with polyhedral moving sets in Hilbert spaces. Based on a detailed analysis of truncated Hausdorff distances between moving polyhedra, we derive new existence and uniqueness theorems for sweeping trajectories corresponding to various classes of control functions acting in moving sets. Then we establish quantitative stability results, which provide efficient estimates on the sweeping trajectory dependence on controls and initial values. Our final topic, accomplished in finite-dimensional state spaces, is deriving new necessary optimality and suboptimality conditions for sweeping control systems with endpoint constrains by using constructive discrete approximations
Existence of similarity profiles for diffusion equations and systems
Get PDFWe study the existence of self-similar profiles for diffusion equations and reaction-diffusion systems on the real line, where different nontrivial limits are imposed at both sides of infinity. The theses profiles solve a coupled system of nonlinear ODEs that can be treated by monotone operator theory