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Analysis of kernel mirror prox for measure optimization
Get PDFBy choosing a suitable function space as the dual to the non-negative measure cone, we study in a unified framework a class of functional saddle-point optimization problems, which we term the Mixed Functional Nash Equilibrium (MFNE), that underlies several existing machine learning algorithms, such as implicit generative models, distributionally robust optimization (DRO), and Wasserstein barycenters. We model the saddle-point optimization dynamics as an interacting Fisher-Rao-RKHS gradient flow when the function space is chosen as a reproducing kernel Hilbert space (RKHS). As a discrete time counterpart, we propose a primal-dual kernel mirror prox (KMP) algorithm, which uses a dual step in the RKHS, and a primal entropic mirror prox step. We then provide a unified convergence analysis of KMP in an infinite-dimensional setting for this class of MFNE problems, which establishes a convergence rate of in the deterministic case and in the stochastic case, where is the iteration counter. As a case study, we apply our analysis to DRO, providing algorithmic guarantees for DRO robustness and convergence
On the Existence and Asymptotic Stability of Two-Dimensional Periodic Solutions with an Internal Transition Layer in a Problem with a Finite Advection
No full textWe consider a periodic boundary value problem for a singularly perturbed reaction-advection-diffusion equation in the case of a two-dimensional space variable. We construct a new interior layer-type formal asymptotics which includes an approximation of the location of the interior layer, investigate the order-preserving properties of the operators generating the asymptotics, and propose a modified procedure to obtain asymptotic lower and upper solutions. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution with an interior layer and estimate the accuracy of its asymptotics. We also prove the asymptotic Lyapunov stability of this solution
Function Phononic Crystals
No full textWe propose a novel type of phononic crystal for which the materials parameters are continuous functions of space coordinates without discontinuities corresponding to a seamless fusion of the constituent materials within the crystal lattice. With the help of an adaptation of this fundamental approach, we extend the well-established concept of phononic crystals, allowing an investigation of the transition from conventional phononic crystals with a regulated step-like parameter function to the realm of so-called function phononic crystals. Our study is based on a first-principle theory assisted by high-performance computer simulations and focuses on an understanding of the effects of a deviation from the typical parameter step function on the phononic density of states (DOS). Our exploration of the DOS reveals a characteristic rapid convergence: even a slight deviation from an ideal step function has the potential to induce radical changes in the band structure leading to the emergence of desirable features, especially multiple complete phononic band gaps
Large Eddy Simulations of the Interaction Between the Atmospheric Boundary Layer and Degrading Arctic Permafrost
No full textArctic permafrost thaw holds the potential to drastically alter the Earth's surface in Northern high latitudes. We utilize high-resolution large eddy simulations to investigate the impact of the changing surfaces onto the neutrally stratified atmospheric boundary layer (ABL). A stochastic surface model based on Gaussian Random Fields modeling typical permafrost landscapes is established in terms of two land cover classes: grass land and open water bodies, which exhibit different surface roughness length and surface sensible heat flux. A set of experiments is conducted where two parameters, the lake areal fraction and the surface correlation length, are varied to study the sensitivity of the boundary layer with respect to surface heterogeneity. Our key findings from the simulations are the following: The lake areal fraction has a substantial impact on the aggregated sensible heat flux at the blending height where surface heterogeneities become horizontally homogenized. The larger the lake areal fraction, the smaller the sensible heat flux. This result gives rise to a potential feedback mechanism. When the Arctic dries due to climate heating, the interaction with the ABL may accelerate permafrost thaw. Furthermore, the blending height shows significant dependency on the correlation length of the surface features. A longer surface correlation length causes an increased blending height. This finding is of relevance for land surface models concerned with Arctic permafrost as they usually do not consider a heterogeneity metric comparable to the surface correlation length
Gibbs measures for hardcore-SOS models on Cayley trees
Get PDFWe investigate the finite-state p-solid-on-solid model, for p = ∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k = 2, 3 and increasing coupling strength, the number of translation-invariant Gibbs measures behaves as 1 → 3 → 5 → 6 → 7. This phase diagram is qualitatively similar to the one observed for three-state p-SOS models with p > 0 and, in the case of k = 2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous
Pontryagin’s Principle for Some Probabilistic Control Problems
No full textIn this paper we investigate optimal control problems perturbed by random events. We assume that the control has to be decided prior to observing the outcome of the perturbed state equations. We investigate the use of probability functions in the objective function or constraints to define optimal or feasible controls. We provide an extension of differentiability results for probability functions in infinite dimensions usable in this context. These results are subsequently combined with the optimal control setting to derive a novel Pontryagin’s optimality principle
Proof of off-diagonal long-range order in a mean-field trapped Bose gas via the Feynman--Kac formula
Get PDFWe consider the non-interacting Bose gas of many bosons in dimension larger than two in a trap in a mean-field setting with a vanishing factor in front of the kinetic energy. The semi-classical setting is a particular case and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman--Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose?Einstein condensation) for the vanishing factor above a certain threshold and non-occurrence of ODLRO for below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman--Kac formula, the order parameter that Feynman suggested in [Fe53]. For small values of the factor, we prove that all loops have the minimal length one, and for large ones we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson--Dirichlet distribution
Time-periodic behaviour in one- and two-dimensional interacting particle systems
Get PDFWe provide a class of examples of interacting particle systems on , for , that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure
A Descent Algorithm for the Optimal Control of ReLU Neural Network Informed PDEs based on Approximate Directional Derivatives
No full textWe propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that a direct smoothing of the ReLU network with the aim to make use of classical numerical solvers can have certain disadvantages, namely potentially introducing multiple solutions for the corresponding state equation. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown