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Strong stationarity conditions for optimal control problems governed by a rate-independent evolution variational inequality
Get PDFWe prove strong stationarity conditions for optimal control problems that are governed by a prototypical rate-independent evolution variational inequality, i.e., first-order necessary optimality conditions in the form of a primal-dual multiplier system that are equivalent to the purely primal notion of Bouligand stationarity. Our analysis relies on recent results on the Hadamard directional differentiability of the scalar stop operator and a new concept of temporal polyhedricity that generalizes classical ideas of Mignot. The established strong stationarity system is compared with known optimality conditions for optimal control problems governed by elliptic obstacle-type variational inequalities and stationarity systems obtained by regularization
Square waves and Bykov T-points in a delay algebraic model for the Kerr--Gires--Tournois interferometer
Get PDFWe study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system
A note on the monomer-dimer model
Get PDFWe consider the monomer-dimer model, whose realisations are spanning sub-graphs of a given graph such that every vertex has degree zero or one. The measure depends on a parameter, the monomer activity, which rewards the total number of monomers. We consider general correlation functions including monomer-monomer correlations and dimer-dimer covariances. We show that these correlations decay exponentially fast with the distance if the monomer activity is strictly positive. Our result improves a previous upper bound from van den Berg and is of interest due to its relation to truncated spin-spin correlations in classical spin systems. Our proof is based on the cluster expansion technique
Impact of Turbulence Modeling on the Simulation of Blood Flow in Aortic Coarctation (Dataset)
No full textInput data (geometry and inflow) for simulations as well as a selection of the output dat
Data and code from the paper "On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems"
No full textThis is a collection of data files, used in the publication "On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems". It contains all scripts to actually train PINNs to approximate the solution to two-dimensional convection-diffusion problems. Moreover, all networks that have been trained for this publications and the script to evaluate the results are provided
Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime
Get PDFWe study a multidimensional stochastic differential equation with additive noise: where the drift is integrable in space and time, and is either a fractional Brownian motion or an -stable process. We show weak existence of solutions to this equation under the optimal condition on integrability indices of , going beyond the subcritical Krylov-Röckner (Prodi-Serrin-Ladyzhenskaya) regime. This extends the recent results of Krylov (2020) to the fractional Brownian and Lévy cases. We also construct a counterexample to demonstrate the optimality of this condition. Our methods are built upon a version of the stochastic sewing lemma of Lê and the John--Nirenberg inequality
Concentration of a high dimensional sub-gaussian vector
Get PDFThis note describes the concentration phenomenon for a high dimensional sub-gaussian vector . In the Gaussian case, for any linear operator , it holds and with ; see \cite{laurentmassart2000}. This implies concentration of the squared norm around its expectation provided that is sufficiently large. An extension of this result to a non-gaussian case is a nontrivial task even under sub-gaussian behavior of , especially if the entries of cannot be assumed independent and Hanson-Wright type bounds do not apply. The results of this paper extend the Gaussian deviation bounds and support the concentration phenomenon for using recent advances in Laplace approximation from \cite{SpLaplace2022} and \cite{katsevich2023tight}. The results are illustrated by the case when is an i.i.d. sum
Exploiting Higher-order Derivates in Convex Optimization Methods
No full textExploiting higher-order derivatives in convex optimization is known at least since 1970's. In each iteration higher-order (also called tensor) methods minimize a regularized Taylor expansion of the objective function, which leads to faster convergence rates if the corresponding higher-order derivative is Lipschitz-continuous. Recently a series of lower iteration complexity bounds for such methods were proved, and a gap between upper an lower complexity bounds was revealed. Moreover, it was shown that such methods can be implementable since the appropriately regularized Taylor expansion of a convex function is also convex and, thus, can be minimized in polynomial time. Only very recently an algorithm with optimal convergence rate was proposed for minimizing convex functions with Lipschitz -th derivative. For convex functions with Lipschitz third derivative, these developments allowed to propose a second-order method with convergence rate , which is faster than the rate of existing second-order methods
Generative models for the deformation of industrial shapes with linear geometric constraints: Model order and parameter space reductions
Get PDFReal-world applications of computational fluid dynamics often involve the evaluation of quantities of interest for several distinct geometries that define the computational domain or are embedded inside it. For example, design optimization studies require the realization of response surfaces from the parameters that determine the geometrical deformations to relevant outputs to be optimized. In this context, a crucial aspect to be addressed are the limited resources at disposal to computationally generate different geometries or to physically obtain them from direct measurements. This is the case for patient-specific biomedical applications for example. When additional linear geometrical constraints need to be imposed, the computational costs increase substantially. Such constraints include total volume conservation, barycenter location and fixed moments of inertia. We develop a new paradigm that employs generative models from machine learning to efficiently sample new geometries with linear constraints. A consequence of our approach is the reduction of the parameter space from the original geometrical parametrization to a low-dimensional latent space of the generative models. Crucial is the assessment of the quality of the distribution of the constrained geometries obtained with respect to physical and geometrical quantities of interest. Non-intrusive model order reduction is enhanced since smaller parametric spaces are considered. We test our methodology on two academic test cases: a mixed Poisson problem on the 3d Stanford bunny with fixed barycenter deformations and the multiphase turbulent incompressible Navier-Stokes equations for the Duisburg test case with fixed volume deformations of the naval hull
Building hierarchies of semiclassical Jacobi polynomials for spectral methods in annuli
Get PDFWe discuss computing with hierarchies of families of (potentially weighted) semiclassical Jacobi polynomials which arise in the construction of multivariate orthogonal polynomials. In particular, we outline how to build connection and differentiation matrices with optimal complexity and compute analysis and synthesis operations in quasi-optimal complexity. We investigate a particular application of these results to constructing orthogonal polynomials in annuli, called the generalised Zernike annular polynomials, which lead to sparse discretisations of partial differential equations. We compare against a scaled-and-shifted Chebyshev--Fourier series showing that in general the annular polynomials converge faster when approximating smooth functions and have better conditioning. We also construct a sparse spectral element method by combining disk and annulus cells, which is highly effective for solving PDEs with radially discontinuous variable coefficients and data