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Existence of energy-variational solutions to hyperbolic conservation laws
No full textWe introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions
Interaction-force transport gradient flows
Get PDFThis paper presents a new gradient flow dissipation geometry over non-negative and probability measures. This is motivated by a principled construction that combines the unbalanced optimal transport and interaction forces modeled by reproducing kernels. Using a precise connection between the Hellinger geometry and the maximum mean discrepancy (MMD), we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and improved empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy
A globalized inexact semismooth Newton method for nonsmooth fixed-point equations involving variational inequalities
Get PDFWe develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general functional analytic setting and allows for inexact function evaluations and Newton steps. Moreover, if a certain contraction assumption holds, we show that it is possible to globalize the algorithm by means of the Banach fixed-point theorem and to ensure q-superlinear convergence to the problem solution for arbitrary starting values. By means of a localization technique, our Newton method can also be used to determine solutions of fixed-point equations that are only locally contractive and not uniquely solvable. We apply our algorithm to a quasi-variational inequality which arises in thermoforming and which not only involves the obstacle problem as a source of nonsmoothness but also a semilinear PDE containing a nondifferentiable Nemytskii operator. Our analysis is accompanied by numerical experiments that illustrate the mesh-independence and q -superlinear convergence of the developed solution algorithm
A neural network approach to learning solutions of a class of elliptic variational inequalities
Get PDFWe develop a weak adversarial approach to solving obstacle problems using neural networks. By employing (generalised) regularised gap functions and their properties we rewrite the obstacle problem (which is an elliptic variational inequality) as a minmax problem, providing a natural formulation amenable to learning. Our approach, in contrast to much of the literature, does not require the elliptic operator to be symmetric. We provide an error analysis for suitable discretisations of the continuous problem, estimating in particular the approximation and statistical errors. Parametrising the solution and test function as neural networks, we apply a modified gradient descent ascent algorithm to treat the problem and conclude the paper with various examples and experiments. Our solution algorithm is in particular able to easily handle obstacle problems that feature biactivity (or lack of strict complementarity), a situation that poses difficulty for traditional numerical methods
Spectral bounds for the operator pencil of an elliptic system in an angle
Get PDFThe model problem of a plane angle for a second-order elliptic system with Dirichlet, mixed, and Neumann boundary conditions is analyzed. The existence of solutions is, for each boundary condition, reduced to solving a matrix equation. Leveraging these matrix equations and focusing on Dirichlet and mixed boundary conditions, optimal bounds on these solutions are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results in linear elasticity
Bounded functional calculus for divergence form operators with dynamical boundary conditions
Get PDFWe consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a matter of fact, the elliptic operator and its semigroup act simultaneously in the interior and on the boundary. We show that the elliptic operator has a bounded holomorphic calculus in Lebesgue spaces if the coefficients satisfy a p-adapted ellipticity condition. A major challenge in the proof is that different parts of the spatial domain of the operator have different dimensions. Our strategy relies on extending a contractivity criterion due to Nittka and a non-linear heat flow method recently popularized by Carbonaro--Dragicevic to our setting
Bernstein-type and Bennett-type inequalities for unbound matrix martingales
Get PDFWe derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued supermartingale processes with unbounded observations. Specifically, we assume that the - Orlicz (quasi-)norms of their difference process are bounded for some . As corollaries, we prove an empirical version of Bernstein's inequality and an extension of the bounded differences inequality, also known as McDiarmid's inequality
Bound-preserving PINNs for steady-state convection-diffusion-reaction problems
Get PDFNumerical approximations of solutions of convection-diffusion-reaction problems should take only physically admissible values. Provided that bounds for the admissible values are known, this paper presents several approaches within PINNs and -VPINNs for preserving these bounds. Numerical simulations are performed for convection-dominated problems. One of the proposed approaches turned out to be superior to the other ones with respect to the accuracy of the computed solutions
Stochastic control with signatures
Get PDFThis paper proposes to parameterize open loop controls in stochastic optimal control problems via suit- able classes of functionals depending on the driver's path signature, a concept adopted from rough path integration theory. We rigorously prove that these controls are dense in the class of progressively mea- surable controls and use rough path methods to establish suitable conditions for stability of the controlled dynamics and target functional. These results pave the way for Monte Carlo methods to stochastic optimal control for generic target functionals and dynamics. We discuss the rather versatile numerical algorithms for computing approximately optimal controls and verify their accurateness in benchmark problems from Mathematical Finance
Optimal boundary control of the isothermal semilinear Euler equation for gas dynamics on a network
No full textThe analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush–Kuhn–Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived