Out-of-core Constrained Delaunay Tetrahedralizations for Large Scenes

Tetrahedralization algorithms are used for many applications such as Ray Tracing and Finite Element Methods. For most of the applications, constrained tetrahedralization algorithms are chosen because they can preserve input triangles. The constrained tetrahedralization algorithms developed so far might suffer from a lack of memory. We propose an out-of-core near Delaunay constrained tetrahedralization algorithm using the divide-and-conquer paradigm to decrease memory usage. If the expected memory usage is below the user-defined memory limit, we tetrahedralize using TetGen. Otherwise, we subdivide the set of input points into two halves and recursively apply the same idea to the two halves. When compared with the TetGen, our algorithm tetrahedralizes the point clouds using less amount of memory but takes more time and generates tetrahedralizations that do not satisfy the Delaunay criterion at the boundaries of the merged regions. We quantify the error using the aspect-ratio metric. The difference between the tetrahedralizations that our approach produce and the Delaunay tetrahedralization are small and the results are acceptable for most applications

First-order conditions for the optimal control of learning-informed nonsmooth PDEs

In 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

Bifurcations and intermittency in coupled dissipative kicked rotors.

We investigate the emergence of complex dynamics in a system of coupled dissipative kicked rotors and show that critical transitions can be understood via bifurcations of simple states. We study multistability and bifurcations in the single-rotor model, demonstrating how these give rise to a variety of coexisting spatial patterns in a coupled system. A combined order parameter is introduced to characterize different spatial patterns and to reveal the coexistence of chaotic and regular attractors. Finally, we illustrate an intermittent phenomenon near the onset of chaos

Hellinger--Kantorovich gradient flows: Global exponential decay of entropy functionals

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger--Kantorovich (HK) geometry, which unifies transport mechanism of Otto--Wasserstein, and the birth-death mechanism of Hellinger (or Fisher--Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals under Otto--Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures---where the typical log-Sobolev arguments fail---we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the Polyak--Łojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning

First- and second-order optimality conditions in the sparse optimal control of Cahn--Hilliard systems

This paper deals with the sparse distributed control of viscous and nonviscous Cahn--Hilliard systems. We report on results concerning first-order necessary and second-order sufficient optimality conditions that have recently established by the authors. The analysis covers both the cases when the nonlinear double well potential governing the evolution is of either regular or logarithmic type. A major difficulty originates from the sparsity-enhancing term in the cost functional which typically is nondifferentiable

On the stability and efficiency of high-order parallel algorithms for 3D wave problems

In this work, we investigate the stability conditions for four new high-order ADI type schemes proposed to solve 3D wave equations with a non-constant sound speed coefficient. This analysis is mainly based on the spectral method, therefore a basic benchmark problem is formulated with a constant sound speed coefficient. For a case of general non-constant coefficient the stability analysis is done by using the energy method. Our main conclusion states that the selected ADI type schemes use different factorization operators (mainly due to the need to approximate the artificial boundary conditions on the split time levels), but the general structure of the stability factors are similar for all schemes and thus the obtained CFL conditions are also very similar. The second goal is to compare the accuracy and efficiency of the selected ADI solvers. This analysis also includes parallel versions of these schemes. Two schemes are selected as the most effective and accurate

A Posteriori Error Control for Stochastic Galerkin FEM with High-Dimensional Random Parametric PDEs

PDEs with random data are investigated and simulated in the field of Uncertainty Quantification (UQ), where uncertainties or (planned) variations of coefficients, forces, domains and boundary conditions in differential equations formally depend on random events with respect to a pre-determined probability distribution. The discretization of these PDEs typically leads to high-dimensional (deterministic) systems, where in addition to the physical space also the (often much larger) parameter space has to be considered. A proven technique for this task is the Stochastic Galerkin Finite Element Method (SGFEM), for which a review of the state of the art is provided. Moreover, important concepts and results are summarized. A special focus lies on the a posteriori error estimation and the derivation of an adaptive algorithm that controls all discretization parameters. In addition to an explicit residual based error estimator, also an equilibration estimator with guaranteed bounds is discussed. Under certain mild assumptions it can be shown that the successive refinement produced by such an adaptive algorithm leads to a sequence of approximations with guaranteed convergence to the true solution. Numerical examples illustrate the practical behavior for some common benchmark problems. Additionally, an adaptive algorithm for a problem with a non-affine coefficient is shown. By transforming the original PDE a convection-diffusion problem is obtained, which can be treated similarly to the standard affine case

Quantum dynamics of coupled excitons and phonons in chain-like systems: Tensor train approaches and higher-order propagators

We investigate tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing memory consumption and computational costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train-based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of solutions is found only when the maximum number of ranks exceeds a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions commutating within each of the groups. In particular, the fourth-order Yoshida-Neri and the eighth-order Kahan-Li symplectic composition yield results close to machine precision. Similar results are found for fourth and eighth order global Krylov scheme. However, the computational effort currently restricts the use of these four propagators to rather short chains, which also applies to propagators based on the time-dependent variational principle, typically used for matrix product states. Yet, another class of propagators involves explicit, time-symmetrized Euler integrators. Especially, the fourth-order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for splitting or variational schemes

Graph-to-local limit for the nonlocal interaction equation

We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localizing infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretization for the equation under study

Mathematical Modeling of Blood Flow in the Cardiovascular System

This chapter gives a short overview of the mathematical modeling of blood flow at different resolutions, from the large vessel scale (three-dimensional, one-dimensional, and zero-dimensional modeling) to microcirculation and tissue perfusion. The chapter focuses first on the formulation of the mathematical modeling, discussing the underlying physical laws, the need for suitable boundary conditions, and the link to clinical data. Recent applications related to medical imaging are then discussed, in order to highlight the potential of computer simulation and of the interplay between modeling, imaging, and experiments in order to improve clinical diagnosis and treatment. The chapter ends presenting some current challenges and perspectives