Abstract

Recently, neural networks (NN) with an infinite number of layers have been introduced. Especially for these very large NN the training procedure is very expensive. Hence, there is interest to study their robustness with respect to input data to avoid unnecessarily retraining the network.

Typically, model-based statistical inference methods, e.g. Bayesian neural networks, are used to quantify uncertainties. Here, we consider a special class of residual neural networks and we study the case, when the number of layers can be arbitrarily large. Then, kinetic theory allows to interpret the network as a dynamical system, described by a partial differential equation. We study the robustness of the mean-field neural network with respect to perturbations in initial data by applying UQ approaches on the loss functions.

Reference

Foundations of Data Sience, Vol. 2, No. 3, pp. 257-278, 2020

Abstract

Functionals that penalize bending or stretching of a surface play a key role in geometric and scientific computing, but to date have ignored a very basic requirement: in many situations, surfaces must not pass through themselves or each other. This paper develops a numerical framework for optimization of surface geometry while avoiding (self-)collision. The starting point is the tangent-point energy, which effectively pushes apart pairs of points that are close in space but distant along the surface. We develop a discretization of this energy for triangle meshes, and introduce a novel acceleration scheme based on a fractional Sobolev inner product. In contrast to similar schemes developed for curves, we avoid the complexity of building a multiresolution mesh hierarchy by decomposing our preconditioner into two ordinary Poisson equations, plus forward application of a fractional differential operator. We further accelerate this scheme via hierarchical approximation, and describe how to incorporate a variety of constraints (on area, volume, etc.). Finally, we explore how this machinery might be applied to problems in mathematical visualization, geometric modeling, and geometry processing.

Reference

accepted for SIGGRAPH Asia 2021; to be published in ACM Trans. Graph

Download

arXiv: 2107.01664

Abstract

Both self-avoidance and self-contact of geometric objects can be modeled using repulsive energies that separate isotopy classes. Giving rise to nonlocal operators, they are interesting objects in their own right. Moreover, their analytical structure allows for devising numerical schemes enjoying robust features such as energy stability. This workshop aimed at discussing recent trends in this matter, including potential applications to modeling.

Download

Oberwolfach Reports, Volume 17, Issue 4, 2020, pp. 1823–1856

Abstract

Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This paper develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energy, which provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent, based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima—independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, non-crossing spline interpolation, flow visualization, and robotic path planning.

Reference

to appear in ACM Trans. Graph.

Download

arXiv:2006.07859

Abstract

We discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty}$-topology for piecewise-linear interpolation; and in (ii) the $W^{2,p}$-topology, $p \in [ 2, \infty [⁠$, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.

Reference

IMA Journal of Numerical Analysis, 2020

Abstract

In Gerhard et al. (2015a) a new class of adaptive Discontinuous Galerkin schemes has been introduced for shallow water equations, including the particular necessary properties, such as well-balancing and wetting-drying treatments. The adaptivity strategy is based on multiresolution analysis using multiwavelets in order to encode information across different mesh resolution levels. In this work, we follow-up on the previous proof-of-concept to thoroughly explore the performance, capabilities and weaknesses of the adaptive numerical scheme in the two-dimensional shallow water setting, under complex and realistic problems. To do so, we simulate three well-known and frequently used experimental benchmark tests in the context of flood modelling, ranging from laboratory to field scale. The real and complex topographies result in complex flow fields which pose a greater challenge to the adaptive numerical scheme and are computationally more ambitious, thus requiring a parallelised version of the aforementioned scheme. The benchmark tests allow to examine in depth the resulting adaptive meshes and the hydrodynamic performance of the scheme. We show that the scheme presented by Gerhard et al. (2015a) is accurate, i.e., allows to capture simultaneously large and very small flow structures, is robust, i.e., local grid refinement is controlled by just one parmeter that is auotmatically chosen and is more efficient in terms of the adaptive meshes than other shallow-water adaptive schemes achieving higher resolution with less cells.

Reference

Advances in Water Resources, 103559, 2020

Abstract

Gas flow in a pipeline network can be described by a hyperbolic balance law within each pipe along with coupling conditions at the node. For equilibrium or near equilibrium flows it is essential to design well-balanced schemes, in order to avoid spurious oscillations in the solution. Recently Chertock, Herty, and Ozcan introduced a well-balanced central Upwind scheme for 2 × 2 systems of balance laws. Here, we extend the scheme to model coupling conditions at intersection of pipes and compressor stations, thus resulting in a well-balanced scheme across the network.

Download

Conference proceedings “XVII International Conference (HYP2018) on Hyperbolic Problems: Theory, Numerics, Applications”, American Institute of Mathematical Sciences, 2020, 538-545

AIMS on Applied Mathematics, Vol. 10

Abstract

We prove a local variant of Einstein’s formula for the effective viscosity of dilute suspensions, that is, $\mu^\prime=\mu ({{1+\frac 5 2\phi+o(\phi)}})$, where $\phi$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $\phi=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.

Reference

SIAM J. Math. Anal. 52 (2020), no. 3, 2561–2591

Abstract

Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\Omega \subset \mathbb{R}^d⁠, d \in { 2,3 }⁠$, we investigate a fully discrete approximation scheme, using a spatial mixed finite element approximation on general shape-regular simplicial meshes combined with backward Euler time-stepping. We consider the case when the velocity field belongs to the space of solenoidal functions contained in $L^\infty(0,T;L^2 (\Omega)^d) \cap L^q (0,T;W \frac{1,q}{0}(\Omega)^d)$ with $q \in (2d/(d+2),\infty)⁠$, which is the maximal range of $q$ with respect to existence of weak solutions. In order to facilitate passage to the limit with the discretization parameters for the sub-range $q \in (2d/(d+2),(3d+2) / (d+2))$⁠, we introduce a regularization of the momentum equation by means of a penalty term, and first show convergence of a subsequence of approximate solutions to a weak solution of the regularized problem; we then pass to the limit with the regularization parameter. This is achieved by the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness techniques. For $q \geq (3d+2) / (d+2)$ convergence of a subsequence of approximate solutions to a weak solution can be shown directly, without the regularization term.

Reference

IMA J. Numer. Anal. 40 (2020), no. 2, 801–849

Abstract

We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic but conjectured assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.

Download

arXiv:2008.01801

accepted at SIAM J. Numer. Anal.

Abstract

We discuss random hyperbolic conservation laws and introduce a novel formulation interpreting the stochastic variables as additional spatial dimensions with zero flux. The approach is compared with established non-intrusive approaches to random conservation laws. In the scalar case, an entropy solution is proven to exist if and only if a random entropy solution for the original problem exists. Furthermore, existence and numerical convergence of stochastic moments is established. Along with this, the boundedness of the $L^1$-error of the stochastic moments by the $L^1$-error of the approximation is proven. For the numerical approximation a Runge-Kutta discontinuous Galerkin method is employed and a multi-element stochastic collocation is used for the approximation of the stochastic moments. By means of grid adaptation the computational effort is reduced in the spatial as well as in the stochastic directions, simultaneously. Results on Burger’s and Euler equation are validated by several numerical examples and compared to Monte Carlo simulations.

Download

www.igpm.rwth-aachen.de/forschung/preprints/514

Abstract

Previous works have developed boundary conditions that lead to the -boundedness of solutions to the linearised moment equations. Here we present a spatial discretization that preserves the -stability by recovering integration-by-parts over the discretized domain and by imposing boundary conditions using a simultaneous-approximation-term (SAT). We develop three different forms of the SAT using: (i) characteristic splitting of moment equation’s boundary conditions; (ii) decoupling of moments in moment equations; and (iii) characteristic splitting of Boltzmann equation’s boundary conditions. We discuss how the first two forms differ in terms of their usage and implementation. We show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary, and we argue that even though it provides stability, it prescribes the incorrect number of boundary conditions. Using benchmark problems, we compare the accuracy of moment solutions computed using different SATs. Our numerical experiments also provide new insights into the convergence of moment approximations to the Boltzmann equation’s solution.

Reference

J. Comput. Phys. 407 (2020), 109243, 31 pp.

Abstract

In [Commun. Pure Appl. Math., 2 (1949), pp. 331–407], Grad proposed a Hermite series expansion for approximating solutions to kinetic equations that have an unbounded velocity space. However, for initial boundary value problems, poorly imposed boundary conditions lead to instabilities in Grad’s Hermite expansion, which could result in nonconverging solutions. For linear kinetic equations, a method for posing stable boundary conditions was recently proposed for (formally) arbitrary order Hermite approximations. In the present work, we study $L^2$-convergence of these stable Hermite approximations and prove explicit convergence rates under suitable regularity assumptions on the exact solution. We confirm the presented convergence rates through numerical experiments involving the linearized BGK equation of rarefied gas dynamics.

Reference

SIAM J. Numer. Anal. 58 (2020), no. 2, 1164–1194

Abstract

We are interested in solving the Boltzmann equation of chemically reacting rarefied gas flows using the Grad’s-14 moment method. We first propose a novel mathematical model that describes the collision dynamics of chemically reacting hard spheres. Using the collision model, we present an algorithm to compute the moments of the Boltzmann collision operator. Our algorithm is general in the sense that it can be used to compute arbitrary order moments of the collision operator and not just the moments included in the Grad’s-14 moment system. For a first-order chemical kinetics, we derive reaction rates for a chemical reaction outside of equilibrium thereby, extending the Arrhenius law that is valid only in equilibrium. We show that the derived reaction rates (i) are consistent in the sense that at equilibrium, we recover the Arrhenius law and (ii) have an explicit dependence on the scalar fourteenth moment, highlighting the importance of considering a fourteen moment system rather than a thirteen one. Through numerical experiments we study the relaxation of the Grad’s-14 moment system to the equilibrium state.

Reference

Phys. A 574 (2021), Paper No. 125874, 19 pp.

Download

arXiv:2009.03013

Abstract

In the present paper we investigate generalizations of O’Hara’s Möbius energy on curves [1], to Möbius-invariant energies on non-smooth subsets of $\mathbb{R}^n$ of arbitrary dimension and co-dimension. In particular, we show under mild assumptions on the local flatness of an admissible possibly unbounded set $\Sigma\subset \mathbb{R}^n$ that locally finite energy implies that $\Sigma$ is, in fact, an embedded Lipschitz submanifold of $\mathbb{R}^n$ - sometimes even smoother (depending on the a priorily given additional regularity of the admissible set). We also prove, on the other hand, that a local graph structure of low fractional Sobolev regularity on a set $\Sigma$ is already sufficient to guarantee finite energy of $\Sigma$. This type of Sobolev regularity is exactly what one would expect in view of Blatt’s characterization [2] of the correct energy space for the Möbius energy on closed curves. Our results hold in particular for Kusner and Sullivan’s cosine energy $E_\textnormal{KS}$ [3] since one of the energies considered here is equivalent to $E_\textnormal{KS}$.

[1] O’Hara, J. Energy of a knot. Topology 30, 2 (1991), 241–247.

[2] Blatt, S. Boundedness and regularizing effects of O’Hara’s knot energies. J. Knot Theory Ramifications 21, 1 (2012), 1250010, 9.

[3] Kusner, R. B., and Sullivan, J. M. M¨obius energies for knots and links, surfaces and submanifolds. In Geometric topology (Athens, GA, 1993), vol. 2 of AMS/IP Stud. Adv. Math. Amer. Math. Soc., Providence, RI, 1997, pp. 570–604.

Download

arXiv:2010.03906

Abstract

This dissertation deals with minimisation problems related to magnetohydrodynamics. The main part of this thesis is divided in 5 chapters. In the first chapter it is shown that the magnetic energy under the so called helicity constraint admits a global minimiser for each prescribed value of the helicity and that all such minimisers are Beltrami fields, i.e. eigenvector fields of the curl operator. This result is already known in the settings of compact manifolds without boundary and bounded domains with smooth boundary in $\mathbb{R}^3$. We generalise these results to the setting of abstract, compact manifolds with boundary and by this we filled a gap in the literature. In the second chapter we consider the problem of finding a domain of prescribed volume for which the minimal energy in a given helicity class becomes minimal among all other minimal energies, in the same helicity class, of domains of the same volume. This problem was considered in the literature in the setting where the ambient space is $\mathbb{R}^3$. We generalise the known results to the setting where the ambient space is any Riemannian manifold and derive a second variation inequality, which contains terms involving the geometry of the ambient space. The question of whether or not an optimal domain exists is still open and subject of current research. In the third chapter the focus lies on the field line dynamics and zero set structure of Beltrami fields, hence in particular of energy minimising vector fields. Our most important results are on the one hand the observation that the restriction of a Beltrami field to the boundary, assuming it is tangent to it, on a simply connected, compact manifold is always a gradient field, which provides us with a good understanding of the boundary field line behaviour. On the other hand we show that the Hausdorff dimension of the zero set is at most one. This upper bound is already known for the zero set in the interior of the manifold. We show that this upper bound stays valid if we include the zeros on the boundary. Such a result appears to be new in the literature. In the fourth chapter we consider again the minimisation problem from the first chapter, but add an additional symmetry constraint. Results concerning the existence of rotationally symmetric Beltrami fields on domains in $\mathbb{R}^3$ are known in the literature. We generalise these results to the setting of abstract manifolds and develop new arguments to deal with general Killing vector fields. The last chapter deals with minimisation problems on $\mathbb{R}^3$. It is easy to see that the original energy functional does not admit any global minimisers if we consider it on the whole 3-space. That is why we consider two related energies under the helicity constraint. For the first energy it is shown, in view of Lions' “concentration compactness principle”, that the only possible obstruction for the existence of global minimisers are dichotomy effects. For the second energy we derive necessary conditions for global as well as local minimisers.

Reference

https://publications.rwth-aachen.de/record/809159

Abstract

The numerical solutions to nonlinear hyperbolic balance laws at (or near) steady state may develop spurious oscillations due to the imbalance between flux and source terms. In the present article, we study a high order well-balanced discontinuous Galerkin (DG) scheme for balance law with subsonic flow, which preserves equilibrium solutions of the flow exactly, and also provides non-oscillatory solutions for flow near equilibrium. The key technique is to reformulate the DG scheme in terms of global equilibrium variables which remain constant in space and time, and are obtained by rewriting the balance law in conservative form. We show that the proposed scheme is well-balanced and validate the scheme for various flows given by 2 × 2 hyperbolic balance law. We also extend the scheme to flows on networks, particularly to include the coupling conditions at nodes of the network.

Reference

J. Comput. Phys. 429 (2021), 110011

Download

https://www.igpm.rwth-aachen.de/forschung/preprints/507

Abstract

We present a positive and a negative stabilization result for a semilinear model of gas flow in pipelines. For feedback boundary conditions we obtain an unconditional stabilization result in the absence and conditional instability in the presence of the source term. We also obtain unconditional instability for the corresponding quasilinear model given by the isothermal Euler equations

Download

opus4.kobv.de

Abstract

We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein’s formula this effect is of the order of the particle volume fraction $\phi$. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like $|x|^{-2}$). Nevertheless, for well-prepared initial configurations and $\phi \to 0$, we show that the microscopic dynamics is approximated to order $\phi^2|log \phi|$ by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein’s formula. We provide quantitative estimates both for convergence of the densities in the $p$-Wasserstein distance for all $p$ and for the fluid velocity in Lebesgue spaces in terms of the $p$-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray.

Reference

Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 2021

Download

arXiv:2008.04813

Abstract

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian $3$-manifold $(\bar{M},g)$ with (possibly empty) boundary and a smooth flow of isometries $\phi_t:\bar{M}\rightarrow \bar{M}$ we show that, if $\bar{M}$ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field $\bm{X}$, tangent to the boundary, which is an eigenfield of curl and satisfies $(\phi_t)_{*}\bm{X}=\bm{X}$, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and $(\bar{M},g)$ has non-empty boundary, then Arnold’s structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable $3$-manifolds with and without boundary.

Reference

Differential Geometry and its Applications, Volume 78, 2021, 101801,ISSN 0926-2245

Download

arXiv:2006.14507

Abstract

This paper deals with the Aw-Rascle-Zhang model for traffic flow on uni-directional road networks. For the conservation of the mass and the generalized momentum, we construct weak solutions for Riemann problems at the junctions. We particularly focus on a novel approximation to the homogenized pressure by introducing an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approximation is able to approximate the homogenized pressure sufficiently well. The difference of the new approach compared with the Lighthill- Whitham-Richards model is also illustrated.

Download

arXiv:2005.12060

Abstract

In this paper we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected $3$-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected $1$-dimensional submanifolds with (possibly empty) boundary and tame knots. Further we consider the question of how complicated these tame knots can possibly be. To this end we prove that on the standard (open) solid toroidal annulus in $\mathbb{R}^3$, there exist for any pair $(p,q)$ of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric there exists a real analytic Beltrami field, corresponding to the eigenvalue $+1$ of the curl operator, whose zero set is precisely given by a standard $(p,q)$-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

Reference

J. Geom. Anal. (2021)

Download

arXiv:2005.07620

Abstract

We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded $1$-manifolds diffeomorphic to $\mathbb{R}$, which approach the zero set as time goes to $\pm \infty$. We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to $\pm\infty$. During the course of the proof we will in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably $1$-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most $1$. As a consequence we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.

Reference

Ann. Global Anal. Geom. 60 (2021), pages 65–82

Download

arXiv:2005.06590

Abstract

On the set of dissipative solutions to the multi-dimensional isentropic Euler equations we introduce a quasi-order by comparing the acceleration at all times. This quasi-order is continuous with respect to a suitable notion of convergence of dissipative solutions. We establish the existence of minimal elements. Minimizing the acceleration amounts to selecting dissipative solutions that are as close to being a weak solution as possible.

Download

arXiv:2005.03570

Abstract

Under general assumptions on the velocity field, it is possible to construct a flow that is forward untangled. Once such a flow has been selected, the associated transport problem is well-posed.

Download

arXiv:1912.06815

Abstract

We study the convergence of gradient flows related to learning deep linear neural networks (where the activation function is the identity map) from data. In this case, the composition of the network layers amounts to simply multiplying the weight matrices of all layers together, resulting in an overparameterized problem. The gradient flow with respect to these factors can be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$ matrices endowed with a suitable Riemannian metric. We show that the flow always converges to a critical point of the underlying functional. Moreover, we establish that, for almost all initializations, the flow converges to a global minimum on the manifold of rank $k$ matrices for some $k \leq r$.

Download

arXiv:1910.05505

Abstract

We introduce new coupling conditions for isentropic flow on networks based on an artificial density at the junction. The new coupling condition can be formally derived from a kinetic model by imposing a condition on energy dissipation. Existence and uniqueness of solutions to the generalized Riemann and Cauchy problem are proven. The result for the generalized Riemann problem is globally in state space. Furthermore, non-increasing energy at the junction and a maximum principle on the Riemann invariants are proven. Our approach generalizes to full gas dynamics.

Download

arXiv:2004.09184

Abstract

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.

Reference

J. Differential Equations 269 (2020), no. 2, 1192–1225

Download

arXiv:1908.03110

Abstract

Using the method of moments to approximate the solution of kinetic equations has become a standard technique especially in the context of solving the Boltzmann equation to model rarefied gases. One attractive example is the maximum-entropy closure, which, however, is computationally barely affordable. In this paper, we will combine the maximum-entropy approach with the quadrature method of moments (QMOM) – two methods which have in some sense diametric properties, thereby introducing the “Entropic Quadrature” (EQ) closure. EQ calculates a sparse quadrature-based reconstruction of the unknown velocity distribution like the QMOM where the physical meaningful maximum-entropy principle is employed as a criterion for selecting a quadrature among different quadratures fitting the given moments. We will discuss the construction of the closure and give several numerical examples of its performance in one and two dimensions.

Reference

J. Comput. Phys. 401 (2020), 108992, 25 pp.

Abstract

Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods, such as Stochastic Galerkin (SG), lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the generalized polynomial chaos (gPC) coefficients of the SG approximation, which allows a numerically cheap reduction of oscillations. The derived filter is based on Lasso regression which sets small gPC coefficients of high order to zero. We adaptively and automatically choose the filter strength to obtain a zero-valued highest order moment. The filtered SG method is tested for Burgers' and the Euler equations. Results show a reduction of oscillations at shocks, which leads to an improved approximation of expectation values and the variance compared to SG and IPM.

Reference

J. Comput. Phys. 403 (2020), 109073, 19 pp.

Download

arXiv:1808.00819

Abstract

We are interested in existence of gradient flows for shape functionals especially for first Laplacian eigenvalues. We introduce different techniques to prove existence and use different formulations for gradient flows. We apply a compactness argument to prove existence of generalized minimizing movements for Dirichlet and Robin boundary conditions with respect to several common metrics. Moreover, we use Brunn-Minkowski inequalities to prove α-convexity and existence of contraction semi-groups for Dirichlet boundary conditions on convex bodies. Finally, we give a proof of α-convexity for Robin boundary conditions by the second domain variation since we do not have a Brunn-Minkowski inequality in this case.

Download

arXiv:2003.01671

Abstract

Stochastic quantities of interest are expanded in generalized polynomial chaos expansions using stochastic Galerkin methods. An applicationto hyperbolic differential equations does in general not transfer hyperbolicity to the coefficients of the truncated series expansion. For the Haar basis and for piecewise linear multiwavelets we present convex entropies for the systems of coefficients of the one-dimensional shallow water equations by using the Roe variable transform. This allows to obtain hyperbolicity, wellposedness and energy estimates.

Reference

Commun. Comput. Phys., 27 (2020), pp. 639-671