Abstract

Quantifying the output uncertainties of a computational model is a key step to making the model predictive. Hyperbolic conservation laws are a class of models with many applications, but also with specific challenges (such as discontinuous solutions, or non-uniqueness of weak solutions). In this course, we will briefly review theory and numerical methods for hyperbolic conservation laws, in view of uncertain data. We also give an overview about different methods for uncertainty quantification, before taking a closer look at a specific class of entropy-based stochastic Galerkin methods, and ways to make these methods competitive.

Abstract

We consider the polarization of a cell in response to an outer signal. The mathematical model consists of a diffusion equation in the inner volume coupled to a reaction diffusion system on the cell membrane. In a certain asymptotic limit we rigorously prove the convergence towards a generalized obstacle problem. In term of this limit system we derive conditions for the onset of polarization.

(This is joint work with Anna Logioti, Barbara Niethammer and Juan Velazquez)

Abstract

Often in dynamic optimal control problems with a long time horizon, in a large neighburhood of the middle of the time interval the optimal control and the optimal state are very close to the solution of a static control problem that is derived from the dynamic optimal control problems by omitting the information about the initial state and possibly a desired terminal state.

This can be shown in different situations, for example under exact controllability assumptions or with the assumption of nodal profile exact controllability, as studied by Tatsien Li and his group. In this situation, we obtain the turnpike property with interior decay, that has been discussed in the paper Mathematics of Control, Signals, and Systems volume 33, pages 237–258 (2021).

Abstract

Neural Differential Equations (NDEs) demonstrate that neural networks and differential equations are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures (e.g. residual networks, recurrent networks, StyleGAN2, coupling layers) are discretisations. By treating differential equations as a learnt component of a differentiable computation graph, then NDEs extend current physical modelling techniques whilst integrating tightly with current deep learning practice.

NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. They are particularly suitable for tackling dynamical systems, time series problems, and generative problems.

This talk will offer a dedicated introduction to the topic, with examples including neural ordinary differential equations (e.g. to model unknown physics), neural controlled differential equations (“continuous recurrent networks”; e.g. to model functions of time series), and neural stochastic differential equations (e.g. to model time series themselves). If time allows I will discuss other recent work, such as novel numerical neural differential equation solvers. This talk includes joint work with Ricky T. Q. Chen, Xuechen Li, James Foster, and James Morrill.

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.

Abstract

The optimal transportation of one measure into another, leading to the notion of their Wasserstein distance, is a problem in the calculus of variations with a wide range of applications. The regularity theory for the optimal map is subtle and was revolutionized by Caffarelli. This approach relies on the fact that the Euler-Lagrange equation of this variational problem is given by the Monge-Ampère equation. The latter is a prime example of a fully nonlinear (degenerate) elliptic equation, amenable to comparison principle arguments.

We present a purely variational approach to the regularity theory for optimal transportation, introduced with M. Goldman and refined with M. Huesmann. Following De Giorgi’s philosophy for the regularity theory of minimal surfaces, it is based on the approximation of the displacement by a harmonic gradient, through the construction of a variational competitor. This leads to a “one-step improvement lemma”, and feeds into a Campanato iteration on the $C^{1,\alpha}$-level for the optimal map, capitalizing on affine invariance.

On the one hand, this allows to re-prove the $\epsilon$-regularity result (Figalli-Kim, De Philippis-Figalli) bypassing Caffarelli’s celebrated theory. This also extends to general cost functions, which is joint work with M. Prodhomme and T. Ried.

On the other hand, due to its robustness and low-regularity approach, it can be used to study the popular problem of matching two independent Poisson point processes. For example, it can be used to prove non-existence of a stationary cyclically monotone coupling, which is joint work with M. Huesmann and F. Mattesini.

Abstract

We revisit the variational characterization of diffusion as entropic gradient flux, established by Jordan, Kinderlehrer, and Otto in [1], and provide for it a probabilistic interpretation based on stochastic calculus. It was shown in [1] that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the minimum rate of entropy dissipation along the Fokker-Planck flow and measure exactly the deviation from this minimum that corresponds to any given perturbation. As a bonus of the perturbation analysis, we derive the so-called HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).

Joint work with I. Karatzas and B. Tschiderer.

References

[1] R. Jordan, D. Kinderlehrer, and F. Otto (1998) The variational formula of the Fokker-Planck equation. SIAM Journal on Mathematical Analysis 29, no. 1, 1–17.

Abstract

In this talk we consider the pressureless Euler system in dimension greater than or equal to two. Several works have been devoted to the search of solutions which satisfy the following adhesion or sticky particle principle: if two particles of the fluid do not interact, then they move freely keeping constant velocity, otherwise they join with velocity given by the balance of momentum. For initial data given by a finite number of particles pointing each in a given direction, in general dimension, it is easy to show that a global sticky particle solution always exists and is unique. In dimension one, sticky particle solutions have been proved to exist and be unique. In dimension greater or equal than two, it was shown that as soon as the initial data is not concentrated on a finite number of particles, it might lead to non-existence or non-uniqueness of sticky particle solutions.

In collaboration with S. Bianchini, we show that even though the sticky particle solutions are not well-posed for every measure-type initial data, there exists a comeager set of initial data in the weak topology giving rise to a unique sticky particle solution. Moreover, for any of these initial data the sticky particle solution is unique also in the larger class of dissipative solutions (where trajectories are allowed to cross) and is given by a trivial free flow concentrated on trajectories which do not intersect. In particular for such initial data there is only one dissipative solution and its dissipation is equal to zero. Thus, for a comeager set of initial data the problem of finding sticky particle solutions is well-posed, but the dynamics that one sees is trivial. Our notion of dissipative solution is lagrangian and therefore general enough to include weak and measure-valued solutions.

Abstract

We will start with a very basic introduction into the principles of Mathematica as programming/computing environment. This includes the use of lists, definition of functions, solving of equations and plotting. The audience can try most of the examples themselves in real time with the online cloud service. The course will then focus on interactive computing, prototyping numerical methods (with finite difference as example) and visualization of simulation data.

Abstract

Many systems of shallow water type arise in the modeling of thin layer flows. The sources can take several forms, and for each of them the question of building a well-balanced scheme comes out. We are interested here in the case when it is required to build a numerical scheme that preserves two nontrivial families of steady states at rest. Such a scheme can be called multi well-balanced. In order to apply the reconstruction method one has to manage with the two families at the same time. I will show how it can be possible while at the same time verifying a semi-discrete entropy inequality. Two examples of entropy satisfying multi well-balanced schemes will be given: the shallow water MHD system with topography, and a shallow water system with internal variable and topography.

Abstract

We will present a method to efficiently compute Wasserstein gradient flows. The approach is based on a generalization of the back-and-forth method that we developed with Matt Jacobs to solve optimal transport problems. The gradient flow is evolved by solving the dual problem to the JKO scheme: in general, this dual problem is much better behaved than the primal problem. This allows to efficiently run large scale gradient flows simulations for a large class of internal energies including singular and non-convex energies. This is joint work with Matt Jacobs and Wonjun Lee.

Abstract

In this talk, I present a phase-field theory for enriched continua, exposing the geometry of gradient flows. We begin the theory with a set of postulate balances on nonsmooth open surfaces to characterize the fundamental fields. By considering nontrivial interactions inside the body, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a generalized Swift–Hohenberg equation-a second-grade phase-field equation-and its conserved version, the generalized phase-field crystal equation-a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. Configurational forces are a generalization of Newtonian forces to describe the kinetics and kinematics of manifolds. We conclude with the presentation of a comprehensive and thermodynamically consistent set of boundary conditions.

Based on: Espath & Calo, Phase-field gradient theory. 2021, ZAMP. DOI: 10.1007/s00033-020-01441-2.

Abstract

This course aims at presenting some of the ideas behind the (conditional) existence and (weak-strong) uniqueness theory for distributional solutions to mean curvature flow. Focusing on the simple two-phase case, i.e., the evolution of a closed hypersurface, allows for a self-contained and concise presentation, which is accessible for graduate students with some background in PDEs and basic measure theory.

The first lecture provides an overview, basic examples, exercises, and some computational tools. In the second lecture, distributional solutions and a (conditional) closure theorem are presented. If time permits, a relation to the viscosity solution in the two-phase case will be explained. The last lecture is devoted to the weak-strong uniqueness principle in the class of distributional solutions.

Abstract

This course aims at presenting some of the ideas behind the (conditional) existence and (weak-strong) uniqueness theory for distributional solutions to mean curvature flow. Focusing on the simple two-phase case, i.e., the evolution of a closed hypersurface, allows for a self-contained and concise presentation, which is accessible for graduate students with some background in PDEs and basic measure theory.

The first lecture provides an overview, basic examples, exercises, and some computational tools. In the second lecture, distributional solutions and a (conditional) closure theorem are presented. If time permits, a relation to the viscosity solution in the two-phase case will be explained. The last lecture is devoted to the weak-strong uniqueness principle in the class of distributional solutions.

Abstract

This course aims at presenting some of the ideas behind the (conditional) existence and (weak-strong) uniqueness theory for distributional solutions to mean curvature flow. Focusing on the simple two-phase case, i.e., the evolution of a closed hypersurface, allows for a self-contained and concise presentation, which is accessible for graduate students with some background in PDEs and basic measure theory.

The first lecture provides an overview, basic examples, exercises, and some computational tools. In the second lecture, distributional solutions and a (conditional) closure theorem are presented. If time permits, a relation to the viscosity solution in the two-phase case will be explained. The last lecture is devoted to the weak-strong uniqueness principle in the class of distributional solutions.

Abstract

Nuclear fusion reactor design crucially depends on numerical simulation. The plasma can usually be modeled using fluid equations (for mass, momentum and energy). However, the reactor also contains neutral (non-charged) particles (which are important in its operation), of which both the position and velocity distribution is important. This leads to a Boltzmann-type transport equation that needs to be discretised with a Monte Carlo method. One then obtains a coupled finite-volume/Monte-Carlo simulation, of which the results possess both a bias and a variance. In this talk, I introduce the problems associated with the simulation of the plasma edge region in a fusion reactor. I discuss how to couple a finite volume discretisation of the plasma equations with a Monte Carlo simulation of the neutral particles, and show how the Monte Carlo errors affect convergence of steady state computations and reliability of gradient computations (necessary during optimization).

Abstract

We discuss examples of constructing solutions for conservation laws wi-thout spatial BV bounds. The convergence of spectral viscosity and finite volume approximationsare prototype examples for such BV-free scalar constructions. The first example involves hyper-viscosity limits for 1D scalar equationswith convex fluxes : existence of entropy solutions follows by compensatedcompactness arguments, based on one entropy production bound. A second example deals with solutions of 2D scalar equations : we showthat a judicious choice of two entropies entropy bounds will suffice. We then raise several open questions : here comes a third ’example’ ofEuler alignment system.

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Abstract

Understanding the behaviour of flows at the nanoscale holds the key to unlocking a myriad of emerging technologies. However, accurate experimental observation is complex due to the small spatio-temporal scales of interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.

At such scales, the classical Navier-Stokes paradigm no longer provides an accurate description of the flow physics; however, microscopic models such as molecular dynamics (MD) become computationally intractable for most flows of practical interest. In this talk talk I will consider the influence of thermal fluctuations, which we will see are key to understanding counter-intuitive phenomena in nanoscale interfacial flows. A `top down’ framework that incorporates thermal noise is provided by fluctuating hydrodynamics and we shall use this model to gain insight into interfacial nanoflows such as drop coalescence, jet breakup and thin film rupture, using MD as a benchmark.

See also here.

Abstract

We will discuss recent results on intrusive stochastic Galerkin expansion for hyperbolic transport problems with a focus on the role of entropy. Recent results on hyperbolicity and well-posedness of the expanded hyperbolic systems will be presented and numerical results will be discussed. A focus is on 2x2 nonlinear hyperbolic systems as appearing in shallow water or gas flow problems.

Abstract

Since the solution of hyperbolic conservation laws typically exhibits discontinuities, efficient numerical schemes will employ locally refined discretizations that dynamically adapt to the solution. To trigger grid refinement and coarsening an appropriate indicator is needed. Due to the lack of a stable variational formulation for the problem at hand, we apply a multiresolution analysis (MRA) based on multiwavelets to perform data compression. The MRA is combined with a standard discontinuous Galerkin (DG) scheme to end up with an adaptive DG scheme. The framework of both the MRA and the DG scheme will be presented in some detail. The performance of the resulting scheme will be discussed by means of numerous testcases in 1D and 2D for scalar as well as systems of conservation laws. We conclude with a brief outlook how to employ the adaptive framework for the investigation of stochastic conservation laws.