Talks (titles and abstracts)
Yann Brenier: Solving initial value problems by space-time convex optimization
I will explain a possible strategy to recover solutions of nonlinear evolution PDEs by space-time convex optimization based on their weak formulation, the simplest examples being the Burgers equation and the quadratic porous medium equations.
Alfred Bruckstein: Global Swarming Behaviors from Myopic Interactions
We discuss several myopic-sensing-based interactions in swarms of simple mobile agents that lead to useful global behaviors in multi-agent robotic tasks. The inspiration for such systems comes from amazing feats performed by ant colonies, schools of fish and starling flocks. The mathematical tools for analyzing such problems interestingly combine graph theory with continuous dynamic systems and various types of optimization processes.
Russel Caflisch: Optimization of the Boltzmann Equation
The kinetics of rarefied gases and plasmas are described by the Boltzmann equation and numerically approximated by the Direct Simulation Monte Carlo (DSMC) method. We present an optimization method for DSMC, derived from an augmented Lagrangian. After a forward (in time) solution of DSMC, adjoint variables are found by a backwards solver. They are equal to velocity derivatives of an objective function, which can then be optimized. This is joint work with Yunan Yang (Cornell) and Denis Silantyev (U Colorado, Colorado Springs).
José Antonio Carrillo: Global minimizers of Interaction Energies
I will review the existence and uniqueness of global minimizers for interaction energy functionals. Euler-Lagrange equations in the infinity wasserstein distance will be discussed. Based on linear convexity/concavity arguments, qualitative properties of the global minimizers will also be treated. Anisotropic singular potentials appearing in dis- locations will be shown to have rich qualitative properties with loss of dimension and ranges of explicit minimizers. A large part of the course will be based on several works in collaboration with Ruiwen Shu (University of Oxford).
Alina Chertock: A New Locally Divergence-Free Path-Conservative Central-Upwind Scheme for Ideal and Shallow Water Magnetohydrodynamics
In this talk, we present a new second-order unstaggered semi-discrete path-conservative central-upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, does not rely on any (approximate) Riemann problem solver, and robustly produces high-resolution and non-oscillatory results. The scheme is derived from the Godunov-Powell nonconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the modified systems with the evolution equations for the corresponding derivatives of the magnetic field components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic field, which guarantees a non-oscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. We test the proposed PCCU scheme on several benchmarks for ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution but also to preserve the positivity of computed quantities such as density, pressure, and water depth.
Albert Cohen: Stable nonlinear inversion application and application to interface reconstruction from cell-averages
In this lecture, we first discuss measures of optimal dimensionality reduction that may be thoughts as benchmark for efficient numerical methods in forward simulation or inverse problem. While the concept of n-width, introduced in 1936 by Kolmogorov, is well taylored to linear methods, the study of analogous concepts for nonlinear approximation is still the object of ongoing research. We then present a general framework for solving inverse problems using nonlinear approximation spaces. The main principles build up on the so called Parametrized Background Data Weak method (PBDW), which can be thought as a linear counterpart. As an application we study the reconstruction of sharp interfaces from cell average at coarse resolutions for which linear methods are known to be uneffective. We discuss the convergence rates of these reconstructions and their optimality.
Pierre Degond: Macroscopic dynamics of systems of self-propelled rigid-bodies
Collective dynamics in systems of self-propelled particles has stimulated intense mathematical research in the recent years. Many different models have been proposed but most of them rely on point particles. In practice, particles often have more complex geometrical structures. Here, we will consider particles as rigid bodies whose body attitude is described by an orthonormal frame. Particles tend to align their frame with those of their neighbours. A hydrodynamic model will be derived when the number of particles is large. This hydrodynamic model is new: its unknowns are a scalar and a rotation matrix respectively describing the density and mean body attitude. Previously, the hydrodynamic limit was derived in dimension 3 only, or for a different model of BGK type. The Fokker-Planck case in dimension n>3 is considerably more complex. Concepts from Lie-group theory are central to this study. In this talk, we will highlight the main difficulties of this derivation and hint to their solution.
Ronald DeVore: An analysis of PINNS for numerically solving elliptic differential equations
We analyze PINNS as a numerical method for solving elliptic partial differential equations. This analysis identifies changes that should be made to the PINNS loss function in order to guarantee the accuracy of this method for approximating the solution of the PDE in the energy norm. Rates of convergence are also proved.
Qiang Du: Models with nonlocal interactions on bounded domains
We consider nonlocal integrodifferential equations on bounded domains with nonlocal interactions of a finite range. Such problems have received much attention in various applications. We begin by reviewing some earlier works on problems involving nonlocal boundary conditions. We then present recent studies on local boundary conditions. In particular, we discuss the design of boundary localization and the properties of boundary localized convolutions. We establish the desired nonlocal Green's identity and the well-posedness of nonlocal variational problems and coupled local-nonlocal models via local interface conditions.
Björn Engquist: Towards Seamless Numerical Homogenization
The computational challenges from multiscale differential equations have inspired development of a variety of methodologies. Many of them require scale separation but there are also efforts to reduce the dependence on substantial scale gaps even for techniques with sublinear computational complexity. One such class of techniques is the, so called, seamless methods for multiscale dynamical systems. We will prove equivalence between such dynamical systems and one dimensional elliptic multiscale problems. A multidimensional generalization leads to a new class of numerical methods for PDE homogenization problems based on rescaling. This is partially similar to classical techniques involving rescaling as, for example, turbulence modeling, artificial compressibility for incompressible flow and the Car-Parinello technique for molecular dynamics. We will give error estimates and simple numerical examples.
Benjamin Gess: From large deviations around porous media, to PDEs with irregular coefficients, to gradient flow structures
We consider the large deviations of the rescaled zero-range process about its hydrodynamic limit, the porous medium equation. This leads to the analysis of the skeleton equation, an energy-critical, degenerate parabolic-hyperbolic PDE with irregular drift. In this talk, we present a robust well-posedness theory for such PDEs based on concepts of renormalized solutions, the equation's kinetic form, and commutator estimates. The relationship of these large deviations principles to a formal gradient flow interpretation of the porous medium equation will be demonstrated by deducing an entropy dissipation equality from the large deviations and reversibility.
Helge Holden: The Camassa-Holm equation with transport noise
We will discuss recent work regarding the Camassa—Holm equation with transport noise, more precisely, the equation \(u_t+uu_x+P_x+\sigma u_x \circ dW=0\) and \(P-P_{xx}=u^2+u_x^2/2\). In particular, we will show existence of a weak, global, dissipative solution of the Cauchy initial-value problem on the torus. This is joint work with L. Galimberti (King’s College), K.H. Karlsen (Oslo), and P.H.C. Pang (NTNU/Oslo).
Thomas Hou: Potentially singular behavior of 3D incompressible Navier-Stokes equations
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present some numerical evidence that the 3D Navier-Stokes equations develop nearly self-similar singular scaling properties with maximum vorticity increased by a factor of \(10^7\). This potentially singular behavior is induced by a potential finite time singularity of the 3D Euler equations. Unlike the Hou-Luo blowup scenario, the potential singularity of the 3D Euler and Navier-Stokes equations occurs at the origin. We have applied several blowup criteria to study the potentially singular behavior of the Navier-Stokes equations. The Beale-Kato-Majda blow-up criterion, the blowup criteria based on the growth of enstrophy and negative pressure, the Ladyzhenskaya-Prodi-Serrin regularity criteria all seem to imply that the Navier-Stokes equations develop potentially singular behavior. Finally, we present some new numerical evidence that a variant of the axisymmetric Navier-Stokes equations with time dependent fractional dimension develops nearly self-similar blowup with maximum vorticity increased by a factor of \(10^{30}\).
Shi Jin: Quantum Computation of partial differential equations and related problems
Quantum computers have the potential to gain algebraic and even up to exponential speed up compared with its classical counterparts, and can lead to technology revolution in the 21st century. Since quantum computers are designed based on quantum mechanics principle, they are most suitable to solve the Schrodinger equation, and linear PDEs (and ODEs) evolved by unitary operators. The most efficient quantum PDE solver is quantum simulation based on solving the Schrodinger equation. It will be interesting to explore what other problems in scientific computing, such as ODEs, PDEs, and linear algebra that arise in both classical and quantum systems, can be handled by quantum simulation.
We will present a systematic way to develop quantum simulation algorithms for general differential equations. Our basic framework is dimension lifting, that transfers nonlinear PDEs to linear ones, and linear ones to Schrodinger type PDEs. For non-autonomous PDEs and ODEs, or Hamiltonian systems with time-dependent Hamiltonians, we also add an extra dimension to transform them into autonomous PDEs that have only time-independent coefficients, thus quantum simulations can be done without using the cumbersome Dyson’s series and time-ordering operators. Our formulation allows both qubit and qumode (continuous-variable) formulations, and their hybridizations, and provides the foundation for analog quantum computing.
Alexander Kiselev: Regularity of vortex and SQG patches
Patch solutions are an important class of special singular solutions to the 2D Euler or surface quasi-geostrophic (SQG) equations that model evolution of regions of vorticity with sharp boundaries (like hurricanes) or sharp temperature fronts in atmosphere. I will discuss recent progress on regularity properties of vortex and SQG patches. In particular, I will present an example of a vortex patch with continuous initial curvature that immediately becomes infinite but returns to \(C^2\) class at all integer times only without being time periodic. The proof involves derivation of a new system describing the patch evolution in terms of arc-length and curvature. A similar approach leads to discovery of strong ill-posdness of the SQG patches in all but \(L^2\) based spaces, or spaces of infinitely smooth functions. The talk is based on a work joint with Xiaoyutao Luo.
Alexander Kurganov: Central-Upwind Schemes with Reduced Numerical Dissipation
Central-upwind schemes are Riemann-problem-solver-free Godunov-type finite-volume schemes, which are, in fact, non-oscillatory central schemes with a certain upwind flavor: derivation of the central-upwind numerical fluxes is based on the one-sided local speeds of propagation, which can be estimated using the largest and smallest eigenvalues of the Jacobian.
I will introduce two new classes of central-upwind schemes with reduced numerical dissipation. First, we will use a sub-cell resolution at the projection step to enhance the resolution of contact waves, which are typically badly affected by excessive numerical dissipation present in numerical methods. The second approach is based on the utilization of the local characteristic decomposition for the modification of the numerical diffusion of the central-upwind schemes. Both approaches help to significantly reduce the amount of numerical dissipation present in central-upwind schemes without risking large spurious oscillation. Applications to several hyperbolic systems of conservation laws will be discussed.
Qin Li: Speeding up gradient flows on probability measure space
In the past decade, there has been a significant shift in the types of mathematical objects under investigation, moving from vectors and matrices in the Euclidean spaces to functions residing in Hilbert spaces, and ultimately extending to probability measures within the probability measure space. Many questions that were originally posed in the context of linear function spaces are now being revisited in the realm of probability measures. One such question is to the efficiently find a probability measure that minimizes a given objective functional.
In Euclidean space, we devised optimization techniques such as gradient descent and introduced momentum-based methods to accelerate its convergence. Now, the question arises: Can we employ analogous strategies to expedite convergence within the probability measure space?
In this presentation, we provide an affirmative answer to this question. Specifically, we present a series of momentum-inspired acceleration method under the framework of Hamiltonian flow, and we prove the new class of method can achieve arbitrary high-order of convergence. This opens the door of developing methods beyond standard gradient flow.
Pierre-Louis Lions: Large random matrices, Coulomb and Riesz gases: equations and control
Hailiang Liu: Global dynamics and photon loss in the Kompaneets equation
The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose-Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose-Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as times tends to infinity, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in \(L^1\). None of these results impose a boundary condition at the zero-energy boundary. This is a joint work with Joshua Ballet, Gautam Iyer, David Levermore, and Robert Pego.
Pierangelo Marcati: Mathematical problems in Quantum Hydrodynamics
I will briefly review some classical aspects of quantum mechanics both from the point of view of De Broglie Bohm and in the theory of effective equations in the perspective of Feynman, Landau, Gross and Pitaevskii within the theory of Superfluids and Bose Einstein's condensation. We show how it is possible to construct a coherent mathematical theory, which provides a rigorous mathematical framework and allows for the presence of quantum vortices, within a hydrodynamic description. The theory (in 3-D and 2-D) is based on a polar factorization tool and works for general weak solutions with finite energy and without additional regularity requirements. I will describe some recent 1-D and 2-D results obtained in collaboration with Paolo Antonelli and Hao Zheng, where we introduce various new tools, such as wave function lifting and generalized chemical potential, which allows to develop a genuinely hydrodynamics theory. Namely when initial data are not provided by the Madelung transform.
Antoine Mellet: Phase separation and free boundary problems for aggregation-diffusion phenomena
We investigate phase separation phenomena that arise as a result of the competition between nonlocal attraction and local repulsion in classical aggregation-diffusion models (we will mainly discuss the Patlak-Keller-Segel model). We will present some recent results showing that when a large population of organisms is observed at some appropriate scales (in time and space), a sharp interface appears, separating regions of high and low density. The derivation of free boundary problems to describe the motion of this interface shows that the collective dynamics is driven, at macroscopic scales, by surface tension and contact angle phenomena.
Helena Nussenzveig Lopes: Inviscid limit in 2D avoids inviscid dissipation and anomalous dissipation
We say inviscid dissipation occurs when a vanishing viscosity limit does not satisfy energy balance. A closely related phenomenon is anomalous dissipation, where, in the limit of vanishing viscosity, the total dissipation does not vanish. In this talk we will discuss recent results on avoiding these phenomena in 2D incompressible flows, with and without forcing.
Benoît Perthame: Structured equations in biology; relative entropy, Monge-Kantorovich distance
Models arising in biology are often written in terms of Ordinary Differential Equations. The celebrated paper of Kermack-McKendrick (1927), founding mathematical epidemiology, showed the necessity to include parameters in order to describe the state of the individuals, as time elapsed after infection. During the 70s, many mathematical studies were developed when equations are structured by age, size, more generally a physiological trait. The renewal, growth-fragmentation are the more standard equations.
The talk will present structured equations, show that a universal generalized relative entropy property is available in the linear case, which imposes relaxation to a steady state under non-degeneracy conditions. In the nonlinear cases, it might be that periodic solutions occur, which can be interpreted in biological terms, e.g., as network activity in the neuroscience.
When the equations are conservation laws, a variant of the Monge-Kantorovich distance (called Fortet-Mourier distance) also gives a general non-expansion property of solutions.
Denis Serre: Compensated integrability on tori ; a priori estimate for space-periodic gas flows
We extend our theory of Compensated Integrability of positive symmetric tensors, to the case where the domain is the product of a linear space \({\mathbb R}^k\) and of a torus \({\mathbb R}^m/\Lambda\), \(\Lambda\) being a lattice of \({\mathbb R}^m\). We apply our abstract results in two contexts, in which \(k=1\) is associated with a time variable, while \(m=d\) is a space dimension. On the one hand to \(d\)-dimensional gas dynamics, governed by the Euler equations, when the initial data is space-periodic~; we obtain an a priori space-time estimate of our beloved quantity \(\rho^{1\over d}p\). On the other hand to hard spheres dynamics in a periodic box \(L{\mathbb T}_d\). We obtain a weighted estimate of the average number of collisions per unit time, provided that the "linear density" \(Na/L\) (\(N\) particles of radius \(a\)) is below some threshold.
Roman Shvydkoy: Environmental Averaging: unified framework for alignment systems and its implementation to kinetic models
In this talk we will discuss a new analytical framework which unifies many variants of alignment models such as Cucker-Smale, Motsch-Tadmor, multi-flocks, etc. The purpose of this framework is to understand functional properties of the core components of the alignment force -- communication strength and density-averaged operation -- and their effect on the collective dynamics of a particular differential system. One application of this framework is the proof of global relaxation for a class of Fokker-Planck-alignment models which includes the classical Cucker-Smale protocol.
Edriss Titi: On a generalization of the Bardos-Tartar conjecture for nonlinear dissipative PDEs
In this talk I will show that every solution of a KdV-Burgers-Sivashinsky type equation blows up in the energy space, backward in time, provided the solution does not belong to the global attractor. This is a phenomenon contrast to the backward behavior of the 2D Navier-Stokes equations, subject to periodic boundary condition, studied by Constantin, Foias, Kukavica and Majda, but analogous to the backward behavior of the Kuramoto-Sivashinsky equation discovered by Kukavica and Malcok. I will also discuss the backward behavior of solutions to the damped driven nonlinear Schrödinger equation, the complex Ginzburg-Landau equation, and the hyperviscous Navier-Stokes equations. In addition, I will provide some physical interpretation of various backward behaviors of several perturbations of the KdV equation by studying explicit cnoidal wave solutions. Furthermore, I will discuss the connection between the backward behavior and the energy spectra of the solutions. The study of backward behavior of dissipative evolution equations is motivated by a conjecture of Bardos and Tartar which states that the solution operator of the two-dimensional Navier-Stokes equations maps the phase space into a dense subset in this space.
Emil Wiedemann: Non-Deterministic Solution Concepts in Fluid Dynamics
As more and more ill-posedness results have been shown for fluid PDEs (not only by convex integration!), the idea to solve the Cauchy problem by some unique weak or entropy solution has become questionable. Instead, non-deterministic solution concepts such as measure-valued or statistical have sparked much recent research interest. They also seem to be more in line with well-known theories of turbulence, which are typically statistical. I will give an overview of such generalised solution concepts, including their weak-strong stability, their relation to more conventional solutions, and questions of existence.