Analysis of some nonlinear PDEs from multi-scale geophysical applications
Abstract: We investigate PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon give rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable. With this technique, the nonlinear interaction of vortical and wave dynamics is O(epsilon) when averaged in time. Initial data are ill-prepared, i.e. not restricted to be near geostrophic balance.
Two-layer shallow-water flows: recent mathematical results and the modelling of mixing
Abstract: We study the problem of long waves at the interface of two fluid layers of different densities and present three main results: (i) In the Boussinesq limit, the equations of mixed-type are well posed for times up to breaking if the initial data is in the hyperbolic region of phase space. The physical interpretation of this result is that a type of nonlinear stability holds for sufficiently large Richardson number (ii) In a non-Boussinesq formulation the result has to be modified, and there may be initially "stable" data that undergo shear instabilities. (iii) Foe breaking waves we propose a simple reformulation of the problem, with appropriate choices of conservation laws, that allow for the mixing of the two fluids. We apply this reformulation to the lock exchange problem.
Frequentist coverage of adaptive nonparametric Bayesian credible sets
We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of 'self-similar' parameters, in the sense of high probability of coverage of such parameters. On the negative side we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the Sobolev regularity scale.
Based on joint work with Botond Szabo and Aad van der Vaart
Numerical integration of SDEs with nonglobally Lipschitz coefficients
The talk will start with introduction to the problem of approximating solutions to regular SDEs which coefficients can grow faster than a linear function at infinity. In particular, the solution to this problem in the case of weak-sense approximations, the concept of rejecting exploding trajectories (CRE), will be recalled. CRE was introduced in [Milstein, Tretyakov SIAM J. Num. An. 2005] and it allows us to apply any usual method of weak approximation to SDEs with nonglobally Lipschitz coefficients. Then more recent results on mean-square SDE approximations in the nonglobally Lipschitz case will be considered including a special balanced-type scheme proposed in [Tretyakov, Zhang SIAM J. Num. An. 2013].
This balanced scheme is apparently the first explicit mean-square approximation for which convergence with order was proved when both drift and diffusion coefficients can grow faster than a linear function at infinity. Some numerical experiments comparing various mean-square schemes convergent in the nonglobally Lipschitz case will also be presented.
On some quasi-linear elliptic problems involving critical nonlinearities
We present some recent representation theorems for the Palais-Smale sequences associated to some classes of quasi-linear elliptic boundary-value problems. Applications will be given to prove existence and multiplicity results, in the spirit of Brezis and Nirenberg (Comm Pure Appl Math 36, 437–477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209–212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253–294, 1988).
Hierarchical Tensor Product Representations for the solution of High-Dimensional PDEs
The numerical solution of high-dimensional problems like differential equations in high dimensions is one of the most challenging problems in numerical mathematics, due to the Curse of Dimensionality. Typical problems of that kind are e.g. parametric PDE's arising from uncertainty quantification, Fokker Planck equation, many particle Schrödinger equation etc. Low rank tensor product approximation, in analogy to low rank matrix approximation, offer a possibility to circumvent the curse of dimensionality. However beyond the case d=2 (matrix case), fundamental mathematical problems appear. Recently introduced hierarchical Tucker representation (e.g Hackbusch et al.) offer new perspectives. We will compare these techniques with traditional tensor product approximations. As an improvement of the Tucker format, we will show that, for given ranks, the hierarchical tensors form a differentiable manifold. For solving optimization problems within a prescribed format, e.g. i) $L_2$-approximation, ii) linear equations or iii) eigenvalue problems, as well as dynamical PDE's, corresponding differential equations of gradient flow, and projected gradient methods will be developed. A simple optimization approach (ALS) based on alternating directions provides an efficient numerical tool, which will be demonstrated on several examples. A convergene theory is under development.
Tensor network techniques, e.g. matrix product states (MPS), have been developed in quantum physics. They are intimately related to the hierarchical tensor resentations. For example the alternating linear scheme (ALS) resembles the density matrix renormalization group algorithm (DMRG). Recent research tries to combine expertise from quantum physics with numerical mathematics.
Unexpected thermodynamic properties of some exact far-from-equilibrium solutions in molecular dynamics
We describe a time-dependent invariant manifold of the equations of molecular dynamics (MD). The manifold is independent of the description of the atomic forces within a general framework: it is exactly the same manifold whether the atoms are those of steel, water or air. Some version of this manifold is inherited by all accepted models of continuum mechanics, as well as the Boltzmann equation. Some of the MD solutions correspond to certain far-from equilibrium, unsteady flows, while others are naturally associated with the dynamics of particular nanostructures. Both simulations and a study of the Boltzmann equation suggest the presence of certain thermodynamic relations that are completely unexpected in these far-from-equilibrium situations. We describe these relations. Joint work with Kaushik Dayal and Stefan Muelller.
Adaptive discontinuous Galerkin methods for mass transfer through semi-permeable membranes
A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime. A posteriori error bounds and adaptive algorithms for the above class of models will also be discussed, as well as the potential of using other 'exotic' discretisations.
Rigorous Numerical Upscaling of Elliptic Multiscale Problems at High Contrast
We discuss the possibility of numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. Within the general framework of variational multiscale methods, we present a new approach based on novel quasi-interpolation operators with local approximation properties in L2 independent of the contrast. These quasi-interpolation operators have first been developed in the context and used in the analysis of robust domain decomposition methods. The analysis uses novel weighted Poincare inequalities and an abstract Bramble-Hilbert lemma. We show that for some relevant classes of high-contrast coefficients, optimal convergence without pre-asymptotic effects caused by microscopic scales or by the high contrast in the coefficient is possible. Ideas on how to extend the method and the analysis to more general coefficients will be discussed. Classes of coefficients that remain critical are characterized via numerical experiments.
Variational methods for lattice systems
I describe some variational models that arise from consideration of a physical flavour inspired by spin systems or atomistic interactions, and their analysis using techniques of homogenization, integral representation and variational flow.
Locally periodic homogenization and elastic properties of the exoskeleton of a lobster
Many biological or industrial composite materials comprise non-periodic microscopic structures, e.g. fibrous microstructure with varied orientation of fibres in exoskeletons or industrial filters, space-dependent perforations in concrete, non-periodic distribution of cells in a tissue.
An interesting and important for applications special case of non-periodic microstructures is so called locally periodic microstructure, where spatial changes of the microstructure are observed on a scale smaller than the size of the considered domain but larger than the characteristic size of the microstructure.
In this talk I would like to present the generalisation of the well-known in the homogenization theory periodic two-scale convergence to locally periodic situations. The developed theory is then applied to derive macroscopic elastic properties of the exoskeleton of a lobster, comprising plywood structure. The plywood structure is a non-periodic microstructure characterised by the superposition of gradually rotated planes of parallel aligned fibres.
Mesh grid alignment and the solution of the Monge Ampere equation
Mesh generation is an important part of the numerical solution of many PDEs. If the PDE has evolving structure on small scales then it is often essential that the mesh adapts to resolve these scales. One way of doing this is to move the mesh points into regions where greater resolution is needed. Such movement can be regarded as the action of a map from a regular mesh into an adapted mesh. Thus mesh generation can be studied in terms of the properties of this map and the
differential equations that it satisfies.
In this talk I will look at a class of such maps derived from solutions of the (fully nonlinear) Monge Ampere equation. I will show that such maps can be generated easily and moreover the global regularity of the mesh can be understood in terms of the regularity of the solution of the Monge Ampere equation. In particular I will demonstrate that such meshes align themselves very well with underlying features of the solution and thus are effective in approximating these features.
A new self-consistent atomistic - phase field model and its application to Germanium nano-crystallization
In this talk we present a new multiscale model for phase transformation in a heat bath at constant temperature. The model consists on a thermodynamically consistent phase field model that reproduces exactly the interface energetics and kinetics of atomistically computed crystallization fronts. As an additional feature, the interface thickness may be chosen arbitrarily large while preserving this exact atomistic-to-continuum coupling, thus delivering a highly efficient multiscale computational model. By means of this multiscale approach, we study the interplay between nucleation and growth in the nano-crystallization of amorphous Ge. We find simple scaling laws between the mean radius of crystallized Ge grains, the nucleation rate and the time of crystallization.