Séminaire de calcul scientifique du CERMICS

Examples of Error Estimates and Adaptivity for ROM of time-dependent PDEs

Reduced-order models (ROMs) have received lots of attention for simulating dynamical systems at a reduced cost. Numerical simulations have highlighted the efficiency of model-order reduction for many applications. When a dimension reduction occurs, estimating the approximation error between the full-order model and the reduced-order model solutions is a natural question to address.
This talk will present theoretical results on a priori and a posteriori error estimations and numerical experiments for selecting snapshots adaptively. In particular error bounds for the semi-discrete wave equation will be derived in the continuous setting (when the whole trajectory is known) and in the discrete setting when the Newmark average-acceleration scheme is used on the second-order semi-discrete equation. An a posteriori error indicator for linear time-invariant dynamical systems will be proposed in terms of a Krylov-based exponential integrator and an a posteriori residual-based estimate. An adaptive selection of snapshots will be presented for simulating incompressible Navier-Stokes flows over a range of physical parameters.

Consistency of stepwise uncertainty reduction strategies for Gaussian processes

In the first part of the talk, we will introduce spatial Gaussian processes. Spatial Gaussian processes are widely studied from a statistical point of view, and have found applications in many fields, including geostatistics, climate science and computer experiments. Exact inference can be conducted for Gaussian processes, thanks to the Gaussian conditioning theorem. Furthermore, covariance parameters can be estimated, for instance by Maximum Likelihood.
In the second part of the talk, we will introduce a class of iterative sampling strategies for Gaussian processes, called 'stepwise uncertainty reduction' (SUR). We will give examples of SUR strategies which are widely applied to computer experiments, for instance for optimization or detection of failure domains. We will provide a general consistency result for SUR strategies, together with applications to the most standard examples.

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Balancing of stochastic supply and demand in networked systems

We are constantly faced with situations in which we need to make decisions based on a partial knowledge of our environment. Our decisions influence decisions of other decision makers we interact with. How to model and analyze systems in which decisions need to be made in real-time, based on a limited knowledge of the overall interconnected system? If we are the designers of such systems, how to derive scalable algorithms that meet the global objectives?
In this talk, we will consider the problem of real-time balancing of stochastic supply and demand. Two concrete applications will be discussed, leading to two different types of models and objectives. The first problem is inspired by the kidney paired donation program by UNOS (United Network for Organ Sharing). We will consider a discrete-time bipartite matching model with random arrivals of units of supply and demand. This is a variant of a multi-class queueing system in which the notion of service is replaced by the instantaneous matching of compatible items. A control policy determines which items are matched at each time. The focus will be on an infinite-horizon average-cost optimal control problem for the weighted sum of queue sizes.
The second problem is balancing generation and demand in power grids with high percentage of renewable generation. We will highlight some of the major issues and summarize our recent results on distributed control of flexible loads.
This talk is based on the joint work with V. Gupta (Chicago Booth), J. Mairesse (CNRS, LIP6), and S. Meyn (Univ. of Florida).
Related papers:
- A. Busic, V. Gupta, J. Mairesse. Stability of the bipartite matching model. Advances in Applied Probability, 45(2):351-378, 2013.
- A. Busic, S. Meyn. Approximate optimality with bounded regret in dynamic matching models. ArXiv preprint. 2015.
- A. Busic, S. Meyn. Distributed Randomized Control for Demand Dispatch. 55th IEEE Conference on Decision and Control (CDC). 2016.

On a Cross-Diffusion model for Multiple Species with Nonlocal Interaction and Size Exclusion

In this talk we study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate cross-diffusion system, for which we provide a global existence result as well as a uniqueness proof in the case of equal diffusivities. The analysis is motivated by the formulation of this system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness-by-entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero.

Optimisation de tournées de véhicules pour la collecte de déchets après inondation avec des données incertaines

Les inondations génèrent de grandes quantités de déchets ménagers, et la collecte rapide de ces déchets est une composante importante pour le retour à un fonctionnement normal dans un territoire. Nos recherches s'inscivent dans un projet dont le but est de proposer des méthodes pour optimiser la collecte de déchets après une inondation et de les rendre accessibles aux gestionnaires locaux. Cette présentation porte sur un algorithme d'optimisation de tournées de véhicules de collecte qui tient compte d'incertitudes dans les données (quantités de déchets à enlever). Cet algorithme calcule simutanément une nappe de réponses optimales pour un ensemble de valeurs des données d'entrées incertaines.

Efficient methods for the estimation of homogenized coefficients

We consider discrete divergence-form equation with random coefficients on Z^d. The coefficients are independent, identically distributed and bounded away from 0 and infinity. As is well-known, such equations homogenize over large scale, and I will focus on the problem of computing the homogenized coefficients at precision p given a realization of the coefficients. As will be explained, no algorithm can perform such a task in less than p^(-2) operations. I will present new methods that essentially achieve this lower bound, up to logarithmic factors.

Gap Safe screening rules for sparsity enforcing penalties

In high dimensional regression context, sparsity enforcing penalties have proved useful to regularize the data-fitting term. A recently introduced technique called screening rules, leverage the expected sparsity of the solutions by ignoring some variables in the optimization, hence leading to solver speed-ups. When the procedure is guaranteed not to discard features wrongly the rules are said to be safe. We propose a unifying framework that can cope with generalized linear models regularized with standard sparsity enforcing penalties such as l1 or l1/l2 norms. Our technique allows to discard safely more variables than previously considered safe rules, particularly for low regularization parameters. Our proposed Gap Safe rules (so called because they rely on duality gap computation) can cope with any iterative solver but is particularly well suited to block coordinate descent for many standard learning tasks: Lasso, Sparse-Group Lasso, multi-task Lasso, binary and multinomial logistic regression, etc. For all such tasks and on all tested datasets, we report significant speed-ups compared to previously proposed safe rules.
This is joint work with Eugene Ndiaye, Olivier Fercoq and Alexandre Gramfort.

Implémentation de méthodes d'échantillonnage d'évènements rares dans un code de simulations moléculaires : algorithmes "Spatial Averaging" et "Infinite Swapping"

Bien que les capacités de calcul augmentent de manière continue, il est parfois toujours difficile d'échantillonner avec précision l'espace des conformations atomiques d'un système moléculaire. En effet, certains changements de conformation, tels que la réorganisation d'un cluster d'atomes, le repliement d'une protéine ou d'une chaîne de peptides, la diffusion d'un ligand, etc. impliquent souvent de nombreuses configurations intermédiaires séparées bar des barrières d'énergie conséquentes, ce qui résulte en une faible probabilité d'observer les transitions pendant une simulation : on parle alors d'évènements rares.
Pour les systèmes où les différents puits de potentiel (minima) sont relativement bien connectés, des simulations suffisamment longues et utilisant des méthodes standardisées -- telles qu'une dynamique moléculaire, ou une méthode de Monte Carlo Metropolis-Hastings -- peuvent être efficaces. Néanmoins, des méthodes spécialement optimisées sont souvent nécessaires afin d'échantillonner suffisamment les espaces caractérisés par une faible probabilité d'observation.
L'algorithme Spatial averaging MC (SA-MC) appartient à la famille des méthodes de Monte Carlo optimisées ; un nouvel ensemble de densités de probabilité biaisé est construit, afin d'améliorer l'échantillonnage des évènements rares.
La méthode Infinite Swapping (INS) fait partie de la famille des algorithmes de Replica Exchange ; elle utilise une stratégie de symétrisation afin de combiner des densités de probabilité venant de plusieurs simulations effectuées en parallèle à différentes températures, afin de créer un espace plus facile à échantillonner.
Pendant ce séminaire je vais détailler chaque méthode, expliquer la manière dont elles furent implémentées dans le programme de simulations moléculaires CHARMM et illustrer l'efficacité respective de chaque algorithme avec différents systèmes moléculaires.

Statistical approaches to forcefield calibration and prediction uncertainty in molecular simulation (slides)

The use of molecular simulation as a predictive tool requires to estimate the uncertainty associated with the predicted value for a given property. Among the sources of the simulation results uncertainty, the one arising from the forcefield has long been ignored. The forcefield contains all the information about the potential energy of a molecular system, coming from interatomic interactions, which are encrypted in parameters that are commonly calibrated in order to reproduce some (uncertain) experimental data. Although the results of a simulation depends on the values of the forcefield parameters, it is only very recently that careful investigations of the effect of their uncertainties have been undertaken [1-4]. The main reason for this is the difficulty to estimate forcefield parameters uncertainties, that necessitates an extensive exploration of parameter space, incompatible until very recently with the computer time of molecular simulations.

In recent years, we have explored various calibration strategies and calibration models within the Bayesian framework [5]. We have studied a simple two-parameters Lennard-Jones potential for Argon, for which the calibration can be done using cheap analytical expressions, allowing for a thorough exploration of the parameter space. We have shown that prediction uncertainty, albeit very small, is larger than characteristic statistical simulation uncertainty [4]. For more complex systems, more parameters have to be calibrated and, in absence of analytical models, the calibration process requires to run long molecular simulations. In order to face these issues, we have used kriging metamodels and optimal infilling strategies to limit the number of molecular simulations to be performed during the calibration process. We have benchmarked this methodology on the water TIP4P forcefield [6].

[1] Rizzi, F., et al. Multiscale Model. Simul , 10:1428, 2012.

[2] Rizzi, F., et al. Multiscale Model. Simul., 10: 1460, 2012.

[3] Angelikopoulos, P., et al. J. Chem. Phys., 137: 144103, 2012.

[4]Cailliez, F., Pernot, P. J. Chem. Phys., 134:054124, 2011.

[5]Kennedy, M., O'Hagan, A. J. Roy. Stat. Soc. B, 63:425, 2001.

[6] Cailliez, F., Bourasseau A., Pernot, P. J. Comput. Chem., 35: 130, 2014.

Quelques résultats d'existence et de stabilité d'ondes solitaires pour des modèles de Schrödinger discrets

L'équation de Schrödinger non-linéaire (NLS) possède des solutions particulières données sous la forme d'ondes solitaires qui ont la propriété d'être orbitalement stable. Je discuterai de l'approximation numérique de tels objets, et de leur possible existence et stabilité dans des modèles discret en espace (NLS discret ou DNLS) ou la notion de transport sur une grille discrète est source de difficultés liées aux problèmes d'interpolation et d'aliasing. Je donnerai plusieurs résultats partiels de stabilité orbitale discrète, et discuterai de l'approximation temporelle par des schémas symplectiques. Il s'agit de travaux en communs avec D. Bambusi (Milan), J. Bernier (Rennes), B. Grébert (Nantes), et A. Maspero (Nantes).

Repulsive Optimal Transportation: theory, applications and numerics

In this talk I will introduce the multi-marginal optimal transport (MMOT) problem with repulsive costs (for example the Coulomb cost ). We will see that the set of optimal solutions is richer than the one of standard optimal transport: high non uniqueness holds and fractal-like maps can appear. I will show that this kind of problem naturally arises in the framework of the Density Functional Theory: when the repulsion between electrons largely dominates over the kinetic energy, the Hohenberg-Kohn functional can be reformulated as a MMOT problem with the Coulomb cost. I will finally present some numerical simulations obtained by using a generalization of Sinkhorn's algorithm.

High dimensional sampling with the Unadjusted Langevin Algorithm

Recently, the problem of designing MCMC sampler adapted to high-dimensional distributions and with sensible theoretical guarantees has received a lot of interest. The applications are numerous, including large-scale inference in machine learning, Bayesian nonparametrics, Bayesian inverse problem, aggregation of experts among others. When the density is L-smooth (the log-density is continuously differentiable and its derivative is Lipshitz), we will advocate the use of a ''rejection- free'' algorithm, based on the discretization of the Euler diffusion with either constant or decreasing stepsizes. We will present several new results allowing convergence to stationarity under different conditions for the log-density (from the weakest, bounded oscillations on a compact set and super-exponential in the tails to the log concave). When the density is strongly log-concave, the convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality for Lipschitz functions will be reported. Finally, based on optimzation techniques we will propose new methods to sample from high dimensional distributions. In particular, we will be interested in densities which are not continuously differentiable. Some Monte Carlo experiments will be presented to support our findings.

Penalty-free Nitsche method for interface problems

In the finite element method the weak imposition of boundary conditions or interface conditions has become popular over the past few years. Several approaches can be used for this weak enforcement, the Lagrange multiplier method and the Nitsche's method are widely used for this purpose. Nitsche's method is a penalty based method that has a symmetric and a nonsymmetric version. In this work we consider a nonsymmetric penalty-free Nitsche's method, it can be seen as a Lagrange multiplier method, where the Lagrange multipliers has been replaced by the boundary fluxes of the discrete elliptic operator. This leads to a method that is stable without any unknown parameter and without introducing additional degrees of freedom. The less stiff scheme obtained by eliminating the penalty appears to have some advantages for coupled problems, for instance in multiphysics coupling in continuum mechanics. We apply the penalty-free Nitsche's method to interface problems when the mesh fits the interface (nonconforming domain decomposition) or not (unfitted domain decomposition) for the compressible and incompressible elasticity. An unfitted fluid-structure interaction scheme is also considered. For each case we show theoretically and numerically the convergence of the error.

Entropie et dualité dans les modèles de diffusion croisée

Les modèles de diffusion croisée sont des systèmes d'équations de type paraboliques où le couplage peut se faire à l'intérieur des termes de diffusion (i.e. dans un laplacien). Nous verrons comment pour une classe large de ces systèmes on peut construire des solutions (très) faibles en utilisant une structure d entropie sous jacente. Nous verrons également comment les estimations de dualité introduites par Michel Pierre et Didier Schmitt sont particulièrement adaptées à ce type de problème.