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Dynamical low rank approximation of evolution equations with random parameters

English title Dynamical low rank approximation of evolution equations with random parameters
Applicant Nobile Fabio
Number 146360
Funding scheme Project funding (Div. I-III)
Research institution EPFL - SB - SMA-GE
Institution of higher education EPF Lausanne - EPFL
Main discipline Mathematics
Start/End 01.05.2013 - 30.04.2017
Approved amount 224'348.10
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Keywords (8)

A posteriori error estimation; Dynamical low rank approximation; Navier-Stokes equations; Second order wave equations; Evolution equations with random coefficients; Multivariate polynomial approximation; Non-linear diffusion-reaction equations; Adaptivity

Lay Summary (French)

Lead
La puissance de calcul dont on dispose, ainsi que les techniques avancées de simulation, permettent de résoudre des équations aux dérivées partielles (EDP) décrivant le comportement de systèmes physiques complexes comme la dynamique de structures et fluides, des processus biologiques, etc.Souvent, les paramètres qui apparaissent dans les équations sont partiellement inconnus et des marges d'incertitude doivent être inclues dans la simulation afin d'obtenir des prédictions fiables.
Lay summary

Dans une description probabiliste de l'incertitude des paramètres d'entrée, nous cherchons à développer des techniques numériques efficaces pour calculer, la solution moyenne de l'EDP en temps, les marges d'incertitude associées voire une approximation de la loi de probabilité de la solution à chaque instant temporel.

La méthode étudiée s'appuye sur le fait que l'observation de la collection de toutes possibles solutions de l'EDP, correspondant aux possibles valeurs des paramètres d'entrée, est souvent bien approchée par un sous-espace de faible dimension (approximation de faible rang) qui, pourtant, peut varier considérablement en temps. Nous allons développer des algorithmes qui permettent de construire ce sous-espace de faible dimension et de le faire évoluer dynamiquement en temps, en exploitant la structure de l'équation différentielle.

On appliquera cette technique à des phénomènes de diffusion et réaction non-linéaires ; phénomènes de propagation d'ondes ou des problèmes en dynamique de fluides incompressibles.

Notre but est de comprendre les propriétés mathématiques des équations différentielles décrivant l'évolution dynamiques du sous-espace de faible dimension, d'étudier leurs propriétés d'approximation, développer des stratégies adaptatives basées sur des indicateurs d'erreur a-posteriori permettant d'augmenter ou diminuer automatiquement la dimension du sous-espace à un coût de calcul au plus bas.

La méthodologie qu'on propose est potentiellement très puissante pour
l'approximation d'une classe de EDP dépendantes du temps avec
paramètres aléatoires, dont la solution a essentiellement un faible
rang.  Mise à parte la problématique de la quantification de
l'incertitude, cette méthodologie pourrait avoir aussi un impact très
forte dans le domaine de l'assimilation des donnée pour des systèmes
dynamiques en dimension infinie.

Direct link to Lay Summary Last update: 08.04.2013

Responsible applicant and co-applicants

Employees

Publications

Publication
Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions
Musharbash Eleonora, Nobile Fabio (2018), Dual Dynamically Orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, in Journal of Computational Physics, 354, 135-162.
MATHICSE Technical Report : Symplectic dynamical low rank approximation of wave equations with random parameters
Musharbash Eleonora, Nobile Fabio (2017), MATHICSE Technical Report : Symplectic dynamical low rank approximation of wave equations with random parameters, mathicse technical report, EPFL, electronic.
Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs
Musharbash E., Nobile F., Zhou T. (2015), Error Analysis of the Dynamically Orthogonal Approximation of Time Dependent Random PDEs, in SIAM Journal on Scientific Computing, 37(2), A776-A810.

Collaboration

Group / person Country
Types of collaboration
Prof. Tao Zhou, Institute of Computational Mathematics, Chinese Academy of Sciences China (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
ENUMATH: European Conference on Numerical Mathematics and Advanced Applications Talk given at a conference Dynamical low rank approximation of random time dependent PDEs 25.09.2017 Voss, Norway Nobile Fabio;
Workshop SSSTC Stepping Stone Symposium, “Theoretical and Numerical Analysis of Partial Differential Equations with Emphasis on Applications” Talk given at a conference Dynamical low rank approximation of random time dependent PDEs 29.08.2017 University of Geneva, Switzerland Nobile Fabio;
FoCM2017: workshop “Foundations of Numerical PDEs” Talk given at a conference Dynamical low rank approximation of random time dependent PDEs 10.07.2017 Barcelona, Spain Nobile Fabio;
MOX Seminar Series, Politecnico di Milano Individual talk Dynamical low rank approximation of random time dependent PDEs 29.06.2017 Politecnico di Milano, Italy Nobile Fabio;
UNCECOMP 2017 Talk given at a conference Symplectic Dynamical low rank approximation of wave equations with random parameters 15.06.2017 Rhodes Island, Greece Musharbash Eleonora;
SIMAI 2016 Talk given at a conference Dynamical low rank approximation of time dependent PDEs with random parameters 13.09.2016 Politecnico di Milano, Italy Musharbash Eleonora;
MORCIP 2016 Talk given at a conference Dynamical Low rank approximation of incompressible Navier-Stokes equations with random parameters 19.05.2016 EPFL, Lausanne, Switzerland Musharbash Eleonora;
Swiss Numerics Colloquium 2016 Talk given at a conference Dynamically Orthogonal approximation of time dependent random PDEs 22.04.2016 Fribourg, Switzerland Musharbash Eleonora;
SIAM UQ16 Talk given at a conference Dynamical Low rank approximation of incompressible Navier-Stokes equations with random parameters, 05.04.2016 EPFL, Lausanne, Switzerland Musharbash Eleonora;
Workshop COST EU-MORNET WG1 ROM 2016 Individual talk Dynamical Low rank approximation of incompressible Navier-Stokes equations with random parameters 22.02.2016 SISSA, Trieste, Italy Musharbash Eleonora;
Séminaire de calcul scientifique du CERMICS Individual talk Dynamical Low Rank approximation of random time dependent PDEs 14.10.2015 Ecole des Ponts ParisTech, France Nobile Fabio;
Micro Workshop on Reduced Order Modeling Techniques & Applications Individual talk Dynamical Low Rank approximation of random time dependent PDEs 30.09.2015 EPFL LAUSANNE, Switzerland Nobile Fabio;
2nd GAMM AGUQ Workshop on Uncertainty Quantication Poster Dynamically Orthogonal approximation of time dependent random PDEs 10.09.2015 Technische Universität Chemnitz, Germany Nobile Fabio;
ICIAM 2015 Talk given at a conference Dynamical low rank approximation of incompressible Navier Stokes equations with random parameters 10.08.2014 Beijing, China Musharbash Eleonora;
Swiss Numerics Colloquium 2014 Poster Dynamically Orthogonal approximation of time dependent random PDEs 25.04.2014 Universität Zürich, Switzerland Musharbash Eleonora;
MASCOT NUM 2014 meeting Poster Dynamically Orthogonal approximation of time dependent random PDEs 23.04.2014 ETH ZÜrich, Switzerland Musharbash Eleonora;
SIAM UQ14 Talk given at a conference Dynamical low rank approximation of time dependent PDEs with random data 31.03.2014 Savannah, GE, United States of America Musharbash Eleonora;
Algorithms and Applications Workshop Poster Dynamical low rank approximation of time dependent PDEs with random data 06.01.2014 King Abdullah University of Science and Technology, Saudi Arabia Musharbash Eleonora;


Self-organised

Title Date Place
SIAM UQ16: SIAM conference on Uncertainty Quantication 2016 05.04.2016 EPFL LAUSANNE, Switzerland
SIAM PDE15: Minisymposium Data Assimilation for PDE Models 07.12.2015 ARIZONA, United States of America
NASPDE 2014: Numerical Analysis of Stochastic PDEs 09.09.2014 EPFL LAUSANNE, Switzerland

Associated projects

Number Title Start Funding scheme
172678 Uncertainty Quantification techniques for PDE constrained optimization and random evolution equations. 01.08.2017 Project funding (Div. I-III)
140574 Efficient numerical methods for flow and transport phenomena in heterogeneous random porous media. 01.04.2012 Project funding (Div. I-III)

Abstract

In this project we focus on time dependent partial differential equations with random parameters possibly describing variability in initial conditions, forcing terms, coefficients in the equations, etc.This situation arises in many practical applications and efficient numerical techniques are needed to compute the mean solution together with uncertainty bounds associated to it, or, even more, an approximation of the entire probability law of the solution at each instant of time.In many cases, the collections of all solutions at a given time corresponding to all possible outcomes of the input random processes can be well approximated in a low dimensional subspace (low rank approximation). The main practical difficulty is that such a subspace is, in general, not easy to characterize a priori and might significantly change during the evolution of the system. Therefore, it is worth developing methods that allow to evolve dynamically the subspace exploiting the structure of the equations.In this project we develop dynamical low rank techniques. At each instant of time the solution is written as a linear combination of r (unknown) basis functions u_i, i=1,..., r, with (unknown) coefficients \psi_i that depend on the random parameters. Both thebasis functions and the coefficients in the linear combination are allowed to change in time. This general structure defines a manifold M_r of rank-r functions. The evolution equations are then projected at each time instant onto the tangent manifold, which allowsone to derive differential equations for the time evolution of the basis functions u_i as well as the coefficients \psi_i in the linear combination.We will apply this technique to few specific problems including parabolic diffusion equations, non-linear diffusion-reaction equations, second order hyperbolic wave equations, Navier-Stokes equations for incompressible fluids.We aim, in particular, at investigating the mathematical properties of the resulting dynamical low rank equations and studying their approximation capabilities. We will also develop adaptive strategies, based on a posteriori error indicators, to automatically increase or decrease the number of basis functions needed in the approximation ofthe solution as well as the approximation level needed both in the description of the space dependent basis functions u_i and the parameter dependent coefficients \psi_i.The proposed methodology has great potential as a tool to approximate a class of uncertain time evolving PDEs, whose solutions are effectively low rank. Beside the goal of uncertainty quantification, it could have also a great impact in the development of efficient data assimilation techniques for infinite dimensional dynamical systems.
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