Continuum Mechanics Group
Predictive computational modeling
n The focus of the Continuum Mechanics Group in 2015 was the development of novel models, methodologies and computational tools for quantifying uncertainties and their effect in the simulation of engineering and physics systems. Our work has been directed towards three fronts:
a) the calibration and validation of computational models using experimental data, b) uncertainty propagation in multiscale systems, c) Design/control/ optimization of complex systems under uncertainty.
Prof. Dr. Phaedon-Stelios
Koutsourelakis
Contact
www.contmech.mw.tum.de Phone +49.89.289.16690
A highlight was the organization of the international Symposium on ‘Big Data and Predictive Computational Modeling’ that took place in TUM-IAS during May 18-21
2015. The symposium was sponsored by the European Office of U.S. Aerospace Research Development (EOARD), the TUM-IAS and the Department of Mecha- nical Engineering. It included plenary and
keynote talks from pre-eminent scientists
in applied mathematics, computational physics/chemistry, computer science and engineering.
The event received considerable interest from various communities and an article on it appeared in Volume 48, Number 6
of SIAM news. Videos and slides from the symposium can be found at: http://www. tum-ias.de/bigdata2015/program.html
Excerpt from SIAM news article (Volume 48, Number 6) on ‘Big Data and Predictive Computatio- nal Modeling’
Nonlinear Inverse Problems with Applications in Medical Diagnostics
This project is concerned with the numer- ical solution of model-based, Bayesian inverse problems. We are particularly interested in cases where the cost of each likelihood evaluation (forward-model call)
is expensive and the number of unknown (latent) variables is high. This is the setting of many problems in computational phy- sics where forward models with nonlinear PDEs are used and the parameters to
be calibrated involve spatio-temporarily varying coefficients, which upon discre- tization, give rise to a high-dimensional vector of unknowns.
Our motivating application is biome- chanics where several studies have shown that the identification of material parameters from deformation data can lead to earlier and more accurate diag-
nosis of various pathologies. This project develops Bayesian computational tools that can lead not only to point-estimates of the material properties, but also to a rigorous quantification of the confidence
in those estimates. A significant role in this
effort is played by novel dimensionality reduction techniques which can identify a sparse set of features. One consequence of the ill-posedness of inverse problems
is multi-modality i.e. the possibility of multiple, distinct solutions that fit the data comparably well. We developed adaptive strategies that are capable of accurately capturing the presence of multiple modes.
Coarse-Graining in Equilibrium Statistical Mechanics
(a) Ground truth (reference) (b) Multimodal posterior mean. Points on the boundary have non-zero probability of belonging to healthy
tissue (blue) or tumor (red)
Identification of material para- meters using nonlinear, incom- pressible elasticity models.
Coarse-grained (CG) models provide a
computationally efficient means to study large numbers of atoms over extended spatio-temporal scales. Existing strategies rely on mapping degrees of freedom from micro- to macro-scale, whereas properties are commonly determined as certain
point-estimates in the macro-scale. Various methodologies that have been proposed do not account for the informa- tion loss which unavoidably takes place during coarse-graining. The joint project
‘Predictive Materials Modeling’ with Hans Fischer Senior Fellow Prof. N. Zabaras (Director of Warwick Centre for Predictive Modeling, University of Warwick) addres- ses the formulation and development of
(a) Coarse-graining of water molecules
an enhanced, predictive multiscale frame- work. Our reformulation of coarse-graining follows a data-driven, Bayesian paradigm based on generative probabilistic models. It builds upon: a) a coarse macroscopic description, and b) a probabilistic lifting operator, reconstructing micro configu- rations from coarse states. This affords a more flexible definition of CG variables,
the rigorous quantification of uncertainties
associated with using limited training data and various degrees of information loss. Furthermore, it is capable of iden- tifying sparse representations for the CG potential which reveal qualitative, physical features of the CG model.
(b) Proposed coarse-graining scheme
Reduced-order Modeling for Uncertainty Quantification
(a) True (reference) solution manifold in three-dimensional phase space.
Reduced-basis construction for Kraichnan-Orszag nonlinear ODE with parameters/inputs being the initial conditions and time.
(b) Mixture of local, reduced-basis identified from data. Each color corresponds to a different reduced-basis set associated with a particular subdomain of the input space.
As the physical problems become more complex and the mathematical models more involved, current computational methods prove increasingly inefficient,
especially in tasks requiring numerous solutions as in uncertainty quantification. In the latter case, the difficulty is further amplified by the large number of input parameters (random variables). This project employing advanced, statistical learning tools in order to develop probabi-
listic reduced-basis models. These consist of mixtures of reduced-bases which can
be used to project the governing equa- tions in order to obtain descriptions of significantly lower dimension. The tools developed can learn reduced-bases sets from a limited number of full-scale runs and can simultaneously partition the input
and output space so that different response regimes can be computed as well as the most important/sensitive inputs can be
identified.
Topology optimization problem. Deterministic (left) vs. Stochastic (right) solutions. The stochastic version accounts for random vari- ability in the material properties
Design in the Presence of Uncertainty
(a) Deterministic solution (b) Stochastic solution (mean)
This project is concerned with a lesser- studied problem in the context of
model-based, uncertainty quantification
(UQ), that of optimization/design/control under uncertainty. The solution of such problems is hindered not only by the
usual difficulties encountered in UQ tasks
(e.g. the high computational cost of each forward simulation, the large number of random variables) but also by the need
to solve a nonlinear optimization prob- lem involving large numbers of design variables and potentially constraints. We propose a framework that is suitable for a class of such problems and is based on the idea of recasting them as proba- bilistic inference tasks. To that end, we
developed a variational Bayesian (VB) formulation and an iterative VB expecta- tion maximization scheme that is capable of identifying a local maximum as well as a low-dimensional set of directions in the design space, along which, the objective exhibits the largest sensitivity. In cases considered the cost of the computations
in terms of calls to the forward model was of the order of 100 or less. The accuracy of the approximations provided was
assessed by information-theoretic metrics.
Research Focus
n Uncertainty quantification
n Random media
n Multiscale formulations
n Bayesian inverse problems
n Design/optimization under uncertainty
Competence
n Computer simulation
n Mathematical modeling of stochastic systems
Infrastructure
n 256core HPC
Courses
B.Sc.
n Uncertainty Quantification in Mechani- cal Engineering (SS)
n Modeling in Structural Mechanics (WS)
Master
n Atomistic Modeling of Materials (WS)
n Bayesian Strategies for Inverse Prob- lems (SS)
n Journal Club Uncertainty Quantification
(WS-SS)
MSE
n Continuum Mechanics (WS)
n Uncertainty Modeling in Engineering
(SS) (Top Teaching Trophy 2014, 2015)
Management
Prof. Phaedon-Stelios Koutsourelakis, Ph.D., Director
Administrative Staff
Cigdem Filker
Research Scientists Dipl.-Ing. Isabell Franck Markus Schoeberl, M.Sc.
Constantin Grigo, M.Sc. (Physics) Maximilian Koschade, B.Sc. Mariella Kast, B.Sc.
Lukas Koestler, B.Sc. Maximilian Soepper, B.Sc.
Publications 2014-15
n I. Franck, P.S. Koutsourelakis. Sparse Variational Bayesian Approximations for Nonlinear Inverse Problems: Applications in Nonlinear Elastography, Computer Methods in Applied Mechanics and Engineering, in press 2015 (to appear Volume 299,
1 February 2016, pp. 215-244).
n P.S. Koutsourelakis. Variational Bayesian Strategies for High-Dimensional, Stochastic Design Problems. Journal of Computational Physics, accepted 2015 (in press).
n I. Franck, P.S. Koutsourelakis. Multimodal,
high-dimensional, model-based, Bayesian inverse problems with applications in biomechanics, in preparation arxiv
n I. Franck, Uncertainty Quantification for Nonlinear
Inverse Problems in High DDimensions. 2nd poster award, 3rd ECCOMAS International Young Investi- gators Conference, Aachen, Germany (poster).
n M. Schoeberl. A Bayesian Approach to Coarse-
Graining. Predictive Multiscale Materials Modelling. Isaac Newton Institute, Cambridge. 2015 (poster).
n M. Schoeberl, N. Zabaras, P.S. Koutsourelakis.
Predictive Coarse-Graining. Predictive Multiscale Materials Modelling. Isaac Newton Institute, Cambridge. 2015 (presentation).
n I.Franck, P.S. Koutsourelakis. Variational Bayesian
Formulations for High-Dimensional Inverse Problems. SIAM CSE 2015, Salt Lake USA. 2015 (presentation).
n P.S. Koutsourelakis. Variational Bayesian Strategies
for High-Dimensional, Stochastic Design Problems. Big Data and Predictive Computational Modeling, May 2015, Germany (presentation).
n M. Schoeberl, N. Zabaras, P.S. Koutsourelakis. Pre-
dictive Coarse-Graining. To be submitted to Journal of Computational Physics 2015 (in preparation).