# Workshops

## Reduced Basis, POD and Reduced Order Methods for model and computational reduction: towards real-time computing and visualization?

### Organisers

- Gianluigi Rozza
*(SISSA, Trieste, Italy)* - Alfio Quarteroni
*(Swiss Federal Institute of Technology Lausanne (EPFL), Switzerland)*

### Supports

### Description

This proposed CECAM workshop will consider a range of model reduction strategies with applications in real-time visualization and computing. The increasing complexity of mathematical models used to predict real-world systems, such as climate or the human cardiovascular system, has led to a need for model reduction, which means developing systematic algorithms for replacing complex models with far simpler ones, that still accurately capture the most important aspects of the phenomena being modelled. Model reduction can be divided into two main approaches: "reduce-then-model" and "discretize-then-reduce". In the former approach the continuous equations representing the underlying physics are first reduced, e.g. by symmetry assumptions that allow us to consider 1D or 2D equations instead of the full 3D equations, before a computational model is derived. In the latter approach a computational model is obtained by discretizing the continuous equations and only then a reduced model is sought. Some subtopics include spatial dimensionality reduction and multiscale modelling frameworks in the "reduce-then-model" category; state space and parameter space reduction -- with a special accent on reduced basis and proper orthogonal decomposition -- in the "discretize-then-reduce" category.

The workshop will emphasize model reduction topics in several themes: 1. design, optimization, and control theory in real-time with applications in engineering; 2. data assimilation, geometry registration, and parameter estimation with a special attention to real-time computing in biomedical engineering and computational physics; 3. real-time visualization of physics-based simulations in computer science; 4. the treatment of high-dimensional problems in state space, physical space, or parameter space; 5. the interactions between different model reduction and dimensionality reduction approaches; 6. the development of general error estimation frameworks which accommodate both model and discretization effects. In this workshop the emphasis is on mathematical models based on both ordinary and partial differential equations. The focus of the workshop is methodological, however, we anticipate a wide range of both academic and industrial problems of high complexity to motivate, stimulate, and ultimately demonstrate the meaningfulness and efficiency of the proposed approaches.

The emphasis on real-time computing applications, and above all real-time visualization, can be seen as a new frontier in scientific computing to assist scientists and engineers during design, construction, manufacturing or production phases, and even medical doctors during surgery or diagnosis.

We focus on recent developments of reduced order modelling applications in transport and mechanics: unsteady and steady heat and mass transfer; acoustics; and solid and fluid mechanics. Of course we do not preclude other domains of inquiry within engineering (e.g., electromagnetics) or even more broadly within the quantitative disciplines (e.g., finance).

We consider input-output relationships held by the solution of partial and/or ordinary differential equations. The input-parameter vector typically characterizes the geometric configuration, the physical properties, and the boundary conditions and sources of a complex system. The outputs of interest might be the maximum system temperature, an added mass coefficient, a crack stress intensity factor, an effective constitutive property, an acoustic waveguide transmission loss, or a channel flowrate or pressure drop. Finally, the field variables that connect the input parameters to the outputs can represent a distribution function, temperature or concentration, displacement, pressure, or velocity.

The methodologies we consider in this workshop are motivated by, optimized for, and applied within two particular contexts: the real-time context (e.g., parameter estimation or control) and the many query context (e.g., design optimization or multimodel/scale simulation). Both these contexts are crucial to computational science and engineering and to more widespread adoption and application of numerical methods for partial and/or ordinary differential equations in engineering practice and education.

The real-time context can be found in engineering situations dealing with in-the-field robust parameter estimation (or inverse problems, or nondestructive evaluation), design and optimization, and control. On the other side the many-query context involves multiscale (temporal, spatial) or multiphysics models in which behavior at a larger scale must “invoke” many spatial or temporal realizations of parametrized behavior at a smaller scale. Particular cases (to which these methods have been applied) include stress intensity factor evaluation within a crack fatigue growth model; calculation of spatially varying cell properties within homogenization theory predictions for macroscale composite properties; assembly and interaction of many similar building blocks in large (e.g., cardio-vascular) biological networks; or molecular dynamics computations based on quantum-derived energies/forces (CECAM "core" activity). In all these cases, the number of input-output evaluations is often measured in the tens of thousands. Both the real-time and many-query contexts present a significant and often unsurmountable challenge to “classical” numerical techniques such as the finite element method (or finite difference, finite volume). These contexts are often much better served by the reduced order modelling techniques (often associated with a posteriori error estimation techniques).

We note, however, that the reduced order modelling methods we consider do not replace, but rather build upon and are measured (as regards accuracy) relative to, a classical discretization technique: the reduced method approximates not the exact solution but rather a “given” finite element discretization of (typically) very large dimension. In short, we pursue an algorithmic collaboration rather than an algorithmic competition. The development of reduced order modelling can perhaps be viewed as a response to the considerations and imperatives described above. In particular, the parametric real-time and many-query contexts represent not only computational challenges, but also computational opportunities.

The state of the art at the moment is moving towards interaction between different reduced order modelling techniques. For example, reduced basis method and proper orthogonal decomposition are combined to solve time-dependent parametrized diffusion-reaction problems with certification of accuracy for the reduced model provided by a posteriori error bounds. Theoretical studies are being carried out to ensure a deeper comprehension of model order reduction and the reliability and the applicability of the methodologies proposed. Parametrization of systems is advancing by proposing new techniques to deal with more complex configurations and more parameters. Techniques to improve the exploration of parameter space (sampling procedures, greedy algorithms) have been refined, combined, and specialized. The new frontier is an accurate and reliable methodology for certified real-time computing and visualization. Here advances made in computer graphics and physics-based simulation communities can be adapted to produce new methodologies satisfying the real-time needs of our target applications.

### References

[1] A Quarteroni, G Rozza, and A Manzoni, Certified Reduced Basis Approximation for Parametrized Partial Differential Equations and Applications. Math in Industry Vol. 1:3, 2011.

[2] G Rozza, DBP Huynh, and AT Patera, Reduced Basis Approximation and A Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations Application to Transport and Continuum Mechanics. Archives of Computational Methods in Engineering, Vol.15(3):229275, 2008.

[3] A. Antoulas, Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, SIAM, Philadelphia, 2005.

[4] L. T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes and B. van Bloemen Waanders (eds.), Real-Time PDE-Constrained Optimization. Computational Science and Engineering, Vol. 3, SIAM, Philadelphia, 2007.

[5] W.H.A. Schilders, H.A. van der Vorst and J. Rommes, Model order reduction: theory, research aspects and applications, Springer, 2008.

[6] D.N. Metaxas, Physics-based deformable models: applications to computer vision, graphics, and medical imaging, Kluwer Academic Publishers, 1997.