CSE 2013: A Conference Within a Conference for MOR Researchers

July 23, 2013

David Amsallem, Bernard Haasdonk, and Gianluigi Rozza

Model order reduction (MOR) is increasingly important in the mathematical and engineering sciences. With the availability of highly accurate discretization and simulation techniques for systems of ordinary and partial differential equations, the demand for accelerated solution of these high-dimensional models has become more and more urgent. This is particularly so in multi-query or real-time simulation scenarios, such as simulation-based control and optimization, statistical investigation, and interactive design, to name just a few.

Acceleration of simulation models is typically achieved through surrogate modeling, based on the projection of high-dimensional quantities, such as the state vector, onto low-dimensional subspaces. MOR procedures aim to achieve good approximation of states or the input–output behaviour of the full model. A typical feature of MOR is a decoupling of the computational procedure: In a possibly expensive "offline" step, the reduced order model is generated using significant computational resources. This can include, for instance, full model simulations for carefully chosen configurations, selected either by numerical experts or by automatic procedures. Simulation of the reduced model is then carried out in an inexpensive "online" step; this enables rapid simulations for many different configurations on less powerful computational platforms.

As a topic inherently linked to simulation science, MOR was discussed at previous SIAM CSE conferences, but its presence in the program increased dramatically at the 2013 conference. MOR was the subject of an invited plenary presentation, by Jan Hesthaven of Brown University, who gave a broad introduction to certified MOR by reduced basis methods (a recording of the talk can be accessed at http://www.siam.org/meetings/presents.php). It was also the theme of 22 minisymposia, a total of 88 presentations, held in the course of the week; eight of the minisymposia were organized by the authors of this article. With a track consisting of several minisymposia held in the same room throughout the week, the sessions constituted something of a conference within a conference. The remainder of this article highlights noteworthy recent developments and research directions featured in the many conference sessions devoted to MOR.

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Several talks were devoted to structure-preserving model reduction. Presenters demonstrated that an appropriate MOR technique should always preserve certain characteristics, such as passivity or Lagrangian structure of the underlying large-scale dynamical system, in order to obtain physical and accurate predictions.

While many MOR techniques are based on simulations that use high-fidelity models, such as finite element or finite volume schemes, speakers also presented novel data-based and data-driven MOR approaches. With these techniques, researchers use experimental data and observations to build reduced order models. Several speakers presented novel approaches based on solution snapshots generated from physical experiments, as well as the use of Loewner matrices, kernel interpolation schemes, and Kalman filtering.

Rapid generation of predictions is one objective of MOR, as is the certification of accuracy and reliability of those predictions by a priori and a posteriori error estimators. Several speakers discussed such error quantification. In particular, a new space–time norm analysis framework guarantees the effectivity and reliability of the resulting error estimators, and the stability of the reduced models for long-time integration of flow problems.

Many presentations highlighted difficulties associated with the efficient reduction of nonlinear problems. In such cases, an additional level of approximation is required to achieve large online speedups. Among the proposed approaches are empirical interpolation, gappy proper orthogonal decomposition, missing-point estimation, and hyper-reduction procedures.

Other speakers proposed novel approaches for reducing the complexity associated with expensive bottlenecks, such as spatial integral evaluations and the computation of euclidean inner products of large-scale vectors.

The parametric dependency of systems is an important question in MOR. Most parametric MOR procedures are limited to the case of small numbers of parameters, as high parameter dimensions are associated with the curse of dimensionality and issues related to effective sampling. Realistic applications, however, are typically characterized by high-dimensional parameter spaces. Three new approaches that can be effective in such scenarios were presented at SIAM CSE: (1) parameter space reduction for inference (e.g., in subsurface contaminant analysis), (2) state space partition by local reduced bases (e.g., in CFD), and (3) coupling of submodels by reduced basis elements, such as "LEGOs" (e.g., in stress analysis of component-based structures).

Many talks at CSE 2013 were devoted to optimal control and optimization with reduced order models.

Several sessions detailed the application of MOR to complex systems in general, and to computational fluid dynamics problems in particular. Examples include flow control problems applied to airplane design (Figure 1) and cardiovascular devices (Figure 2). In addition to CFD, application fields ranged from environmental sciences, building design, and computational neuroscience to the design of semiconductors and car engines. A notable effort to transfer MOR technology to industry was described by a speaker from Akselos, a startup company that provides an easy-to-use interactive platform for real-time simulations in structural and thermal analysis. Currently, the approach makes extensive use of the recent static condensation reduced basis element (SCRBE) method for real-time simulations of complex structures/geometries, using libraries of components and providing these online for download. This platform has been designed for both companies and universities.

Figure 1. Pressure at the surface of a commercial aircraft computed using model order reduction. From K. Washabaugh, D. Amsallem, M. Zahr, and C. Farhat, Nonlinear Model Reduction for CFD Problems Using Local Reduced-Order Bases, 42nd AIAA Fluid Dynamics Conference and Exhibit, June 25–28, 2012, New Orleans, Louisiana.

Figure 2. Shape optimization of a bypass anastomosis (visualization of the velocity field) using the reduced basis method applied to viscous flow equations with the iterative optimization algorithm of vorticity reduction. From A. Manzoni, Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics, PhD Thesis 5402, EPFL, 2012; T. Lassila, A. Manzoni, A. Quarteroni, and G. Rozza, A reduced computational and geometrical framework for inverse problems in haemodynamics, International Journal for Numerical Methods in Biomedical Engineering, in press, 2013.

Despite the advances presented at SIAM CSE 2013 and briefly mentioned here, several challenges remain to be addressed. Among them are the handling of large numbers of parameters, and the definition of efficient parameter sampling and better exploration techniques for the construction of reduced order models. Fluid mechanics continues to offer many interesting challenges for the MOR community, such as the model order reduction of turbulent viscous flows at high Reynolds numbers and the efficient handling of features propagating in flows.

Overall, and in particular from the perspective of MOR, SIAM CSE 2013 can be considered a very successful event, gathering the computational science and engineering community to exchange information about recent developments, open questions, new insights, and interests that could result in further leading applications.

David Amsallem is a researcher in the Department of Aeronautics & Astronautics at Stanford University. Bernard Haasdonk is a junior professor in the Department of Mathematics at Universität Stuttgart. Gianluigi Rozza is a researcher at SISSA, International School for Advanced Studies, MathLab, Trieste.

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