13 research outputs found

    Localized model reduction for parameterized problems

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    In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that have only support on part of the domain and compute a global approximation by a suitable coupling of the local spaces. In detail, we show how optimal local approximation spaces can be constructed and approximated by random sampling. An overview of possible conforming and non-conforming couplings of the local spaces is provided and corresponding localized a posteriori error estimates are derived. We introduce concepts of local basis enrichment, which includes a discussion of adaptivity. Implementational aspects of localized model reduction methods are addressed. Finally, we illustrate the presented concepts for multiscale, linear elasticity and fluid-flow problems, providing several numerical experiments. This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order Reduction. Walter De Gruyter GmbH, Berlin, 2019+

    A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems

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    This paper presents a method for the numerical treatment of reaction-convection-diffusion problems with parameter-dependent coefficients that are arbitrary rough and possibly varying at a very fine scale. The presented technique combines the reduced basis (RB) framework with the recently proposed super-localized orthogonal decomposition (SLOD). More specifically, the RB is used for accelerating the typically costly SLOD basis computation, while the SLOD is employed for an efficient compression of the problem's solution operator requiring coarse solves only. The combined advantages of both methods allow one to tackle the challenges arising from parametric heterogeneous coefficients. Given a value of the parameter vector, the method outputs a corresponding compressed solution operator which can be used to efficiently treat multiple, possibly non-affine, right-hand sides at the same time, requiring only one coarse solve per right-hand side.Comment: 27 pages, 6 figure

    Randomized Local Model Order Reduction

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    In this paper we propose local approximation spaces for localized model order reduction procedures such as domain decomposition and multiscale methods. Those spaces are constructed from local solutions of the partial differential equation (PDE) with random boundary conditions, yield an approximation that converges provably at a nearly optimal rate, and can be generated at close to optimal computational complexity. In many localized model order reduction approaches like the generalized finite element method, static condensation procedures, and the multiscale finite element method local approximation spaces can be constructed by approximating the range of a suitably defined transfer operator that acts on the space of local solutions of the PDE. Optimal local approximation spaces that yield in general an exponentially convergent approximation are given by the left singular vectors of this transfer operator [I. Babu\v{s}ka and R. Lipton 2011, K. Smetana and A. T. Patera 2016]. However, the direct calculation of these singular vectors is computationally very expensive. In this paper, we propose an adaptive randomized algorithm based on methods from randomized linear algebra [N. Halko et al. 2011], which constructs a local reduced space approximating the range of the transfer operator and thus the optimal local approximation spaces. The adaptive algorithm relies on a probabilistic a posteriori error estimator for which we prove that it is both efficient and reliable with high probability. Several numerical experiments confirm the theoretical findings.Comment: 31 pages, 14 figures, 1 table, 1 algorith

    Static condensation optimal port/interface reduction and error estimation for structural health monitoring

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    Having the application in structural health monitoring in mind, we propose reduced port spaces that exhibit an exponential convergence for static condensation procedures on structures with changing geometries for instance induced by newly detected defects. Those reduced port spaces generalize the port spaces introduced in [K. Smetana and A.T. Patera, SIAM J. Sci. Comput., 2016] to geometry changes and are optimal in the sense that they minimize the approximation error among all port spaces of the same dimension. Moreover, we show numerically that we can reuse port spaces that are constructed on a certain geometry also for the static condensation approximation on a significantly different geometry, making the optimal port spaces well suited for use in structural health monitoring

    A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

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    We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology
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