13 research outputs found
Localized model reduction for parameterized problems
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
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
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
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
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