Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems

SUMMARYAn adaptive and efficient approach for constructing reduced-order models (ROMs) that are robust to changes in parameters is developed. The approach is based on a greedy sampling of the underlying high-dimensional model (HDM) together with an efficient procedure for exploring the configuration space and identifying parameters for which the error is likely to be high. Because this exploration is based on a surrogate model for an error indicator, it is amenable to a fast training phase. Furthermore, a model for the exact error based on the error indicator is constructed and used to determine when the greedy procedure reaches a desired error tolerance. An efficient procedure for updating the reduced-order basis constructed by proper orthogonal decomposition is also introduced in this paper, avoiding the cost associated with computing largescale singular value decompositions. The proposed procedure is then shown to require less evaluations of a posteriori error estimators than the classical procedure in order to identify locations of the parameter space to be sampled. It is illustrated on the training of parametric ROMs for three linear and nonlinear mechanical systems, including the realistic prediction of the response of a V-hull vehicle to underbody blasts. Copyright

show abstract

“…In addition, a search procedure is required in order to find the maximizer of (11). This search entails a cost T search .…”

Section: Training Budgetmentioning

confidence: 99%

An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

Paul-Dubois-Taine

Amsallem

2014

Numerical Meth Engineering

141

164

View full text Add to dashboard Cite

show abstract

“…In comparison with the formula used by standard and tensorial POD (38), we notice that the summation spans only the location of DEIM points instead of entire discrete space. For completeness, we recall that n is the size of the full discrete space, k is the size of reduced order model and m is the number of DEIM points.…”

Section: A2 Swe Tensorsmentioning

confidence: 99%

“…Recently Amsallem et al [8] used a two-step gappy POD procedure to decrease the computational complexity of the reduced nonlinear terms in the solution of shape optimization problems. Reduced basis approximation [13,47,78,86,38] is known to be very efficient for parameterized problems and has recently been applied in the context of reduced order optimization [48,70,74,85].…”

Section: Introductionmentioning

confidence: 99%

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

Ştefanescu

Sandu

Navon

2015

Journal of Computational Physics

View full text Add to dashboard Cite

Discrete empirical interpolation method (DEIM) Reduced-order models (ROMs) Shallow water equations Finite difference methodsThis work studies reduced order modeling (ROM) approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models. It is shown that a key requirement for a successful reduced order solution is that reduced order Karush-Kuhn-Tucker conditions accurately represent their full order counterparts. In particular, accurate reduced order approximations are needed for the forward and adjoint dynamical models, as well as for the reduced gradient. New strategies to construct reduced order based are developed for proper orthogonal decomposition (POD) ROM data assimilation using both Galerkin and Petrov-Galerkin projections. For the first time POD, tensorial POD, and discrete empirical interpolation method (DEIM) are employed to develop reduced data assimilation systems for a geophysical flow model, namely, the two dimensional shallow water equations. Numerical experiments confirm the theoretical framework for Galerkin projection. In the case of Petrov-Galerkin projection, stabilization strategies must be considered for the reduced order models. The new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time.

show abstract

“…Missing point estimation and Gauss–Newton with approximated tensors methods are relying upon the gappy POD technique and were developed for the same reason. Reduced basis methods have been recently developed and utilize on greedy algorithms to efficiently compute numerical solutions for parametrized applications .…”

Section: Introductionmentioning

confidence: 99%

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

Ştefanescu

Sandu

Navon

2014

Numerical Methods in Fluids

103

View full text Add to dashboard Cite

This paper introduces tensorial calculus techniques in the framework of POD to reduce the computational complexity of the reduced nonlinear terms. The resulting method, named tensorial POD, can be applied to polynomial nonlinearities of any degree p. Such nonlinear terms have an online complexity of O.k pC1 /, where k is the dimension of POD basis and therefore is independent of full space dimension. However, it is efficient only for quadratic nonlinear terms because for higher nonlinearities, POD model proves to be less time consuming once the POD basis dimension k is increased. Numerical experiments are carried out with a two-dimensional SWE test problem to compare the performance of tensorial POD, POD, and POD/discrete empirical interpolation method (DEIM). Numerical results show that tensorial POD decreases by 76 the computational cost of the online stage of POD model for configurations using more than 300,000 model variables. The tensorial POD SWE model was only 2 to 8 slower than the POD/DEIM SWE model but the implementation effort is considerably increased. Tensorial calculus was again employed to construct a new algorithm allowing POD/DEIM SWE model to compute its offline stage faster than POD and tensorial POD approaches. 498 R.ŞTEFȂNESCU, A. SANDU AND I. M. NAVON truncation does not extend easily for high-order systems, and several grammians approximations were proposed leading to methods such as approximate subspace iteration [9], least squares approximation [10], Krylov subspace methods [11,12], and balanced POD [13]. Among moment matching methods, we mention partial realization [14,15], Padé approximation [16][17][18][19], and rational approximation [20].Although for linear models we are able to produce input-independent highly accurate reduced models, in the case of general nonlinear systems, the transfer function approach is not yet applicable and input-specified semi-empirical methods are usually employed. Recently, some encouraging research results using generalized transfer functions and generalized moment matching have been obtained by [21] for nonlinear model order reduction but future investigations are required.Proper orthogonal decomposition and its variants are also known as Karhunen-Loève expansions [22,23], principal component analysis [24], and empirical orthogonal functions [25] among others. It is the most prevalent basis selection method for nonlinear problems and, among other requirements, relies on the fact that the desired simulation is well simulated in the input collection. Data analysis using POD is conducted to extract basis functions from experimental data or detailed simulations of high-dimensional systems (method of snapshots introduced by [26-28]) for subsequent use in Galerkin projections that yield low-dimensional dynamical models. Unfortunately, the POD Galerkin approach displays a major disadvantage because its nonlinear reduced terms still have to be evaluated on the original state space making the simulation of the reduced order system too expensive. There exist seve...

show abstract

Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems

Cited by 36 publications

References 33 publications

An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

An adaptive and efficient greedy procedure for the optimal training of parametric reduced‐order models

POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations

Contact Info

Product

Resources

About