arXiv subjects

Lihong Feng

Publications and source records attributed to Lihong Feng.

26 records · Page 2Linked to original sources

arXiv2020

An Adaptive Sampling Approach for the Reduced Basis Method

The offline time of the reduced basis method can be very long given a large training set of parameter samples. This usually happens when the system has more than two independent parameters. On the other hand, if the training set includes fewer parameter samples, the greedy algorithm might produce a reduced-order model with large errors at the samples outside of the training set. We introduce a method based on a surrogate error model to efficiently sample the parameter domain such that the training set is adaptively updated starting from a coarse set with a small number of parameter samples. A sharp a posteriori error estimator is evaluated on a coarse training set. Radial Basis Functions are used to interpolate the error estimator over a separate fine training set. Points from the fine training set are added into the coarse training set at every iteration based on a user defined criterion. In parallel, parameter samples satisfying a defined tolerance are adaptively removed from the coarse training set. The approach is shown to avoid high computational costs by using a small training set and to provide a reduced-order model with guaranteed accuracy over a fine training set. Further, we show numerical evidence that the reduced-order model meets the defined tolerance over an independently sampled test set from the parameter domain.

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arXiv2020

Adaptive Basis Construction and Improved Error Estimation for Parametric Nonlinear Dynamical Systems

An adaptive scheme to generate reduced-order models for parametric nonlinear dynamical systems is proposed. It aims to automatize the POD-Greedy algorithm combined with empirical interpolation. At each iteration, it is able to adaptively determine the number of the reduced basis vectors and the number of the interpolation basis vectors for basis construction. The proposed technique is able to derive a suitable match between the reduced basis and the interpolation basis vectors, making the generation of a stable, compact and reliable reduced-order model possible. This is achieved by adaptively adding new basis vectors or removing unnecessary ones, at each iteration of the greedy algorithm. An efficient output error indicator plays a key role in the adaptive scheme. We also propose an improved output error indicator based on previous work. Upon convergence of the POD-Greedy algorithm, the new error indicator is shown to be sharper than the existing ones, implicating that a more reliable reduced-order model can be constructed. The proposed method is tested on several nonlinear dynamical systems, namely, the viscous Burgers' equation and two other models from chemical engineering.

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arXiv2020

On Error Estimation for Reduced-order Modeling of Linear Non-parametric and Parametric Systems

Motivated by a recently proposed error estimator for the transfer function of the reduced-order model of a given linear dynamical system, we further develop more theoretical results in this work. Furthermore, we propose several variants of the error estimator, and compare those variants with the existing ones both theoretically and numerically. It has been shown that some of the proposed error estimators perform better than or equally well as the existing ones. All the error estimators considered can be easily extended to estimate output error of reduced-order modeling for steady linear parametric systems.

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arXiv2020

Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain

Interpolatory methods offer a powerful framework for generating reduced-order models (ROMs) for non-parametric or parametric systems with time-varying inputs. Choosing the interpolation points adaptively remains an area of active interest. A greedy framework has been introduced in Feng et al. [ESAIM: Math. Model. Numer. Anal. 51(6), 2017] and in Feng and Benner [IEEE Trans. Microw. Theory Techn. 67(12), 2019] to choose interpolation points automatically using a posteriori error estimators. Nevertheless, when the parameter range is large or if the parameter space dimension is larger than two, the greedy algorithm may take considerable time, since the training set needs to include a considerable number of parameters. As a remedy, we introduce an adaptive training technique by learning an efficient a posteriori error estimator over the parameter domain. A fast learning process is created by interpolating the error estimator using radial basis functions (RBF) over a fine parameter training set, representing the whole parameter domain. The error estimator is evaluated only on a coarse training set including a few parameter samples. The algorithm is an extension of the work in Chellappa et al. [arXiv e-prints 1910.00298] to interpolatory model order reduction (MOR) in frequency domain. Beyond this work, we use a newly proposed inf-sup-constant-free error estimator in the frequency domain, which is often much tighter than the error estimator using the inf-sup constant.

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arXiv2019

Implicit Higher-Order Moment Matching Technique for Model Reduction of Quadratic-bilinear Systems

We propose a projection based multi-moment matching method for model order reduction of quadratic-bilinear systems. The goal is to construct a reduced system that ensures higher-order moment matching for the multivariate transfer functions appearing in the input-output representation of the nonlinear system. An existing technique achieves this for the first two multivariate transfer functions, in what is called the symmetric form of the multivariate transfer functions. We extend this framework to an equivalent and simplified form, the regular form, which allows us to show moment matching for the first three multivariate transfer functions. Numerical results for three benchmark examples of quadratic-bilinear systems show that the proposed framework exhibits better performance with reduced computational cost in comparison to existing techniques.

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arXiv2019

An Adaptive Pole-Matching Method for Interpolating Reduced-Order Models

An adaptive parametric reduced-order modeling method based on interpolating poles of reduced-order models is proposed in this paper. To guarantee correct interpolation, a pole-matching process is conducted to determine which poles of two reduced-order models correspond to the same parametric pole. First, the pole-matching in the scenario of parameter perturbation is discussed. It is formulated as a combinatorial optimization problem and solved by a branch and bound algorithm. Then, an adaptive framework is proposed to build repository ROMs at adaptively chosen parameter values, which well represent the parameter domain of interest. To achieve this, we propose techniques including a predictor-corrector strategy and an adaptive refinement strategy, which enable us to use larger steps to explore the parameter domain of interest with good accuracy. The framework also consists of regression as an optional post-processing phase to further reduce the data storage. The advantages over other parametric reduced-order modeling approaches are, e.g., compatibility with any model order reduction method, constant size of the parametric reduced-order model with respect to the number of parameters, and capability to deal with complicated parameter dependency.

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arXiv2017

Fast Solution of the Linearized Poisson-Boltzmann Equation with nonaffine Parametrized Boundary Conditions Using the Reduced Basis Method

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to FOM. The discrete empirical interpolation method (DEIM) is applied to the Dirichlet boundary conditions which are nonaffine with the parameter, to reduce the complexity of the reduced order model (ROM). From the numerical results, we notice that the RBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ at an accuracy of $10^{-9}$ and reduces computational time by a factor of approximately $7,600$. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of $20$ for a single iteration of the greedy algorithm.

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arXiv2015

Recycling BiCGSTAB with an Application to Parametric Model Order Reduction

Krylov subspace recycling is a process for accelerating the convergence of sequences of linear systems. Based on this technique, the recycling BiCG algorithm has been developed recently. Here, we now generalize and extend this recycling theory to BiCGSTAB. Recycling BiCG focuses on efficiently solving sequences of dual linear systems, while the focus here is on efficiently solving sequences of single linear systems (assuming non-symmetric matrices for both recycling BiCG and recycling BiCGSTAB). As compared with other methods for solving sequences of single linear systems with non-symmetric matrices (e.g., recycling variants of GMRES), BiCG based recycling algorithms, like recycling BiCGSTAB, have the advantage that they involve a short-term recurrence, and hence, do not suffer from storage issues and are also cheaper with respect to the orthogonalizations. We modify the BiCGSTAB algorithm to use a recycle space, which is built from left and right approximate invariant subspaces. Using our algorithm for a parametric model order reduction example gives good results. We show about 40% savings in the number of matrix-vector products and about 35% savings in runtime.

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