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A comprehensive approach for real-time computations using a database of parametric, linear, projection-based reduced-order models (ROMs) based on arbitrary underlying meshes is proposed. In the offline phase of this approach, the parameter space is sampled and linear ROMs defined by linear reduced operators are pre-computed at the sampled parameter points and stored. Then, these operators and associated ROMs are transformed into counterparts that satisfy a certain notion of consistency. In the online phase of this approach, a linear ROM is constructed in real-time at a queried but unsampled parameter point by interpolating the pre-computed linear reduced operators on matrix manifolds and therefore computing an interpolated linear ROM. The proposed overall model reduction framework is illustrated with two applications: a parametric inverse acoustic scattering problem associated with a mockup submarine, and a parametric flutter prediction problem associated with a wing-tank system. The second application is implemented on a mobile device, illustrating the capability of the proposed computational framework to operate in real-time.

Large scale finite element analysis requires model order reduction for computationally expensive applications such as optimization, parametric studies and control design. Although model reduction for nonlinear problems is an active area of research, a major hurdle is modeling and approximating contact problems. This manuscript introduces a projection-based model reduction approach for static and dynamic contact problems. In this approach, non-negative matrix factorization is utilized to optimally compress and strongly enforce positivity of contact forces in training simulation snapshots. Moreover, a greedy algorithm coupled with an error indicator is developed to efficiently construct parametrically robust low-order models. The proposed approach is successfully demonstrated for the model reduction of several two-dimensional elliptic and hyperbolic obstacle and self contact problems. ∗Corresponding author Email address: maciej.balajewicz@stanford.edu (Maciej Balajewicz) 1Postdoctoral Fellow 2Engineering Research Associate 3Vivian Church Hoff Professor of Aircraft Structures

Summary Reduced-order models for linear time-invariant dynamical systems are considered, and the error between the full-order model and the reduced-order model solutions is characterized. Based on the analytical representation of the error, an a posteriori error indicator is proposed that combines a Krylov-based exponential integrator and an a posteriori residual-based estimate. Numerical experiments illustrate the quality of the error estimator. Copyright © 2014 John Wiley & Sons, Ltd.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve the second-order structure, are studied. Error bounds for the full state variables are derived in the continuous setting (when the whole trajectory is known) and in the discrete setting when the Newmark average-acceleration scheme is used on the second-order semi-discrete equation. When the approximating subspace is constructed using the proper orthogonal decomposition, the error estimates are proportional to the sums of the neglected singular values. Numerical experiments illustrate the theoretical results.

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate difference schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the first demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial differential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

SUMMARY A new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced-order bases is presented. It is particularly suited for problems characterized by different physical regimes, parameter variations, or moving features such as discontinuities and fronts. Instead of approximating the solution of interest in a fixed lower-dimensional subspace of global basis vectors, the proposed model order reduction method approximates this solution in a lower-dimensional subspace generated by most appropriate local basis vectors. To this effect, the solution space is partitioned into subregions, and a local reduced-order basis is constructed and assigned to each subregion offline. During the incremental solution online of the reduced problem, a local basis is chosen according to the subregion of the solution space where the current high-dimensional solution lies. This is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower-dimensional solution space. Because it is also applicable to the process of hyper reduction, the proposed method for nonlinear model order reduction is computationally efficient. Its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid-structure-electric interaction problems. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY A rigorous method for stabilizing projection-based linear reduced-order models without significantly affecting their accuracy is proposed. Unlike alternative approaches, this method is computationally efficient. It requires primarily the solution of a small-scale convex optimization problem. Furthermore, it is nonintrusive in the sense that it operates directly on readily available reduced-order operators. These can be precomputed using any data compression technique including balanced truncation, balanced proper orthogonal decomposition, proper orthogonal decomposition, or moment matching. The proposed method is illustrated with three applications: the stabilization of the reduction of the Computational Fluid Dynamics-based model of a linearized unsteady supersonic flow, the reduction of a Computational Structural Dynamics system, and the stabilization of the reduction of a coupled Computational Fluid Dynamics–Computational Structural Dynamics model of a linearized aeroelastic system in the transonic flow regime. Copyright © 2012 John Wiley & Sons, Ltd.

A rigorous method for interpolating a set of parameterized linear structural dynamics reduced-order models (ROMs) is presented. By design, this method does not operate on the underlying set of parameterized full-order models. Hence, it is amenable to an online real-time implementation. It is based on mapping appropriately the ROM data onto a tangent space to the manifold of symmetric positive-definite matrices, interpolating the mapped data in this space and mapping back the result to the aforementioned manifold. Algorithms for computing the forward and backward mappings are offered for the case where the ROMs are derived from a general Galerkin projection method and the case where they are constructed from modal reduction. The proposed interpolation method is illustrated with applications ranging from the fast dynamic characterization of a parameterized structural model to the fast evaluation of its response to a given input. In all cases, good accuracy is demonstrated at real-time processing speeds. Copyright © 2009 John Wiley & Sons, Ltd.

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.

In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems. (C) 2015 Elsevier Inc. All rights reserved.

The Gauss--Newton with approximated tensors (GNAT) method is a nonlinear model reduction method that operates on fully discretized computational models. It achieves dimension reduction by a Petrov--Galerkin projection associated with residual minimization; it delivers computational efficency by a hyper-reduction procedure based on the `gappy POD' technique. Originally presented in Ref. [1], where it was applied to implicit nonlinear structural-dynamics models, this method is further developed here and applied to the solution of a benchmark turbulent viscous flow problem. To begin, this paper develops global state-space error bounds that justify the method's design and highlight its advantages in terms of minimizing components of these error bounds. Next, the paper introduces a `sample mesh' concept that enables a distributed, computationally efficient implementation of the GNAT method in finite-volume-based computational-fluid-dynamics (CFD) codes. The suitability of GNAT for parameterized problems is highlighted with the solution of an academic problem featuring moving discontinuities. Finally, the capability of this method to reduce by orders of magnitude the core-hours required for large-scale CFD computations, while preserving accuracy, is demonstrated with the simulation of turbulent flow over the Ahmed body. For an instance of this benchmark problem with over 17 million degrees of freedom, GNAT outperforms several other nonlinear model-reduction methods, reduces the required computational resources by more than two orders of magnitude, and delivers a solution that differs by less than 1% from its high-dimensional counterpart.