Model reduction of nonlinear systems

The paper is concerned with the model reduction of nonlinear systems. Such methods are required in all branches of engineering. Often the level of detail in system models can cloud the essential behaviour and hence, methods are required to identify the behaviour of interest to the designer. In this contribution, the particular focus is on the empirical balanced truncation method of model reduction. A standard test case will illustrate the efficacies of the suggested approach. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction for nonlinear problems in circuit simulation

Electrical circuits usually contain nonlinear components. Hence we are interested in MOR methods that can be applied to a system of nonlinear Differential-Algebraic Equations (DAEs). In particular we consider the TPWL (Trajectory PieceWise Linear) and POD (Proper Orthogonal Decomposition) methods. While the first one fully exploits linearity, the last method needs modifications to become efficient in evaluation. We describe a particular technique based on Missing Point Estimation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model reduction and dynamic iteration for coupled nonlinear systems

The dynamical behavior of coupled systems is determined by different interconnected subsystems that are usually governed by entirely different physical laws and often act in different time and space scales. We discuss the simulation of coupled nonlinear systems using dynamic iteration combined with model order reduction. We also study the convergence of this approach and derive error estimates for approximate solutions. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficient nonlinear reanalysis for structural modifications

In this paper an approach for nonlinear reanalysis based on residual increment approximations (RIA) is presented. The method requires only the evaluation of residual vectors, which can be done very fast and efficient. Moreover, the results are improved by using a rational approximation method. The approach is general and can be applied to linear and nonlinear problems with different kind of design modifications. Furthermore, the proposed method is very efficient and easily to implement in existing finite element programs. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A reduction principle for coupled nonlinear parabolic-hyperbolic PDE

We consider an abstract system of coupled nonlinear parabolic-hyperbolic partial differential equations. This system may describe thermoelastic phenomena in a continuum medium. Under some condition we prove the existence of an exponentially attracting invariant manifold for the coupled system and show that this system can be reduce to a single hyperbolic equation with modified nonlinearity.     

Neural network closures for nonlinear model order reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.     

Galerkin-based multi-scale time integration for nonlinear structural dynamics

This paper deals with a GALERKIN-based multi-scale time integration of a viscoelastic rope model. Using HAMILTON's dynamical formulation, NEWTON's equation of motion as a second-order partial differential equation is transformed into two coupled first order partial differential equations in time. The considered finite viscoelastic deformations are described by means of a deformation-like internal variable determined by a first order ordinary differential equation in time. The corresponding multi-scale time-integration is based on a PETROV-GALERKIN approximation of all time evolution equations, leading to a new family of time stepping schemes with different accuracy orders in the state variables. The resulting nonlinear algebraic time evolution equations are solved by a multi-level NEWTON-RAPHSON method. Realizing this transient numerical simulation, we also demonstrates a parallelized solution of the viscous evolution equation in CUDA©. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The method of successive affine reduction for nonlinear minimization

The traditional development of conjugate gradient (CG) methods emphasizes notions of conjugacy and the minimization of quadratic functions. The associated theory of conjugate direction methods, strictly a branch of numerical linear algebra, is both elegant and useful for obtaining insight into algorithms for nonlinear minimization. Nevertheless, it is preferable that favorable behavior on a quadratic be a consquence of a more general approach, one which fits in more naturally with Newton and variable metric methods. We give new CG algorithms along these lines and discuss some of their properties, along with some numerical supporting evidence.I acknowledge partial support by ONR Contract #N00014-76-C-0013 at the Center for Pure & Applied Mathematics, University of California, Berkeley in the dissemination of this article.     

Compensated compactness for nonlinear homogenization and reduction of dimension

We study the limit behaviour of some nonlinear monotone equations, such as: , in a domain which is thin in some directions (e.g. is a plate or a thin cylinder). After rescaling to a fixed domain , the above equation is transformed into: , with convenient operators and . Assuming that and the inverse of have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000): 35B27, 35B40, 74Q15     

Analysis of an alignment algorithm for nonlinear dimensionality reduction

The goal of dimensionality reduction or manifold learning for a given set of high-dimensional data points, is to find a low-dimensional parametrization for them. Usually it is easy to carry out this parametrization process within a small region to produce a collection of local coordinate systems. Alignment is the process to stitch those local systems together to produce a global coordinate system and is done through the computation of a partial eigendecomposition of a so-called alignment matrix. In this paper, we present an analysis of the alignment process, giving conditions under which the null space of the alignment matrix recovers the global coordinate system up to an affine transformation. We also propose a post-processing step that can determine the global coordinate system up to a rigid motion. This in turn shows that Local Tangent Space Alignment method (LTSA) can recover a locally isometric embedding up to a rigid motion. AMS subject classification (2000)  65F15, 62H30, 15A18     

Model reduction techniques for sampled-data systems

An approximately balanced realization of linear finite-dimensional sampled-data systems is proposed. The theoretical support of the approximately balancing algorithm is represented by a result on the asymptotic expansions with respect to the sampling step of the sampled controllability and observability graminas. Reduced order models obtained as singular perturbational approximations of approximately balanced realizations of sampled-data systems are shown to be acceptable solutions to the sampled-data system model reduction problem in the sense that, enjoying some asymptotic properties, they come close to the exact solutions as the sampling step decreases. An example illustrates the results.     

Robust estimation of nonlinear structural models

The problem is considered of identification of nonlinear error-in-variables models under conditions of a heavy contamination of the sample, including the presence of outliers (i.e., anomalous observations). On the basis of robust estimation methods, we propose a development of the algorithms of adjusted and total least squares. This has enabled us to improve the accuracy of the response prediction in the presence of outliers in the sample. The algorithms are used for constructing the Engel curve from the budget survey data. In result, we draw better conclusions about the regularities of the household behavior when the income changes.     

Linear low-rank approximation and nonlinear dimensionality reduction

We present our recent work on both linear and nonlinear data reduction methods and algorithms: for the linear case we discuss results on structure analysis of SVD of column-partitioned matrices and sparse low-rank approximation; for the nonlinear case we investigate methods for nonlinear dimensionality reduction and manifold learning. The problems we address have attracted great deal of interest in data mining and machine learning.     

Model reduction and clustering techniques for crash simulations

Model reduction in car crash simulations is a fairly new research field. In this paper, a possible workflow is presented: Since nonlinear behavior can occur, parts with linear and nonlinear behavior need to be separated with clustering methods such as k-means or spectral clustering. For the latter, a nonlinear reduction technique such as POD-DEIM needs to be applied. A longitudinal chassis beam of a 2001 Ford Taurus is used to examine the different clustering methods. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An interactive weight space reduction procedure for nonlinear multiple objective mathematical programming

To make a decision that is defined by multiple, conflicting objectives it is necessary to know the relative importance of the different objectives. In this paper we present an interactive method and the underlying theory for solving multiple objective mathematical programming problems defined by a convex feasible region and concave, continuously differentiable objective functions. The relative importance of the different objectives for a decision maker is elicited by using binary comparisons of objective function vectors. The method is cognitively easy to use and in test problems has rapidly converged to an optimal solution.     

Sensitivity and stability analysis for nonlinear programming

We give a brief overview of important results in several areas of sensitivity and stability analysis for nonlinear programming, focusing initially on qualitative characterizations (e.g., continuity, differentiability and convexity) of the optimal value function. Subsequent results concern quantitative measures, in particular optimal value and solution point parameter derivative calculations, algorithmic approximations, and bounds. Our treatment is far from exhaustive and concentrates on results that hold for smooth well-structured problems.Research supported by National Science Foundation Grant ECS-86-19859 and Grant N00014-89-J-1537 Office of Naval Research.     

Blow-up analysis for a nonlinear diffusion system

This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.     

Finite element methods for nonlinear shell analysis

Summary Finite element methods for nonlinear shell analysis are analyzed using both the minimum potential energy and the mixed formulations. Existence and local uniqueness of both the exact solutions and the corresponding finite element solutions are proved. Error bounds, which are of the same order as for the corresponding linear problems, are established.     

Involution analysis for nonlinear exterior differential systems

Analysing auxiliary systems for integrability conditions is an indispensable part of many indirect studies of partial differential equations, such as symmetry analysis. An invariant differential geometric approach to integrability analysis is described, using the concept of an involutive exterior differential system. The essential theory is first presented, paying particular attention to the nonlinear case, and then algorithms implementing the central techniques are discussed.