scholarly journals Study of methods for dimension reduction of complex dynamic linear systems models

2018 ◽  
Vol 226 ◽  
pp. 04036
Author(s):  
Yuriy M. Manatskov ◽  
Torsten Bertram ◽  
Danil V. Shaykhutdinov ◽  
Nikolay I. Gorbatenko
Keyword(s):  
Linear Systems ◽  
Main Idea ◽  
Computation Time ◽  
Order Reduction ◽  
Complex Dynamic ◽  
Model Order ◽  
Reduced Order ◽  
Advantages And Disadvantages ◽  
Optimization Stage ◽  
Linear Systems Of Equations

Complex dynamic linear systems of equations are solved by numerical iterative methods, which need much computation and are timeconsuming ones, and the optimization stage requires repeated solution of these equation systems that increases the time on development. To shorten the computation time, various methods can be applied, among them preliminary (estimated) calculation or oversimple models calculation, however, while testing and optimizing the full model is used. Reduced order models are very popular in solving this problem. The main idea of a reduced order model is to find a simplified model that may reflect the required properties of the original model as accurately as possible. There are many methods for the model order reduction, which have their advantages and disadvantages. In this article, a method based on Krylov subspaces and SVD methods is considered. A numerical experiments is given.

scholarly journals Direct Verification of Linear Systems with over 10000 Dimensions

10.29007/dwj1 ◽  
2018 ◽  
Author(s):  
Stanley Bak ◽  
Parasara Sridhar Duggirala
Keyword(s):  
Linear Systems ◽  
Computation Time ◽  
Order Reduction ◽  
Direct Analysis ◽  
Principle Of Superposition ◽  
Basis Matrix ◽  
Time Step ◽  
Model Order ◽  
Large Linear Systems ◽  
Core Part

We evaluate a recently-proposed reachability method on a set of high-dimensional lin- ear system benchmarks taken from model order reduction and presented in ARCH 2016. The approach uses a state-set representation called a generalized star set and the principle of superposition of linear systems to achieve scalability. The method was previously shown to have promise in terms of scalability for direct analysis of large linear systems. For each benchmark, we also compare computing the basis matrix, a core part of the reachabil- ity method, using numerical simulations versus a matrix exponential formulation. The approach successfully analyzes systems with hundreds of dimensions in minutes, and can scale to systems that have over 10000 dimensions with a computation time ranging from tens of minutes to tens of hours, depending on the desired time step.

scholarly journals Model-order reduction of electromagnetic fields in open domains

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. WB61-WB70
Author(s):  
Jörn Zimmerling ◽  
Vladimir Druskin ◽  
Mikhail Zaslavsky ◽  
Rob F. Remis
Keyword(s):  
Model Order Reduction ◽  
Maxwell Equations ◽  
Order Reduction ◽  
Efficient Computation ◽  
Scattering Problems ◽  
Model Order ◽  
Reduced Order ◽  
Advantages And Disadvantages ◽  
Reduction Techniques ◽  
Memory Efficient

We have developed several Krylov projection-based model-order reduction techniques to simulate electromagnetic wave propagation and diffusion in unbounded domains. Such techniques can be used to efficiently approximate transfer function field responses between a given set of sources and receivers and allow for fast and memory-efficient computation of Jacobians, thereby lowering the computational burden associated with inverse scattering problems. We found how general wavefield principles such as reciprocity, passivity, and the Schwarz reflection principle translate from the analytical to the numerical domain and developed polynomial, extended, and rational Krylov model-order reduction techniques that preserve these structures. Furthermore, we found that the symmetry of the Maxwell equations allows for projection onto polynomial and extended Krylov subspaces without saving a complete basis. In particular, short-term recurrence relations can be used to construct reduced-order models that are as memory efficient as time-stepping algorithms. In addition, we evaluated the differences between Krylov reduced-order methods for the full wave and diffusive Maxwell equations and we developed numerical examples to highlight the advantages and disadvantages of the discussed methods.

Model of induction brazing of nonmagnetic metals using model order reduction approach

2018 ◽  
Vol 37 (4) ◽  
pp. 1515-1524 ◽  
Author(s):  
Pavel Karban ◽  
David Pánek ◽  
Ivo Doležel
Keyword(s):  
Heat Treatment ◽  
Model Order Reduction ◽  
Computation Time ◽  
Order Reduction ◽  
Nonlinear Problems ◽  
Model Order ◽  
Content Type ◽  
Weakly Nonlinear ◽  
Induction Brazing ◽  
Multiphysics Problems

Purpose A novel technique for control of complex physical processes based on the solution of their sufficiently accurate models is presented. The technique works with the model order reduction (MOR), which significantly accelerates the solution at a still acceptable uncertainty. Its advantages are illustrated with an example of induction brazing. Design/methodology/approach The complete mathematical model of the above heat treatment process is presented. Considering all relevant nonlinearities, the numerical model is reduced using the orthogonal decomposition and solved by the finite element method (FEM). It is cheap compared with classical FEM. Findings The proposed technique is applicable in a wide variety of linear and weakly nonlinear problems and exhibits a good degree of robustness and reliability. Research limitations/implications The quality of obtained results strongly depends on the temperature dependencies of material properties and degree of nonlinearities involved. In case of multiphysics problems characterized by low nonlinearities, the results of solved problems differ only negligibly from those solved on the full model, but the computation time is lower by two and more orders. Yet, however, application of the technique in problems with stronger nonlinearities was not fully evaluated. Practical implications The presented model and methodology of its solution may represent a basis for design of complex technologies connected with induction-based heat treatment of metal materials. Originality/value Proposal of a sophisticated methodology for solution of complex multiphysics problems established the MOR technology that significantly accelerates their solution at still acceptable errors.

scholarly journals Toward Optimality of Proper Generalised Decomposition Bases

2019 ◽  
Vol 24 (1) ◽  
pp. 30 ◽  
Author(s):  
Shadi Alameddin ◽  
Amélie Fau ◽  
David Néron ◽  
Pierre Ladevèze ◽  
Udo Nackenhorst
Keyword(s):  
Greedy Algorithms ◽  
Order Reduction ◽  
Nonlinear Material ◽  
Proper Generalized Decomposition ◽  
Model Order ◽  
Reduced Order ◽  
Structural Problems ◽  
Low Dimensional ◽  
Value Decomposition ◽  
The Greedy Algorithm

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method.

scholarly journals A Robust Computational Technique for Model Order Reduction of Two-Time-Scale Discrete Systems via Genetic Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Othman M. K. Alsmadi ◽  
Zaer S. Abo-Hammour
Keyword(s):  
Genetic Algorithms ◽  
Time Scale ◽  
Model Order Reduction ◽  
Fitness Function ◽  
Order Reduction ◽  
Discrete Systems ◽  
Computational Technique ◽  
Model Order ◽  
New Approach ◽  
Reduced Order

A robust computational technique for model order reduction (MOR) of multi-time-scale discrete systems (single input single output (SISO) and multi-input multioutput (MIMO)) is presented in this paper. This work is motivated by the singular perturbation of multi-time-scale systems where some specific dynamics may not have significant influence on the overall system behavior. The new approach is proposed using genetic algorithms (GA) with the advantage of obtaining a reduced order model, maintaining the exact dominant dynamics in the reduced order, and minimizing the steady state error. The reduction process is performed by obtaining an upper triangular transformed matrix of the system state matrix defined in state space representation along with the elements ofB,C, andDmatrices. The GA computational procedure is based on maximizing the fitness function corresponding to the response deviation between the full and reduced order models. The proposed computational intelligence MOR method is compared to recently published work on MOR techniques where simulation results show the potential and advantages of the new approach.

scholarly journals Large-Scale Linear Systems from Order-Reduction

10.29007/xk7x ◽  
2018 ◽  
Author(s):  
Hoang-Dung Tran ◽  
Luan Viet Nguyen ◽  
Taylor T Johnson
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Order Reduction ◽  
Single Mode ◽  
Hybrid Automaton ◽  
Model Order ◽  
Verification Tools ◽  
Test Sets ◽  
Benchmark Suite ◽  
And Robotics

This benchmark suite is composed of nine examples of large-scale linear systems, ranging in dimensionality in the tens to the low thousands. The benchmarks are derived from diverse fields such as civil engineering and robotics, and are based on similar existing test sets for model-order reduction algorithms in control and numerical analysis. Each example is provided in the SpaceEx XML model format as single-mode hybrid automaton and are compatible with the HyST model transformation tool to support analysis in other verification tools. Some preliminary reachability analysis results for some of the smaller examples (on the order of tens of dimensions) are presented using SpaceEx.

scholarly journals Suboptimal Control Using Model Order Reduction

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Avadh Pati ◽  
Awadhesh Kumar ◽  
Dinesh Chandra
Keyword(s):  
Model Order Reduction ◽  
Order Reduction ◽  
Higher Order ◽  
Reduced Order Model ◽  
Suboptimal Control ◽  
Order Model ◽  
Model Order ◽  
Markov Parameters ◽  
Reduced Order ◽  
Partial Feedback

A Padé approximation based technique for designing a suboptimal controller is presented. The technique uses matching of both time moments and Markov parameters for model order reduction. In this method, the suboptimal controller is first derived for reduced order model and then implemented for higher order plant by partial feedback of measurable states.

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