Arnoldi-based model order reduction for linear systems with inhomogeneous initial conditions

SUMMARYThis paper addresses the H ∞ model reduction problem of discrete‐time positive linear systems with inhomogeneous initial conditions. For an asymptotically stable positive system with non‐zero initial condition, our goal is to approximate it by a reduced‐order initial‐valued positive system without introducing significant error. We establish a necessary and sufficient condition for the existence of a desired reduced‐order model such that the output error between the original system and the reduced‐order one is bounded by a weighted sum of the magnitude of the input and that of the initial condition. Moreover, based on congruent transformation and the dual form of bounded real lemma, several equivalent conditions are derived in terms of LMIs and an iterative convex optimization algorithm is developed accordingly. Finally, an illustrative example is presented to show the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.

A new model order reduction (MOR) method for large-scale time-delay differential algebra systems is proposed herein. The presented MOR algorithm is based on parametric moment matching and $$\varepsilon $$ -embedding. By selecting an appropriate projection matrix, this process generates reduced-order models that preserve the structure of the original time-delay differential algebra system. Additionally, moment-matching results and error estimations are provided. Finally, two numerical circuits are tested to illustrate the effectiveness of the proposed MOR method.

Three-dimensional electromagnetic methods are fundamental tools for the analysis and design of high-speed systems. These methods often generate large systems of equations, and model order reduction (MOR) methods are used to reduce such a high complexity. When the geometric dimensions become electrically large or signal waveform rise times decrease, time delays must be included in the modeling. Design space optimization and exploration are usually performed during a typical design process that consequently requires repeated simulations for different design parameter values. Efficient performing of these design activities calls for parameterized model order reduction (PMOR) methods, which are able to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as layout or substrate features. We propose a novel PMOR method for neutral delayed differential systems, which is based on an efficient and reliable combination of univariate model order reduction methods, a procedure to find scaling and frequency shifting coefficients and positive interpolation schemes. The proposed scaling and frequency shifting coefficients enhance and improve the modeling capability of standard positive interpolation schemes and allow accurate modeling of highly dynamic systems with a limited amount of initial univariate models in the design space. The proposed method is able to provide parameterized reduced order models passive by construction over the design space of interest. Pertinent numerical examples validate the proposed PMOR approach.

Many new promising model order reduction (MOR) methods and algorithms were developed during the last decade. Industry and academic research institutions intend to test, validate, compare, and use these new promising MOR techniques with their own models. Therefore, an MOR toolbox bridging the gap between theoretical, algorithmic, and numerical developments to an end-user-oriented program, usable by non-experts, was developed called ‘Model Order Reduction of Elastic Multibody Systems’ (Morembs). A C++ implementation as well as a Matlab implementation including an intuitive graphical user interface is available. Import from various FE programs is possible, and the reduced elastic bodies can be exported to a variety of programs to simulate the compact models. In the course of the various projects, many improvements on the algorithmic side were added. As we learned over the years, there is not one ‘optimal’ MOR method. ‘Optimal’ MOR depends on circumstances, like boundary conditions, excitation spectra, further model usage. The toolbox is now used, e.g., in solid mechanics, biomechanics, vehicle dynamics, control of flexible structures, or crash simulations. In all these use cases, the toolbox allows the user to facilitate their well-known modeling and simulation environment. Only the critical MOR process during preprocessing is performed with Morembs, which helps to compare the various MOR techniques to find the most suited one.

We consider the effects of inhomogeneous initial conditions in both the scalar field profile and the extrinsic curvature on different inflationary models. In particular, we compare the robustness of small field inflation to that of large field inflation, using numerical simulations with Einstein gravity in 3+1 dimensions. We find that small field inflation can fail in the presence of subdominant gradient energies, suggesting that it is much less robust to inhomogeneities than large field inflation, which withstands dominant gradient energies. However, we also show that small field inflation can be successful even if some regions of spacetime start out in the region of the potential that does not support inflation. In the large field case, we confirm previous results that inflation is robust if the inflaton occupies the inflationary part of the potential. Furthermore, we show that increasing initial scalar gradients will not form sufficiently massive inflation-ending black holes if the initial hypersurface is approximately flat. Finally, we consider the large field case with a varying extrinsic curvature K, such that some regions are initially collapsing. We find that this may again lead to local black holes, but overall the spacetime remains inflationary if the spacetime is open, which confirms previous theoretical studies.

General orthogonal polynomials are introduced to analyze and approximate the solution of a class of scaled systems. Using the operational matrix of integration, together with the operational matrix of linear transformation, the dynamical equation of a scaled system is reduced to a set of simultaneous linear algebraic equations. The coefficient vectors of the general orthogonal polynomials can be determined recursively by the derived algorithm. An illustrative example is given to demonstrate the validity and applicability of the orthogonal polynomial approximations.

We discuss methods for model order reduction (MOR) of linear systems with many input and output variables, arising in the modeling of linear (sub) circuits with a huge number of nodes and a large number of terminals, like power grids. Our work is based on the approaches SVDMOR and ESVDMOR proposed in recent publications (1; 2; 3; 4; 5). In particular, we discuss efficient numerical algorithms for their implementation. Only by using efficient tools from numerical linear algebra, these methods become applicable for truly large-scale problems.

Linear fractional differential equations with variable coefficients and various types of fractional derivatives can be solved explicitly, using either fixed‐point theorems and successive approximations or Green's functions. In this paper, we consider a multi‐term fractional differential equation with continuous variable coefficients and incommensurate fractional orders, and we construct an explicit solution by a direct method of successive approximations, which is made rigorous by checking the solution function explicitly via substitution. The results are obtained for the most general initial value problem of an inhomogeneous equation with inhomogeneous initial conditions, the first time that an explicit analytical solution has been found for this general problem. Special cases are provided to illustrate the main results, and connections are discussed with previous results in the literature.

The degenerate parabolic variational inequality with mixed boundary conditions and inhomogeneous initial conditions is studied in the case where the corresponding operator can lose the properties of coercivity and continuity in the corresponding Sobolev spaces. By using the Hardy–Poincare inequality, we prove the unique solvability of the original evolutionary variational inequality under the condition that the degenerate weight function is a function of potential type.

This paper proposes a new second-order Model Order Reduction (MOR) method suitable for reducing very large sized RC circuits or RC circuit parts of a non-RC circuit. The MOR is carried out on a partitioned circuit, which enables the use of low-order moments and macromodels of few elements. It is shown that the use of the second-order MOR method in conjunction with lower-order methods improves the overall MOR results and reliability in the case where a circuit contains particularly complex structures.

A new mixed method for relative error model order reduction is proposed. In the proposed method the frequency domain balanced stochastic truncation method is improved by applying the generalized singular perturbation method to the frequency domain balanced system in the reduction procedure. The frequency domain balanced stochastic truncation method, which was proposed in [15] and [17] by the author, is based on two recently developed methods, namely frequency domain balanced truncation within a desired frequency bound and inner-outer factorization techniques. The proposed method in this paper is a carry over of the frequency-domain balanced stochastic truncation and is of interest for practical model order reduction because in this context it shows to keep the accuracy of the approximation as high as possible without sacrificing the computational efficiency and important system properties. It is shown that some important properties of the frequency domain stochastic balanced reduction technique are extended to the proposed reduction method by using the concept and properties of the reciprocal systems. Numerical results show the accuracy, simplicity and flexibility enhancement of the method.

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.

ABSTRACTThe purpose of this paper is the identification of a reduced order model (ROM) from numerical code output by non-intrusive techniques (i.e. not requiring projecting of the governing equations onto the reduced basis modes). In this paper, we perform a comparison between two methods of model order reduction based on dynamic mode decomposition (DMD). The first method is a deterministic (classic) DMD technique endowed with a dynamic filtering criterion of selection of modes used in the ROM model. The second method is an adaptive randomised DMD algorithm (ARDMD) based on a randomised singular value decomposition. This produced an accelerating algorithm, which is endowed with a few additional advantages. In addition, the reduced order model is guaranteed to satisfy the boundary conditions of the full model, which is crucial for surrogate modelling. For numerical illustration, we use the shallow water equations model.

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

A new method for model order reduction for an important class of non-linear dynamical systems is proposed. The reduction technique is based on the frequency balanced reduction method within a desired frequency band and a developed Trajectory Piecewise-Linear (TPWL) model order reduction approach. Both the methods are recently developed techniques and they can increase the accuracy and efficiency of the reduction method. The computational load and memory are significantly reduced when the method is used in the hybrid reduction method. The accuracy enhanced hybrid method is applicable to very large scale non-linear models efficiently and accurately. The numerical results show the accuracy and efficiency enhancement of the proposed method.