Model Order Reduction for Unsteady Aerodynamic Applications

In this work a nonlinear model order reduction approach for unsteady aerodynamic applications is presented. It is based on Proper Orthogonal Decomposition (POD). Given a set of snapshots, which are solutions to the full order model, POD yields an optimal basis for representing reduced order solutions of the governing equations. The idea of the reduced order modeling approach in this work is to formulate each time step of the unsteady equations as a steady problem. This yields the so-called unsteady residual. The unsteady residual is then minimized in the space spanned by the POD basis vectors. As this space is of reduced size, the minimization problem is as well. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

高阶马尔可夫链的降阶模型及其应用

本文通过对传统高阶马尔可夫链模型的状态空间进行阶数重构,导出一个在重构状态空间上的降阶马尔可夫模型。理论分析证明,降阶马尔可夫链模型不但可以描述传统高阶马尔可夫链模型的全部性态,更能表达较传统模型细微的随机结构。然后,应用降阶模型对我国股票指数的动态变化进行实证分析,讨论了阶数的选取和高阶马尔可夫性检验,分析了股票市场内在波动结构。最后,对股指序列作出短期与长期的预测分析。     

Error-Controlled Model Order Reduction of Second Order Systems with Adaptive Choice of Shifts and Order

This contribution summarizes a novel approach to error-controlled model order reduction of second order systems, i. e. a new way to find a reduced order model of specified approximation quality. To this end, starting from a particular state space realization of the model, one performs multiple reduction steps and increments the reduced model until (computationally affordable) rigorous global error bounds fall below the prescribed limit. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Process Model Order Reduction for Controller Design in Discrete Systems

This paper studies the model reduction problem with emphasis on closed loop performance. First, the admissible fixed order models are identified as the ones that are simultaneously stabilizable with the high order process. A complete parametrization of this set in terms of stable parameters that satisfy cubic equality constraints is developed. An optimization problem is then formulated so that its solution identifies the fixed order model that satisfies nominal performance specifications, it is simultaneously stabilizable with the actual process and the actual closed loop optimally satisfies the nominal performance specifications. This nonlinear, infinite dimensional optimization problem is solved by means of exact penalty functions and an asymptotic approximation procedure. An illustrative example is also presented.     

A Parallel Reduction of Hamiltonian Cycle to Hamiltonian Path in Tournaments

We propose a parallel algorithm which reduces the problem of computing Hamiltonian cycles in tournaments to the problem of computing Hamiltonian paths. The running time of our algorithm is O(log n) using O(n²/log n) processors on a CRCW PRAM, and O(log n log log n) on an EREW PRAM using O(n²/log n log log n) processors. As a corollary, we obtain a new parallel algorithm for computing Hamiltonian cycles in tournaments. This algorithm can be implemented in time O(log n) using O(n²/log n) processors in the CRCW model and in time O(log²n) with O(n²/log n log log n) processors in the EREW model.     

Model Order Reduction of a Drill-String-Model with Self-Excited Stick-Slip Vibrations

The dynamical behavior of a drill-string is defined by its small diameter-to-length ratio, which makes the string vulnerable to torsional vibrations. In combination with the nonlinear friction characteristic at the drill bit, this can lead to self-excited stick-slip vibrations which are detrimental to the drilling process. The string can be modeled by the Finite Element Method or as a Multi-Body system to represent the distributed character of the system. The analysis of the resulting high-dimensional model is, however, elaborate and time-consuming. We show that through Galerkin Projection onto the first two Characteristic Functions gained from Karhunen-Loève-Transformation, a reduced system can be obtained which reproduces the essential dynamical properties of the original system, e.g. the stick-slip motion. With the reduced system, the linear stability of the drill-string can be analyzed. We show that by reducing the inertia of the rotary table the system can be stabilized. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Model Order Reduction in Nanoelectronics: Combination of POD and TPWL

In this paper we demonstrate model order reduction of a nonlinear academic model of an inverter chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are combined, the trajectory piecewise linear approach and the proper orthogonal decomposition. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dimension Reduction in Extended Quermass-Interaction Process

Many objects studied in biology, medicine or material sciences create spatial formations of random shape in which we can observe mutual interactions among those objects. In order to analyse the data composed of such patterns, we use the methods of spatial statistics. Recently, extended random-disc Quermass-interaction process was studied, simulated and consequently statistically analysed using MCMC maximum likelihood method (MCMC MLE). However, this analysis brought some problems. First, it was quite time-consuming, secondly, in some special cases, the parameter estimates may undervalue the real parameter values. In this paper, we describe how we can solve these problems by dimension reduction.     

A Connection between Time Domain Model Order Reduction and Moment Matching

We investigate the time domain model order reduction (MOR) framework using general orthogonal polynomials by Jiang and Chen [4] and extend their idea by exploiting the structure of the corresponding linear system of equations. Identifying an equivalent Sylvester equation we show a connection to a rational Krylov subspace, and thus moment matching. This theoretical link between the MOR techniques is illustrated by one numerical example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model Order Reduction and Error Estimation for the Parameter-Dependent Eddy Current Equation

In design optimization many simulations are performed of the same physical phenomena. Using model order reduction methods these physical phenomena can be efficiently computed for various geometries, but there is an error compared to the full order solution. This error is estimated by error estimators in different norms. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Krylov Subspace Methods for Model Order Reduction of Bilinear Discrete-Time Control Systems

In this paper, we propose a projection technique for model order reduction of discrete-time bilinear control systems based on the concept of so-called multimoments. We will make use of an explicit solution formula of the system and consider its Z-transform which allows us to characterize the system output by generalized transfer functions. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction

In this article an efficient numerical method to solve multiobjective optimization problems for fluid flow governed by the Navier Stokes equations is presented. In order to decrease the computational effort, a reduced order model is introduced using Proper Orthogonal Decomposition and a corresponding Galerkin Projection. A global, derivative free multiobjective optimization algorithm is applied to compute the Pareto set (i.e. the set of optimal compromises) for the concurrent objectives minimization of flow field fluctuations and control cost. The method is illustrated for a 2D flow around a cylinder at Re = 100. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于投影寻踪分类模型的各地区节能降耗评价研究

能源作为经济增长中重要的投入要素,对各国经济发展都起着决定性影响。国家十一五规划中明确提出2010年中国单位GDP能源消耗下降20%,这个宏伟目标的实现对于我国经济的可持续发展和改善人民生活环境具有重要意义。本文运用投影寻踪分类模型,对节能降耗综合评价指标体系进行分析,得到最佳投影值及各地区应承担的节能降耗责任大小。与主成分分析法结果的比较表明投影寻踪分类模型对这类指标评价问题的合理性和可行性。     

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

We study the existence of quasi-periodic, invariant tori in a nearly integrable Hamiltonian system of high order proper degeneracy, i.e., the integrable part of the Hamiltonian involves several time scales and at each time scale the corresponding Hamiltonian depends on only part of the action variables. Such a Hamiltonian system arises frequently in problems of celestial mechanics, for instance, in perturbed Kepler problems like the restricted and non-restricted 3-body problems and spatial lunar problems in which several bodies with very small masses are coupled with two massive bodies and the nearly integrable Hamiltonian systems naturally involve different time scales. Using KAM method, we will show under certain higher order non-degenerate conditions of Bruno–Rüssmann type that the majority of quasi-periodic, invariant tori associated with the integrable part will persist after the non-integrable perturbation. This actually concludes the KAM metric stability for such a properly degenerate Hamiltonian system.     

Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integrality-preserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Section 6 reviews lift-and-project cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with lift-and-project cuts. This is an updated and extended version of the paper published in LNCS 2241, Springer, 2001 (as given in Balas, 2001). Research was supported by the National Science Foundation through grant #DMI-9802773 and by the Office of Naval Research through contract N00014-97-1-0196.     

基于时域误差限的大规模系统自适应模型降阶

为满足解大规模动态系统常微分方程组对精度和速度权衡的要求,提出了一种基于误差限的大规模系统自适应模型降阶方法,其中方法的误差分析基于时域最大误差限,降阶方法基于SVD-Krylov子空间的方法.方法既考虑了算法的复杂性,又保证了算法的精度.通过对典型实例分析,结果表明该方法在给定相对误差限10~(-4)下得出的降阶阶数在不同频率下都能给出很好的近似精度,低频1~10Hz平均相对误差为1.1812×10~(-5),高频1~10GHz平均相对误差为5.6408×10~(-5),即在很宽的频率范围内都能满足精度要求.     

Back-Transformation into Physical Configuration Space after Model Order Reduction onto a General Subspace

Model order reduction (MOR) techniques that project onto a general subspace are common practice. In Elastic Multi-Body-Dynamics (EMBD) distinct interface coordinates are required for the interconnection of the elastic structure and the multi-body system. For that reason, a physical interpretation of the coordinates is mandatory, which is not the case for reduced order models in a general subspace. To make alternative MOR techniques accessible for EMBD, a back-projection approach was introduced by [1]. Therein, the system is projected back onto the physical configuration space, which requires the inversion of the master partition of the projection matrix. But the procedure lacks of robustness and generality. A novel approach is introduced by generating additional master coordinates using sensor placement methods, e.g. the Effective-Independence-algorithm (EfI) [2]. By using a rank criterion for the automatic selection of additional coordinates, the improved back-transformation performs properly and without damaging the reduced order model at a fairly small computational overhead, which is demonstrated at the example of a gear box housing. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reparameterization-Invariant Reduction in the Hamiltonian Description of a Relativistic String

We study the time-reparameterization-invariant dynamics of an open relativistic string using the generalized Dirac–Hamilton theory and resolving the constraints of the first kind. The reparameterization-invariant evolution variable is the time coordinate of the string center of mass. Using a transformation that preserves the diffeomorphism group of the generalized Hamiltonian and the Poincaré covariance of the local constraints, we segregate the center-of-mass coordinates from the local degrees of freedom of the string. We identify the time coordinate of the string center of mass and the proper time measured in the string frame of reference using the Levi-Civita–Shanmugadhasan canonical transformation, which transforms the global constraint (the mass shell) in the new momentum such that the Hamiltonian reduction does not require the corresponding gauge condition. Resolving the local constraints, we obtain an equivalent reduced system whose Hamiltonian describes the evolution w.r.t. the proper time of the string center of mass. The Röhrlich quantum relativistic string theory, which includes the Virasoro operators L_n only with n > 0, is used to quantize this system. In our approach, the standard problems that appear in the traditional quantization scheme, including the space–time dimension D = 26 and the tachyon emergence, arise only in the case of a massless string, M² = 0.     

具有位数3和4的双随机循环矩阵中的素元

本文研究了双随机循环矩阵中素元的分类问题．由于任一n阶双随机循环矩阵都可以唯一地表示为移位的n-1次一元多项式,从而可把双随机循环矩阵中素元的分类问题简化为解双随机循环矩阵上的一个方程．应用此原理,本文完全解决了判别具有位数3的n阶双随机循环矩阵是否为素元的问题,并给出了n阶双随机循环矩阵中一类具有位数4的素元．     

The Reduction Method in the Theory of Lie-Algebraically Integrable Oscillatory Hamiltonian Systems

We study the problem of the complete integrability of nonlinear oscillatory dynamical systems connected, in particular, both with the Cartan decomposition of a Lie algebra is the Lie algebra of a fixed subgroup with respect to an involution : G G on the Lie group G, and with a Poisson action of special type on a symplectic matrix manifold.