An efficient computational method for the optimal control problem for the Burgers equation

Pointwise control of the viscous Burgers equation in one spatial dimension is studied with the objective of minimizing the distance between the final state function and target profile along with the energy of the control. An efficient computational method is proposed for solving such problems, which is based on special orthonormal functions that satisfy the associated boundary conditions. Employing these orthonormal functions as a basis of a modal expansion method, the solution space is limited to the smallest lower subspace that is sufficient to describe the original problem. Consequently, the Burgers equation is reduced to a set of a minimal number of ordinary nonlinear differential equations. Thus, by the modal expansion method, the optimal control of a distributed parameter system described by the Burgers equation is converted to the optimal control of lumped parameter dynamical systems in finite dimension. The time-variant control is approximated by a finite term of the Fourier series whose unknown coefficients and frequencies giving an optimal solution are sought, thereby converting the optimal control problem into a mathematical programming problem. The solution space obtained is based on control parameterization by using the Runge–Kutta method. The efficiency of the proposed method is examined using a numerical example for various target functions.     

A method for reduction of human ventricular action potential model

ABSTRACT

Mathematical modelling and computer simulations are important tools in the field of cardiac electrophysiology. High computational costs of complex models make them difficult to apply in large-scale simulations like tissue. Therefore, model reduction are of particular importance in heart studies. In this paper, we introduce a technique for simplification of ventricular cell(VC) complex models. By using this technique, starting with a complex model of human VC including 17state variables, we reduce the number of state variables to two. Our simplified model is compared with the original one via several electrophysiological features and computational efficiency. Results show that the reduced model has acceptable behaviours in single cell and one-dimensional simulation, moreover, is 55 times faster than the original one. As the presented method does not depend on the reference model, it may be applied to every cardiac cell models or each complex excitable dynamical systems with the same dynamics as VC.     

Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems

Correct selection of spatial basis functions is crucial for model reduction for nonlinear distributed parameter systems in engineering applications. To construct appropriate reduced models, modelling accuracy and computational costs must be balanced. In this paper, empirical Gramian-based spatial basis functions were proposed for model reduction of nonlinear distributed parameter systems. Empirical Gramians can be computed by generalizing linear Gramians onto nonlinear systems, which results in calculations that only require standard matrix operations. Associated model reduction is described under the framework of Galerkin projection. In this study, two numerical examples were used to evaluate the efficacy of the proposed approach. Lower-order reduced models were achieved with the required modelling accuracy compared to linear Gramian-based combined spatial basis function- and spectral eigenfunction-based methods.     

Non-gaussian Test Models for Prediction and State Estimation with Model Errors

Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where the statistical ensemble prediction and the real time filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here, a class of statistically exactly solvable non-Gaussian test models is introduced, where a generalized Feynman-Kac formulation reduces the exact behavior of conditional statistical moments to the solution to inhomogeneous Fokker-Planck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and is combined with information theory to address two important issues in the contemporary statistical prediction of turbulent dynamical systems: the coarse-grained ensemble prediction in a perfect model and the improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for the real time prediction and the state estimation.     

An efficient,partitioned ensemble algorithm for simulating ensembles of evolutionary MHD flows at low magnetic Reynolds number

Studying the propagation of uncertainties in a nonlinear dynamical system usually involves generating a set of samples in the stochastic parameter space and then repeated simulations with different sampled parameters. The main difficulty faced in the process is the excessive computational cost. In this paper, we present an efficient, partitioned ensemble algorithm to determine multiple realizations of a reduced Magnetohydrodynamics (MHD) system, which models MHD flows at low magnetic Reynolds number. The algorithm decouples the fully coupled problem into two smaller subphysics problems, which reduces the size of the linear systems that to be solved and allows the use of optimized codes for each subphysics problem. Moreover, the resulting coefficient matrices are the same for all realizations at each time step, which allows faster computation of all realizations and significant savings in computational cost. We prove this algorithm is first order accurate and long time stable under a time step condition. Numerical examples are provided to verify the theoretical results and demonstrate the efficiency of the algorithm.     

Reduced-order modelling of self-excited,time-periodic systems using the method of Proper Orthogonal Decomposition and the Floquet theory

The mathematical models of dynamical systems become more and more complex, and hence, numerical investigations are a time-consuming process. This is particularly disadvantageous if a repeated evaluation is needed, as is the case in the field of model-based design, for example, where system parameters are subject of variation. Therefore, there exists a necessity for providing compact models which allow for a fast numerical evaluation. Nonetheless, reduced models should reflect at least the principle of system dynamics of the original model.

In this contribution, the reduction of dynamical systems with time-periodic coefficients, termed as parametrically excited systems, subjected to self-excitation is addressed. For certain frequencies of the time-periodic coefficients, referred to as parametric antiresonance frequencies, vibration suppression is achieved, as it is known from the literature. It is shown in this article that by using the method of Proper Orthogonal Decomposition (POD) excitation at a parametric antiresonance frequency results in a concentration of the main system dynamics in a subspace of the original solution space. The POD method allows to identify this subspace accurately and to set up reduced models which approximate the stability behaviour of the original model in the vicinity of the antiresonance frequency in a satisfying manner. For the sake of comparison, modally reduced models are established as well.     

二阶耗散动力系统的降维对解长期行为的误差估计

基于非线性动力学理论,对一类高维二阶耗散自治动力系统的降维及其对解的长期行为的影响进行了理论分析．该分析将方程的解投影到控制方程的线性算子的特征向量所张成的完备空间中,并在相空间中引入一距离的概念,方便地解决了缩减后系统与原始系统解之间的误差或距离的描述．基于此距离定义,首先,分析了由于高阶模态的截取对解的长期行为的影响,并推导出了相应的误差估计,该估计表明由于降维对系统长期行为的影响不仅与系统的高阶子空间中的固有频率和阻尼比乘积的最小值有关,并且与高阶子空间中的某一最大固有频率有关．然后,将一般的模态截取视为对原系统的解的一个扰动,对一些文献中由于降维程度的不同而造成解的拓扑性质发生变化的现象进行了定性的解释．     

Spatial Localization for Nonlinear Dynamical Stochastic Models for Excitable Media

Nonlinear dynamical stochastic models are ubiquitous in different areas. Their statistical properties are often of great interest, but are also very challenging to compute. Many excitable media models belong to such types of complex systems with large state dimensions and the associated covariance matrices have localized structures. In this article, a mathematical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous \linebreak mathematical analysis shows that the local effect from the diffusion results in an exponential decay of the components in the covariance matrix as a function of the distance while the global effect due to the mean field interaction synchronizes different components and contributes to a global covariance. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.     

Minimum-order observers for discrete-time systems

We consider the synthesis of a minimum-order state or functional observer for a linear dynamical system. The synthesis problem is solved for completely certain systems of general form and for some classes of uncertain systems. Various approaches are described, which ultimately lead to the same task: finding a minimum-dimension Hurwitz solution for a system of linear equations with a Hankel matrix. For scalar and vector linear systems, prior upper and lower bounds on the observer dimension are derived, which makes it possible to switch to an iterative procedure of finding an optimal solution. The discussion is set out for discrete-time dynamical systems.     

A combination between the reduced basis method and the ANOVA expansion: On the computation of sensitivity indices

We consider a method to efficiently evaluate in a real-time context an output based on the numerical solution of a partial differential equation depending on a large number of parameters. We state a result allowing to improve the computational performance of a three-step RB–ANOVA–RB method. This is a combination of the reduced basis (RB) method and the analysis of variations (ANOVA) expansion, aiming at compressing the parameter space without affecting the accuracy of the output. The idea of this method is to compute a first (coarse) RB approximation of the output of interest involving all the parameter components, but with a large tolerance on the a posteriori error estimate; then, we evaluate the ANOVA expansion of the output and freeze the least important parameter components; finally, considering a restricted model involving just the retained parameter components, we compute a second (fine) RB approximation with a smaller tolerance on the a posteriori error estimate. The fine RB approximation entails lower computational costs than the coarse one, because of the reduction of parameter dimensionality. Our result provides a criterion to avoid the computation of those terms in the ANOVA expansion that are related to the interaction between parameters in the bilinear form, thus making the RB–ANOVA–RB procedure computationally more feasible.     

Efficient solution of large scale Lyapunov and Riccati equations arising in model order reduction problems

Model order reduction of large–scale linear time–invariant systems is an omnipresent task in control and simulation of complex dynamical processes. The solution of large scale Lyapunov and Riccati equations is a major task, e.g., in balanced truncation and related model order reduction methods, in particular when applied to semi–discretized partial differential equations constraint control problems. The software package LyaPack has shown to be a valuable tool in the task of solving these equations since its introduction in 2000. Here we want to discuss recent improvements and extensions of the underlying algorithms and their implementation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model order reduction methods and their possible application in compliant mechanisms

Model order reduction (MOR) is commonly used to approximate large-scale linear time-invariant dynamical systems. A new feed unit based on a compliant mechanism consisting of flexure hinges can be described by a discrete system of n ordinary differential equations. A projection framework using modal and Krylov subspace techniques is applied to reduce the order of the system to lower computational cost and make the model feasible for control, analysis and optimization. Single flexure hinges are investigated numerical, analytical and experimental and compared to reduced models via modal and tangential Krylov subspace methods regarding the first eigenfrequency. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system.Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘reducibility conditions’. If these ‘reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover ‘resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.     

Evaluation of the growth rate of the state vector in a second-order generalized linear stochastic system

A second-order generalized linear stochastic dynamical system is considered. The entries of the system matrix are assumed to be independent and exponentially distributed. Evaluation of the growth rate of the system state vector is reduced to algebraic computations which involve solving an algebraic linear system and evaluating a linear functional for the solution.     

Reduction of overestimation in interval arithmetic simulation of biological wastewater treatment processes

A novel interval arithmetic simulation approach is introduced in order to evaluate the performance of biological wastewater treatment processes. Such processes are typically modeled as dynamical systems where the reaction kinetics appears as additive nonlinearity in state. In the calculation of guaranteed bounds of state variables uncertain parameters and uncertain initial conditions are considered. The recursive evaluation of such systems of nonlinear state equations yields overestimation of the state variables that is accumulating over the simulation time. To cope with this wrapping effect, innovative splitting and merging criteria based on a recursive uncertain linear transformation of the state variables are discussed. Additionally, re-approximation strategies for regions in the state space calculated by interval arithmetic techniques using disjoint subintervals improve the simulation quality significantly if these regions are described by several overlapping subintervals. This simulation approach is used to find a practical compromise between computational effort and simulation quality. It is pointed out how these splitting and merging algorithms can be combined with other methods that aim at the reduction of overestimation by applying consistency techniques. Simulation results are presented for a simplified reduced-order model of the reduction of organic matter in the activated sludge process of biological wastewater treatment.     

A multi-product continuous review inventory system with stochastic demand,backorders, and a budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary or storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when orders arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. This paper models a multi-item stochastic inventory system with backordered shortages when estimation of marginal backorder cost is available, and payment is due upon order arrival. The budget constraint can easily be converted into a storage constraint.     

Transition Manifolds of Complex Metastable Systems

We consider complex dynamical systems showing metastable behavior, but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics.     

An efficient dimension-adaptive uncertainty propagation approach

In recent years, there has been a growing interest in uncertainty propagation, and a wide variety of uncertainty propagation methods exist in literature. In this paper, an uncertainty propagation approach is developed by using high-dimensional model representation (HDMR) and dimension reduction (DR) method technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower-order to higher-order component functions. To save the computational cost, a dimension-adaptive version of the additive decomposition is proposed to detect the important component functions to reduce the terms. The proposed method requires neither the calculation of partial derivatives of response, as in commonly used Taylor expansion/perturbation methods, nor the inversion of random matrices, as in the Neumann expansion method. Two numerical examples show the efficiency and accuracy of the proposed method.