Model Reduction Approach Research Articles

A Multilinear HJB-POD Method for the Optimal Control of PDEs on a Tree Structure

Optimal control problems driven by evolutionary partial differential equations arise in many industrial applications and their numerical solution is known to be a challenging problem. One approach to obtain an optimal feedback control is via the Dynamic Programming principle. Nevertheless, despite many theoretical results, this method has been applied only to very special cases since it suffers from the curse of dimensionality. Our goal is to mitigate this crucial obstruction developing a version of dynamic programming algorithms based on a tree structure and exploiting the compact representation of the dynamical systems based on tensors notations via a model reduction approach. Here, we want to show how this algorithm can be constructed for general nonlinear control problems and to illustrate its performances on a number of challenging numerical tests introducing novel pruning strategies that improve the efficacy of the method. Our numerical results indicate a large decrease in memory requirements, as well as computational time, for the proposed problems. Moreover, we prove the convergence of the algorithm and give some hints on its implementation.

Memoryless Dual-Observer-Based Output Feedback Stabilization of Linear Systems With Input and Output Delays.

This article focuses on the dual-observer-based stabilization of linear systems with delays in both the inputs and outputs. By a model reduction approach the system can be converted to an equivalent linear system without delays, for which a dual-observer-based stabilizing controller can be designed. However, such a controller is infinite dimensional or memory-based. To solve such a problem, a modified memoryless dual-observer-based stabilizing controller is designed, and the closed-loop stability is proven under some additional conditions. Compared with the reduced-order observer-based controller, the dimension of the dual-observer-based controller is smaller if the system has more inputs than outputs. At the same time, the design approach is more challenging in proving closed-loop stability, as for example a more intricate Lyapunov-Krasovskii functional has to be constructed associated with the proposed approach. The proposed approach is applicable to both continuous-time and discrete-time systems. Numerical simulations validate the effectiveness of the proposed method.

An Approximate Bayesian Computation Approach for Embryonic Astrocyte Migration Model Reduction.

During embryonic development of the retina of the eye, astrocytes, a type of glial cell, migrate over the retinal surface and form a dynamic mesh. This mesh then serves as scaffolding for blood vessels to form the retinal vasculature network that supplies oxygen and nutrients to the inner portion of the retina. Astrocyte spreading proceeds in a radially symmetric manner over the retinal surface. Additionally, astrocytes mature from astrocyte precursor cells (APCs) to immature perinatal astrocytes (IPAs) during this embryonic stage. We extend a previously-developed continuum model that describes tension-driven migration and oxygen and growth factor influenced proliferation and differentiation. Comparing numerical simulations to experimental data, we identify model equation components that can be removed via model reduction using approximate Bayesian computation (ABC). Our results verify experimental studies indicating that the choroid oxygen supply plays a negligible role in promoting differentiation of APCs into IPAs and in promoting IPA proliferation, and the hyaloid artery oxygen supply and APC apoptosis play negligible roles in astrocyte spreading and differentiation.

In silico analysis reveals the prospects of renal anisotropy in improving chronic kidney disease detection using ultrasound shear wave elastography.

Renal anisotropy is a complex property of the kidney and often poses a challenge in obtaining consistent measurements when using shear wave elastography to detect chronic kidney disease. To circumvent the challenge posed by renal anisotropy in clinical settings, a dimensionless biomarker termed the 'anisotropic ratio' was introduced to establish a correlation between changes in degree of renal anisotropy and progression of chronic kidney disease through an in silico perspective. To achieve this, an efficient model reduction approach was developed to model the anisotropic property of kidneys. Good agreement between the numerical and experimental data were obtained, as percentage errors of less than 5.5% were reported when compared against experimental phantom measurement from the literature. To demonstrate the applicability of the model to clinical measurements, the anisotropic ratio of sheep kidneys was quantified, with both numerical and derived experimental results reporting a value of .667. Analysis of the anisotropic ratio with progression of chronic kidney disease demonstrated that patients with normal kidneys would have a lower anisotropic ratio of .872 as opposed to patients suffering from renal impairment, in which the anisotropic ratio may increase to .904, as determined from this study. The findings demonstrate the potential of the anisotropic ratio in improving the detection of chronic kidney disease using shear wave elastography.

Reduced-order modeling for time domain analysis of finite periodic structures with absorbing boundary conditions

A reduced-order model of finite periodic structures with absorbing boundary conditions (ABCs) and localized time-dependent excitations is proposed. 1D-periodic structures whose cells can represent any 2D or 3D arbitrary substructures are considered. The key steps for modeling an ABC in the time domain are: (i) expression of the impedance matrix with the wave finite element (WFE) method; (ii) partial rational decomposition of the impedance matrix; (iii) introduction of vectors of supplementary variables to describe the ABC in the time domain. In this paper, a model reduction approach is proposed to speed up the computation of the ABCs. The focus is on the reduction of the number of internal degrees of freedom of substructures, and the reduction of the number of degrees of freedom at the substructure interfaces. The approach allows the consideration of complex substructures with large FE models, and the computation of the time response of periodic structures at a low computational cost. Numerical experiments are carried out to highlight the relevance of the proposed approach. Specifically, the analysis of a metamaterial structure containing nonlinear resonant substructures is carried out with the proposed approach. Results show that, when compared to the linear case, band gaps in periodic structures with nonlinear substructures can be significantly improved.

Control strategies and performance analysis of doubly approximation and large-scale system control using hybrid model reduction approach

Control strategies and performance analysis of doubly approximation and large-scale system control using hybrid model reduction approach

Arsenic Adsorption Isotherms of Sediments from Old Mining Area and Their Implications: A Case Study of the Ron Phibun Area, Thailand

Although tin mining activities in the southern part of Thailand have been abandoned for more than 40 years, arsenic contamination in old tin-mining areas continues to pose a health risk. Fortunately, natural adsorption of arsenic by sediment has reduced the arsenic concentration in water to the maximum allowable level within a 10-kilometre radius of the mine sources. This article attempts to characterize the arsenic adsorption properties of sediment along a water creek which passes through the old tin-mine areas in Ron Phibun district, Nakhon Si Thammarat province, Thailand, where the contamination has been most severe. It was found that Fe tends to be associated with the adsorption capacity of arsenic in sediment. However, pH 4 and pH 7 did not associate with adsorption capacity. At pH 12, the solubility of arsenic increases, making it less likely to be adsorbed into the sediment and more likely to dissolve in water. For the adsorption isotherms of the sediment, the so-called ‘2-step Langmuir/Freundlich models’, with the advance of switching functions (Logistic or Arctan functions), were proposed. Then, based on the principle of parsimony, a stepwise model reduction approach was used to choose the optimal models. It was found that the isotherms were used successfully to describe the historical arsenic-contamination data of the past 14 years (2005 - 2019) and to predict future trends qualitatively. HIGHLIGHTS Two-step Langmuir/Freundlich models with the switching functions were proposed The historical arsenic-contamination data of the past 14 years (2005 - 2019) was examined The model described the historical arsenic-contamination data as a function of pH and metal-complex formation in sediment soils successfully GRAPHICAL ABSTRACT

Another explicit forms of dynamic substructuring approaches using dynamic constraints at joint nodes

ABSTRACT The dynamic substructuring design describes the dynamic responses in the modal, frequency and physical domains using coupling methods at joint nodes. This paper presents dynamic structuring methods to modify existing substructuring methods by merging the concept of interfacial forces in the satisfaction of compatibility conditions at joint nodes. The algorithms correspond to a type of model reduction approaches that use a few modes of substructures with the assumption of pseudo-masses at the joint nodes of the substructures. It was verified through numerical examples that the pseudo-masses rarely affected the dynamic responses of the synthesized structure. A frequency response function-based substructuring method was derived by applying interface forces at joint nodes to frequency response function (FRF). The proposed dynamic substructuring methods are expressed in explicit forms for synthesizing substructures without any numerical schemes. The validity of the proposed method was illustrated using two numerical examples. The limitations of this study are evaluated using numerical examples.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

A model reduction approach for discrete-time linear time-variant systems with delayed inputs

A model reduction approach for discrete-time linear time-variant systems with delayed inputs

Data-driven model reduction approach for active vibration control of cable-strut structures

The cable-strut structures have the features of low stiffness and weak damping, resulting in large vibration response under dynamic excitation. The vibration response of the cable-strut structures can be reduced by active control methods. However, many traditional active control methods need to build a state-space model (SSM) based on finite element model (FEM), and solving the active control force for the high-dimensional SSM is time-consuming. In this paper, the dynamic mode decomposition with control (DMDc) method is proposed to achieve a data-driven reduced-order SSM for active vibration control of cable-strut structures. This proposed method only uses the data of structural response and control force input to build reduced-order SSM and does not rely on the FEM. Linear quadratic regulator (LQR) design using the DMDc-based reduced-order SSM (DMDc-LQR) is more computationally efficient than that using the FEM-based SSM (FEM-LQR). The relationship between the design parameters of the DMDc-LQR controller and the FEM-LQR controller is established, which ensures that the control performance indexes of the DMDc-LQR controller and the FEM-LQR controller are equal. The effectiveness of the proposed DMDc-LQR method is validated using numerical Kiewitt cable dome and tensegrity grid under wind and seismic loads. The results show that the DMDc-LQR controller maintains similar control effect to that of the FEM-LQR controller and has advantage of high computing efficiency as well. In addition, the influence of critical parameters (i.e., data noise and model order) on vibration control effect is comprehensively explored. It is found that increasing the model order is effective to mitigate the noise influence on the DMDc-based reduced-order SSM and ensures the effectiveness of the DMDc-LQR controller.

Computationally Efficient Reduced Order Modeling of DFIG-Based Wind Turbines: A Novel Frequency-Weighted and Limited Model Reduction Approach With Error Bounds

Computationally Efficient Reduced Order Modeling of DFIG-Based Wind Turbines: A Novel Frequency-Weighted and Limited Model Reduction Approach With Error Bounds

Midpoint-Based Balanced Truncation for Switched Linear Systems With Known Switching Signal

We propose a novel model reduction approach for switched linear systems with known switching signal. The class of considered systems encompasses switched systems with mode-dependent state-dimension as well as impulsive systems. Our method is based on a suitable definition of (time-varying) reachability and observability Gramians and we show that these Gramians satisfy precise interpretations in terms of input and output energy. Based on balancing the midpoint Gramians, we propose a piecewise-constant projection based model reduction resulting in a switched linear system of smaller size.

Novel Gramian-Based Structure-Preserving Model Order Reduction for Power Systems With High Penetration of Power Converters

Renewable energy and high voltage direct current connections increase the number of power electronic devices in the grid, changing how power systems operate and introducing new dynamic behavior. For this reason, it is fundamental to understand what modeling details are needed to represent the system in this new context. This paper proposes a novel model reduction approach using frequency-limited controllability and observability Gramians that preserves the link between the physical structure of the system and the state variables. This method allows understanding and analytically determining how vital each state is to the input-output behavior in different frequency ranges. The primary contribution of this paper is an algorithm that can provide intuitive results to help power system experts understand the simulation tools and minimum modeling details needed to perform different stability studies. Two case studies supported the applicability of the developed method, and a tool for MATLAB/Simulink was developed based on the findings, allowing visualization and immediate identification of the model components that are most relevant for the system dynamics under study. This tool can analyze dynamic behaviors where the cause is poorly understood and provide clarity around modeling detail requirements in converter-rich power systems.

Finite Element–Boundary Element Acoustic Backscattering with Model Reduction of Surface Pressure Based on Coherent Clusters

Computing backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shape is a problem of considerable practical significance. The finite element method is commonly applied to the discretization of the target; on the other hand, the boundary element method naturally incorporates the radiation boundary condition at infinity. The coupled model tends to be expensive, primarily due to the need to manipulate large, dense, and complex matrices and to repeatedly solve systems of complex linear algebraic equations of significant size for each frequency of interest. In this article, we develop a model reduction transformation based on the notion of coherence applied to the surface pressures, which considerably reduces the size of the systems to be solved. We found that the proposed model reduction approach delivers acceptably accurate results at a fraction of the cost of the full model. A typical speedup of an order of magnitude was realized in our numerical experiments. Our approach enables backscattering computations with considerably larger models than have been feasible to date.

Incorporating the complex rheological behavior of blood in microvascular network simulations: Two-phase modeling and a model reduction approach

Vessel flow resistance in microvascular network simulations is often computed using the single-phase approximation, where the no-slip boundary condition is imposed on the microvessel wall, and the "in vitro viscosity law" or "the relative hematocrit" relationships proposed by Pries et al. (1992) are utilized. However, these models have limitations due to their reliance on experimental data, which inherently contain measurement errors, thereby affecting their consistency and accuracy. To moderate these limitations, biphasic flow models can be employed, allowing for adaptable rheological descriptions of blood. Nevertheless, this approach presents challenges in terms of automation and computational cost. In order to address these challenges, we propose a monophasic steady-state model that consistently incorporates a wall-slip condition in place of the cell-free layer contribution. Model reduction is achieved by establishing an explicit relationship for the apparent slip length, which is calibrated through the regression of biphasic model results. Subsequently, simulations are conducted on microvessel network structures. The predictions of wall shear stress by the monophasic model exhibit deviations of less than 2 % from the expected values, within the range of vessel radii (10–75 μm) and discharge hematocrits (0.2–0.7). Furthermore, the computational time required is reduced by almost an order of magnitude compared to the biphasic model, rendering it suitable for multiscale biochemical simulations. The developed model closely approximates the flow resistance predictions of the biphasic model, while its modular nature allows for easy implementation of future modifications. Both models successfully reproduce experimental data by Lipowsky et al. (1978). Finally, we derive semi-analytical expressions for the apparent slip length and discuss why it is more appropriate to impose only the "relative hematocrit" relationship as a global constraint for calculating the hemodynamics in microvessels, rather than relying on the "in vitro viscosity law", which provides unreliable predictions at low values of the discharge hematocrit.

From Data to Reduced-Order Models via Generalized Balanced Truncation

This paper proposes a data-driven model reduction approach on the basis of noisy data with a known noise model. Firstly, the concept of data reduction is introduced. In particular, we show that the set of reduced-order models obtained by applying a Petrov-Galerkin projection to all systems explaining the data characterized in a large-dimensional quadratic matrix inequality (QMI) can again be characterized in a lower-dimensional QMI. Next, we develop a data-driven generalized balanced truncation method that relies on two steps. First, we provide necessary and sufficient conditions such that systems explaining the data have common generalized Gramians. Second, these common generalized Gramians are used to construct matrices that allow to characterize a class of reduced-order models via generalized balanced truncation in terms of a lower-dimensional QMI by applying the data reduction concept. Additionally, we present alternative procedures to compute a priori and a posteriori upper bounds with respect to the true system generating the data. Finally, the proposed techniques are illustrated by means of application to an example of a system of a cart with a double-pendulum.

Compact modelling of a thermo-mechanical finite element model of a microelectronic package

Thermo-mechanical behaviour of microelectronic components, when subjected to thermal cycling, plays a vital role in reliability assessments. This motivates the use of accurate numerical simulation tools. However, numerical simulations are time-consuming and thus, model order reduction methodologies are required. Most literature so far, focuses on single-domain model reduction approaches, that is, the reduction of only thermal or structural domain models. This paper extends existing work to order reduction of coupled linearized thermo-mechanical finite element models. Three thermo-mechanical reduction approaches are studied within the framework of the Krylov-basis reduction technique. This enables the examination of the most efficient approach to reduce the one-way coupled problem. The reduced order model is verified through the comparison with the full order model. The results show that the reduced model significantly decreases the computational costs while providing highly accurate results for temperature and deformation. This paper also emphasizes the potential of model order reduction in enabling system-level simulation of microelectronic components.

Efficient and accurate nonlinear model reduction via first-order empirical interpolation

We present a model reduction approach that extends the original empirical interpolation method to enable accurate and efficient reduced basis approximation of parametrized nonlinear partial differential equations (PDEs). In the presence of nonlinearity, the Galerkin reduced basis approximation remains computationally expensive due to the high complexity of evaluating the nonlinear terms, which depends on the dimension of the truth approximation. The empirical interpolation method (EIM) was proposed as a nonlinear model reduction technique to render the complexity of evaluating the nonlinear terms independent of the dimension of the truth approximation. We introduce a first-order empirical interpolation method (FOEIM) that makes use of the partial derivative information to construct an inexpensive and stable interpolation of the nonlinear terms. We propose two different FOEIM algorithms to generate interpolation points and basis functions. We apply the FOEIM to nonlinear elliptic PDEs and compare it to the Galerkin reduced basis approximation and the EIM. Numerical results are presented to demonstrate the performance of the three reduced basis approaches.

Non-local multi-continuum method (NLMC) for Darcy–Forchheimer flow in fractured media

This work presents the application of the non-local multicontinuum method (NLMC) for the Darcy–Forchheimer model in fractured media. The mathematical model describes a nonlinear flow in fractured porous media with a high inertial effect and flow speed. The space approximation is constructed on the sufficiently fine grid using a finite volume method (FVM) with an embedded fracture model (EFM) to approximate lower dimensional fractures. A non-local model reduction approach is presented based on localization and constraint energy minimization. The multiscale basis functions are constructed in oversampled local domains to consider the flow effects from neighboring local domains. Numerical results are presented for a two-dimensional formulation with two test cases of heterogeneity. The influence of model nonlinearity on the multiscale method accuracy is investigated. The numerical results show that the non-local multicontinuum method provides highly accurate results for Darcy–Forchheimer flow in fractured media.