Model Reduction Approach Research Articles

Integrated Stent Models Based on Dimension Reduction: Review and Future Perspectives.

Stent modeling represents a challenging task from both the theoretical and numerical viewpoints, due to its multi-physics nature and to the complex geometrical configuration of these devices. In this light, dimensional model reduction enables a comprehensive geometrical and physical description of stenting at affordable computational costs. In this work, we aim at reviewing dimensional model reduction of stent mechanics and drug release. Firstly, we address model reduction techniques for the description of stent mechanics, aiming to illustrate how a three-dimensional stent model can be transformed into a collection of interconnected one-dimensional rods, called a "stent net". Secondly, we review available model reduction methods similarly applied to drug release, in which the "stent net" concept is adopted for modeling of drug elution. As a result, drug eluting stents are described as a distribution of concentrated drug release sources located on a graph that fully represents the stent geometry. Lastly, new results about the extension of these model reduction approaches to biodegradable stents are also discussed.

A model reduction approach for hyperelastic materials based on Proper Orthogonal Decomposition

AbstractIn this contribution, we present a model reduction approach for hyperelastic materials based on Proper Orthogonal Decomposition. A separation of the solution into time and space dependent functions is performed, which allows to compute reduced solutions within real time. The functionality of the method is demonstrated by a tensile test and is further applied to a stochastic analysis and a multi‐parametric analysis involving non‐linear material behaviour and large deformations. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Model-Based Individualized Treatment of Chemotherapeutics: Bayesian Population Modeling and Dose Optimization.

6-Mercaptopurine (6-MP) is one of the key drugs in the treatment of many pediatric cancers, auto immune diseases and inflammatory bowel disease. 6-MP is a prodrug, converted to an active metabolite 6-thioguanine nucleotide (6-TGN) through enzymatic reaction involving thiopurine methyltransferase (TPMT). Pharmacogenomic variation observed in the TPMT enzyme produces a significant variation in drug response among the patient population. Despite 6-MP’s widespread use and observed variation in treatment response, efforts at quantitative optimization of dose regimens for individual patients are limited. In addition, research efforts devoted on pharmacogenomics to predict clinical responses are proving far from ideal. In this work, we present a Bayesian population modeling approach to develop a pharmacological model for 6-MP metabolism in humans. In the face of scarcity of data in clinical settings, a global sensitivity analysis based model reduction approach is used to minimize the parameter space. For accurate estimation of sensitive parameters, robust optimal experimental design based on D-optimality criteria was exploited. With the patient-specific model, a model predictive control algorithm is used to optimize the dose scheduling with the objective of maintaining the 6-TGN concentration within its therapeutic window. More importantly, for the first time, we show how the incorporation of information from different levels of biological chain-of response (i.e. gene expression-enzyme phenotype-drug phenotype) plays a critical role in determining the uncertainty in predicting therapeutic target. The model and the control approach can be utilized in the clinical setting to individualize 6-MP dosing based on the patient’s ability to metabolize the drug instead of the traditional standard-dose-for-all approach.

Reduced models for sparse grid discretizations of the multi-asset Black-Scholes equation

This work presents reduced models for pricing basket options with the Black-Scholes and the Heston model. Basket options lead to multi-dimensional partial differential equations (PDEs) that quickly become computationally infeasible to discretize on full tensor grids. We therefore rely on sparse grid discretizations of the PDEs, which allow us to cope with the curse of dimensionality to some extent. We then derive reduced models with proper orthogonal decomposition. Our numerical results with the Black-Scholes model show that sufficiently accurate results are achieved while gaining speedups between 80 and 160 compared to the high-fidelity sparse grid model for 2-, 3-, and 4-asset options. For the Heston model, results are presented for a single-asset option that leads to a two-dimensional pricing problem, where we achieve significant speedups with our model reduction approach based on high-fidelity sparse grid models.

Proper generalized decomposition for parameterized Helmholtz problems in heterogeneous and unbounded domains: Application to harbor agitation

Abstract Solving the Helmholtz equation for a large number of input data in an heterogeneous media and unbounded domain still represents a challenge. This is due to the particular nature of the Helmholtz operator and the sensibility of the solution to small variations of the data. Here a reduced order model is used to determine the scattered solution everywhere in the domain for any incoming wave direction and frequency. Moreover, this is applied to a real engineering problem: water agitation inside real harbors for low to mid-high frequencies. The proper generalized decomposition (PGD) model reduction approach is used to obtain a separable representation of the solution at any point and for any incoming wave direction and frequency. Here, its applicability to such a problem is discussed and demonstrated. More precisely, the contributions of the paper include the PGD implementation into a perfectly matched layer framework to model the unbounded domain, and the separability of the operator which is addressed here using an efficient higher-order projection scheme. Then, the performance of the PGD in this framework is discussed and improved using the higher-order projection and a Petrov–Galerkin approach to construct the separated basis. Moreover, the efficiency of the higher-order projection scheme is demonstrated and compared with the higher-order singular value decomposition.

A non‐intrusive model reduction approach for polynomial chaos expansion using proper orthogonal decomposition

SummaryIn this paper, a non‐intrusive stochastic model reduction scheme is developed for polynomial chaos representation using proper orthogonal decomposition. The main idea is to extract the optimal orthogonal basis via inexpensive calculations on a coarse mesh and then use them for the fine‐scale analysis. To validate the developed reduced‐order model, the method is implemented to: (1) the stochastic steady‐state heat diffusion in a square slab; (2) the incompressible, two‐dimensional laminar boundary‐layer over a flat plate with uncertainties in free‐stream velocity and physical properties; and (3) the highly nonlinear Ackley function with uncertain coefficients. For the heat diffusion problem, the thermal conductivity of the slab is assumed to be a stochastic field with known exponential covariance function and approximated via the Karhunen–Loève expansion. In all three test cases, the input random parameters are assumed to be uniformly distributed, and a polynomial chaos expansion is found using the regression method. The Sobol's quasi‐random sequence is used to generate the sample points. The numerical results of the three test cases show that the non‐intrusive model reduction scheme is able to produce satisfactory results for the statistical quantities of interest. It is found that the developed non‐intrusive model reduction scheme is computationally more efficient than the classical polynomial chaos expansion for uncertainty quantification of stochastic problems. The performance of the developed scheme becomes more apparent for the problems with larger stochastic dimensions and those requiring higher polynomial order for the stochastic discretization. Copyright © 2015 John Wiley & Sons, Ltd.

A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

We propose and test the first Reduced Radial Basis Function Method for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.

Model reduction for delay differential equations with guaranteed stability and error bound

In this paper, a structure-preserving model reduction approach for a class of delay differential equations is proposed. Benefits of this approach are, firstly, the fact that the delay nature of the system is preserved after reduction, secondly, that input–output stability properties are preserved and, thirdly, that a computable error bound reflecting the accuracy of the reduction is provided. These results are applicable to large-scale linear delay differential equations with constant delays, but also extensions to a class of nonlinear delay differential equations with time-varying delays are presented. The effectiveness of the results is evidenced by means of an illustrative example.

Forecasting inflation and GDP growth using heuristic optimisation of information criteria and variable reduction methods

Forecasting macroeconomic variables using many predictors is considered. Model selection and model reduction approaches are compared. Model selection includes heuristic optimisation of information criteria using: simulated annealing, genetic algorithms, MC3 and sequential testing. Model reduction employs the methods of principal components, partial least squares and Bayesian shrinkage regression. The problem of unbalanced datasets is discussed and potential solutions are suggested. An out-of-sample forecasting exercise provides evidence that these methods are useful in predicting the growth rates of quarterly GDP and monthly inflation.

Special Issue on Model Reduction

Model reduction, in the sense of the approximation of a numerical solution into a subspace of much smaller dimension than that associated with usual semi-discretization or discretization methods, is rapidly becoming an indispensable tool for many areas of computational sciences and engineering. These include, to name only a few, computational-based design and optimization, statistical analysis, optimal control, and embedded computing. It is also essential for all scenarios where real-time simulation responses are desired. Linear model reduction has deep roots in many engineering fields such as structural dynamics and electronic circuit analysis. For applications with a single or no parameter, it has matured to the point where it is embraced today by many practitioners. By comparison, parametric and nonlinear model reduction methods are still in their infancy. Nevertheless, giant strides have been recently made in these two areas. Consequently, these two topics have gained importance for many applications where high-dimensional, nonlinear, parametric simulations remain computationally prohibitive. For this reason, the main objective of this Special Issue is to disseminate some of the recent advances made in parametric and nonlinear model reduction methods and highlight their potential for making a strong impact on the broad field of computational engineering. To this effect, this issue contains 13 papers 1-13 covering several important aspects of the mathematical theory, numerical algorithms, and computational strategies underlying parametric and nonlinear model reduction approaches. Nine of these papers focus on the parametric aspect of the problem in the linear case. They emphasize two main strategies for constructing a globally accurate parametric reduced-order model (ROM): the offline/online strategy where a ROM is built during a computationally intensive offline phase and exploited in real-time during an online phase and an alternative strategy where a parametric ROM is iteratively constructed, exploited, and updated as needed. The four other papers address more specifically the topic of nonlinear model reduction using the notion of a hierarchy of approximations. Three papers focus also on data assimilation and system monitoring—a rapidly growing field where model reduction is essential—three on numerical stability, and three on accuracy analysis and error estimators. In a field where advances can often give, rightfully or wrongfully, the impression of ‘more of the same’, the editors hope that this Special Issue will bring to the readers a fresh perspective on an important direction where computational engineering should go to continue shining in the future.

Sparse Generalized Multiscale Finite Element Methods and their applications

In a number of previous papers, local (coarse grid) multiscale model reduction techniques are developed using a Generalized Multiscale Finite Element Method. In these approaches, multiscale basis functions are constructed using local snapshot spaces, where a snapshot space is a large space that represents the solution behavior in a coarse block. In a number of applications (e.g., those discussed in the paper), one may have a sparsity in the snapshot space for an appropriate choice of a snapshot space. More precisely, the solution may only involve a portion of the snapshot space. In this case, one can use sparsity techniques to identify multiscale basis functions. In this paper, we consider two such sparse local multiscale model reduction approaches. In the first approach (which is used for parameter-dependent multiscale PDEs), we use local minimization techniques, such as sparse POD, to identify multiscale basis functions, which are sparse in the snapshot space. These minimization techniques use l1 minimization to find local multiscale basis functions, which are further used for finding the solution. In the second approach (which is used for the Helmholtz equation), we directly apply l1 minimization techniques to solve the underlying PDEs. This approach is more expensive as it involves a large snapshot space; however, in this example, we cannot identify a local minimization principle, such as local generalized SVD. All our numerical results assume the sparsity and we discuss this assumption for the snapshot spaces. Moreover, we discuss the computational savings provided by our approach. The sparse solution allows a fast evaluation of stiffness matrices and downscaling the solution to the fine grid since the reduced dimensional solution representation is sparse in terms of local snapshot vectors. Numerical results are presented, which show the convergence of the proposed method and the sparsity of the solution.

A System Level Model Reduction Approach for Flexible Multibody Systems with Parametric Uncertainties.

Stochastic analysis of flexible multibody system for uncertain parameters typically requires a large number of simulation runs for example for Monte-Carlo simulation. However, as the computational load of a regular flexible multibody model is typically rather high, this is often infeasible. A solution to this high computational load is model reduction, but regular model reduction approaches for flexible multibody simulation do not maintain the parameter dependency. This leads to a new model reduction for each parameter which also leads to high computational costs. The current work presents a novel system level model reduction technique for parameterized flexible multibody simulation. The proposed approach is a parameterized version of the Global Modal Parameterization method. In this approach a system level model reduction of the flexible mechanism is performed in which a configuration dependent projection space is used. For the parameterized approach, affine parameter dependence is assumed. In this case the parameter dependency can be externalized and is exactly preserved through the model reduction. The accuracy of the proposed approach is demonstrated through a numerical validation. The model is used for a Monte-Carlo simulation of mechanism with uncertain parameters and delivers accurate probabilistic distributions for the motion of the mechanisms at a highly reduced cost compared to the original model. The proposed approach is shown to provide reliable results with a computational load which is reduced from days to hours.

The Unifying Feature of Projection in Model Order Reduction

Abstract This paper considers the problem of model order reduction of linear systems with the emphasis on the common features of the main approaches. One of these features is the unifying role of operator projection in model reduction. It is shown how projections are implemented for different methods of model reduction and what their properties are. The other common feature is the subspaces where projections are defined. The main approaches for model reduction which are considered in the paper are balanced truncation, proper orthogonal decomposition and the Lanczos procedure from the Krylov subspace methods. It is shown that the range spaces of system gramians for balanced truncation and the range space of the reachability and observability matrices for the Lanczos procedure coincide. The connection between balanced truncation and the proper orthogonal decomposition method is also established. Therefore, the methods for model reduction are similar in terms of general operational principles, and differ mostly in their technical implementation. Several numerical examples are considered showing the validity of the proposed conjectures.

Global–local nonlinear model reduction for flows in heterogeneous porous media

Abstract In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equations, we employ the GMsFEM to represent the solution on a coarse grid with multiscale basis functions and apply proper orthogonal decomposition on a coarse grid. Computing the GMsFEM solution involves calculating the residual and the Jacobian on a fine grid. As such, we use local and global empirical interpolation concepts to circumvent performing these computations on the fine grid. The resulting reduced-order approach significantly reduces the flow problem size while accurately capturing the behavior of fully-resolved solutions. We consider several numerical examples of nonlinear multiscale partial differential equations that are numerically integrated using fully-implicit time marching schemes to demonstrate the capability of the proposed model reduction approach to speed up simulations of nonlinear flows in high-contrast porous media.

A model reduction approach to numerical inversion for a parabolic partial differential equation

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss–Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.

Local Stabilization of Nonlinear Systems Through the Reduction Model Approach

We study a general class of nonlinear systems with input delays of arbitrary size. We adapt the reduction model approach to prove local asymptotic stability of the closed loop input delayed systems, using feedbacks that may be nonlinear. Our Lyapunov-Krasovskii functionals make it possible to determine estimates of the basins of attraction for the closed loop systems.

Coprime factor model reduction for discrete-time uncertain systems

This paper presents a contractive coprime factor model reduction approach for discrete-time uncertain systems of LFT form with norm bounded structured uncertainty. A systematic approach is proposed for coprime factorization and contractive coprime factorization of the underlying uncertain systems. The proposed coprime factor approach overcomes the robust stability restriction on the underlying systems which is required in the balanced truncation approach. Our method is based on the use of LMIs to construct the desired reduced dimension uncertain system model. Closed-loop robustness is discussed under additive coprime factor perturbations.

SY28-4 * CLINICAL APPLICATION OF MOTIVATIONAL INTERVIEWING AT OKAYAMA PSYCHIATRIC MEDICAL CENTER

Although research has proven motivational interviewing (MI) to be effective in addiction treatment, MI is not yet wildly used in Japan. When I started the treatment of addiction in 2006, MI was not used in our psychiatric medical center. I thought MI would solve the problems I had had in clinical interviewing with people with addiction. I studied the methods of MI during seminars and workshops including one by William R. Miller. In our psychiatric medical center, over the past 4 years, my colleagues and I have regularly held meetings where we role play therapist/client aspects of MI once a month, as well as workshops by Japanese MINT (the Motivational Interviewing Network of Trainers) members once a year. MI can be used in outpatient and inpatient clinical settings, as well as group therapy. I think that MI is essential in treatment, not only for substance use disorders, but also process addiction–in which we currently do not have many programs dealing with this type of disorder. In addition, MI is very compatible with the trans-theoretical model and harm reduction approach. In this presentation, I will introduce a case of an alcohol dependent patient who, at this time, had not achieved abstinence from alcohol, but through positive engagement has been moving forward in treatment.

Power Conserving Bond Graph Based Modal Representations and Model Reduction of Lumped Parameter Systems

Dynamic analysis is extensively used to study the behavior of continuous and lumped parameter linear systems. In addition to the physical space, analyses can also be performed in the modal space where very useful frequency information of the system can be extracted. More specifically, modal analysis can be used for the analysis and controller design of dynamic systems, where reduction of model complexity without degrading its accuracy is often required. The reduction of modal models has been extensively studied and many reduction techniques are available. The majority of these techniques use frequency as the metric to determine the reduced model. This paper describes a new method for calculating modal decompositions of lumped parameter systems with the use of the bond graph formulation. The modal decomposition is developed through a power conserving coordinate transformation. The generated modal decomposition model is then used as the basis for reducing its size and complexity. The model reduction approach is based on the previously developed model order reduction algorithm (MORA), which uses the energy-based activity metric in order to generate a series of reduced models. The activity metric was originally developed for the generic case of nonlinear systems; however, in this work, the activity metric is adapted for the case of linear systems with single harmonic excitation. In this case closed form expressions are derived for the calculation of activity. An example is provided to demonstrate the power conserving transformation, calculation of the modal power and the elimination of unimportant modes or modal elements.

Asymmetric Pruning for Learning Cascade Detectors

Cascade classifiers are one of the most important contributions to real-time object detection. Nonetheless, there are many challenging problems arising in training cascade detectors. One common issue is that the node classifier is trained with a symmetric classifier. Having a low misclassification error rate does not guarantee an optimal node learning goal in cascade classifiers, i.e., an extremely high detection rate with a moderate false positive rate. In this work, we present a new approach to train an effective node classifier in a cascade detector. The algorithm is based on two key observations: 1) Redundant weak classifiers can be safely discarded; 2) The final detector should satisfy the asymmetric learning objective of the cascade architecture. To achieve this, we separate the classifier training into two steps: finding a pool of discriminative weak classifiers/features and training the final classifier by pruning weak classifiers which contribute little to the asymmetric learning criterion (asymmetric classifier construction). Our model reduction approach helps accelerate the learning time while achieving the pre-determined learning objective. Experimental results on both face and car data sets verify the effectiveness of the proposed algorithm. On the FDDB face data sets, our approach achieves the state-of-the-art performance, which demonstrates the advantage of our approach.