Model Reduction Research Articles

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

Randomized greedy magic point selection schemes for nonlinear model reduction

An established way to tackle model nonlinearities in projection-based model reduction is via relying on partial information. This idea is shared by the methods of gappy proper orthogonal decomposition (POD), missing point estimation (MPE), masked projection, hyper reduction, and the (discrete) empirical interpolation method (DEIM). The selected indices of the partial information components are often referred to as “magic points.” The original contribution of the work at hand is a novel randomized greedy magic point selection. It is known that the greedy method is associated with minimizing the norm of an oblique projection operator, which, in turn, is associated with solving a sequence of rank-one SVD update problems. We propose simplification measures so that the resulting greedy point selection has the following main features: (1) The inherent rank-one SVD update problem is tackled in a way, such that its dimension does not grow with the number of selected magic points. (2) The approach is online efficient in the sense that the computational costs are independent from the dimension of the full-scale model. To the best of our knowledge, this is the first greedy magic point selection that features this property. We illustrate the findings by means of numerical examples. We find that the computational cost of the proposed method is orders of magnitude lower than that of its deterministic counterpart. Nevertheless, the prediction accuracy is just as good if not better. When compared to a state-of-the-art randomized method based on leverage scores, the randomized greedy method outperforms its competitor.

The nearest graph Laplacian in Frobenius norm

AbstractWe address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the problem as convex quadratic optimization problems with a special structure: one based on the active set method and the other on direct computation of Karush–Kuhn–Tucker points. The proposed algorithms can be applied to system identification and model reduction problems involving Laplacian dynamics. We demonstrate that these algorithms possess lower time complexities and the finite termination property, unlike the interior point method and V‐FISTA, the latter of which is an accelerated projected gradient method. Our numerical experiments confirm the effectiveness of the proposed algorithms.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Parameterization of nonlinear aeroelastic reduced order models via direct interpolation of Taylor partial derivatives

The identification of optimally sparse Taylor partial derivatives presents a new opportunity in efficient nonlinear model reduction for complex aeroelastic systems. Unfortunately, for this class of reduced order model (ROM), the robustness that is observed in the linear regime to parameters including; dynamic pressure, control hinge linear stiffness, or even freeplay, can be quickly compromised in the nonlinear regime. In this paper, the nonlinear sensitivity of selected critical parameters is addressed by interpolating a library of nonlinear unsteady aerodynamic ROMs across a compact subspace in dynamic pressure and freeplay magnitude. The ROM, based on Lagrange interpolation of sparse higher-order Taylor partial derivatives, demonstrates excellent precision in modelling high amplitude transonic limit cycle oscillations for an all-movable wing with freeplay, capturing the LCO region (up to 96% of the linear flutter boundary), and for a range of freeplay values.

Achieving Occam's razor: Deep learning for optimal model reduction.

All fields of science depend on mathematical models. Occam's razor refers to the principle that good models should exclude parameters beyond those minimally required to describe the systems they represent. This is because redundancy can lead to incorrect estimates of model parameters from data, and thus inaccurate or ambiguous conclusions. Here, we show how deep learning can be powerfully leveraged to apply Occam's razor to model parameters. Our method, FixFit, uses a feedforward deep neural network with a bottleneck layer to characterize and predict the behavior of a given model from its input parameters. FixFit has three major benefits. First, it provides a metric to quantify the original model's degree of complexity. Second, it allows for the unique fitting of data. Third, it provides an unbiased way to discriminate between experimental hypotheses that add value versus those that do not. In three use cases, we demonstrate the broad applicability of this method across scientific domains. To validate the method using a known system, we apply FixFit to recover known composite parameters for the Kepler orbit model and a dynamic model of blood glucose regulation. In the latter, we demonstrate the ability to fit the latent parameters to real data. To illustrate how the method can be applied to less well-established fields, we use it to identify parameters for a multi-scale brain model and reduce the search space for viable candidate mechanisms.

Macro-micro decomposition for consistent and conservative model order reduction of hyperbolic shallow water moment equations: a study using POD-Galerkin and dynamical low-rank approximation

Geophysical flow simulations using hyperbolic shallow water moment equations require an efficient discretization of a potentially large system of PDEs, the so-called moment system. This calls for tailored model order reduction techniques that allow for efficient and accurate simulations while guaranteeing physical properties like mass conservation. In this paper, we develop the first model reduction for the hyperbolic shallow water moment equations and achieve mass conservation. This is accomplished using a macro-micro decomposition of the model into a macroscopic (conservative) part and a microscopic (non-conservative) part with subsequent model reduction using either POD-Galerkin or dynamical low-rank approximation only on the microscopic (non-conservative) part. Numerical experiments showcase the performance of the new model reduction methods including high accuracy and fast computation times together with guaranteed conservation and consistency properties.

A stochastic approximation for the finite-size Kuramoto–Sakaguchi model

We perform a stochastic model reduction of the Kuramoto–Sakaguchi model for finitely many coupled phase oscillators with phase frustration. Whereas in the thermodynamic limit coupled oscillators exhibit stationary states and a constant order parameter, finite-size networks exhibit persistent temporal fluctuations of the order parameter. These fluctuations are caused by the interaction of the synchronised oscillators with the non-entrained oscillators. We present numerical results suggesting that the collective effect of the non-entrained oscillators on the synchronised cluster can be approximated by a Gaussian process. This allows for an effective closed evolution equation for the synchronised oscillators driven by a Gaussian process which we approximate by a two-dimensional Ornstein–Uhlenbeck process. Our reduction reproduces the stochastic fluctuations of the order parameter and leads to a simple stochastic differential equation for the order parameter.

Tracing the Evolution of Sound in Six Generations of the Chevrolet Corvette

This study presents the noise characteristics of six generations of the Chevrolet Corvette, an iconic American sports car that has undergone technological leaps and design innovations over the years, redefining performance and handling. The research conducted as part of an undergraduate noise control project is meant to study the auditory experiences of drivers, occupants, and pedestrians, considering the implications of improved cabin noise reduction in newer models while examining the changes surrounding powertrain and exhaust expectations in various generations of the model. Focusing on the evolution of sound pressure levels, the study analyzes both interior and exterior sound pressure level data to extract insights into the acoustic metamorphosis of the Corvette. This exploration enhances the understanding of the difference between sound and the Corvette driving experience. The research's significance extends to the community of Corvette enthusiasts, contributing to the historical narrative of this sports car and impacting automotive culture. Anticipation regarding the exterior experience for pedestrians aligns with sentiments about the powertrain and exhaust personality. The study concludes that stock Corvettes, even older models, do not significantly endanger hearing, mainly due to road noise. The interior sound at various speeds and accelerations can reach close to 90dBA periodically. Also, aftermarket exhaust systems hint at the possibility of extremely high sound pressure levels.

PSDM: A parametrized structural dynamic modeling method based on digital twin for performance prediction

Digital twin (DT) technology is a powerful tool that accurately represents physical entities and provides real-time monitoring and reliability analysis capabilities for decision-makers and managers. However, modeling complex systems in structural health monitoring using DTs can be repetitive and tedious with existing approaches. To address these issues, we propose a new approach called the Parametrized Structural Dynamic Modeling(PSDM) method. This approach effectively parametrizes dynamic systems in structural health monitoring by leveraging Proper Orthogonal Decomposition (POD) to represent and analyze complex structural behaviors. It also incorporates model reduction and kernel functions. By integrating physics-based insights and advanced modeling methodologies, the PSDM method aims to bridge the gap between the physical and digital domains, enabling accurate and efficient analysis of complex structural systems. To verify its accuracy and efficiency, we apply the PSDM method to two engineering cases: the wing cantilever beam and the telehandler boom. The results demonstrate that the online computational cost of the PSDM method is lower than Finite Element Methods (FEM), thereby improving the computational efficiency of DT modeling technology for complex-large structures. Thus, this research presents a feasible method for implementing structural reliability analysis for engineering equipment.

Dynamic Model Reduction for Viscously Damped Structures with Statically Determinate Interfaces

A novel model reduction method for viscously damped structures with statically determinate interfaces, such as spacecraft flexible appendages, is proposed. The paper presents a derivation of the complete complex modal expansion of the interface dynamic stiffness of these structures. Based on the identity relation for all complex modes, which is obtained during the derivation, it is found that the interface acceleration impedance can be expressed as a rational fraction with high accuracy using only low-order complex modes. Using this rational fraction as an approximation model, numerical results of the frequency response can be fitted. The fitted interface acceleration impedance can be applied to real-time control as a reduced model in the form of a transfer function. Furthermore, it can be transformed into the form of system matrices by introducing auxiliary variables, which then participate in the dynamic analysis of the assembly. The reduction process circumvents complex modal analysis and necessitates only the results of frequency responses. Thanks to the powerful ability of conventional finite element software to perform frequency response analysis, this reduction method can be used for large-scale complex models in actual engineering applications.

Feasibility and safety of percutaneous intramyocardial septal cryoablation: A canine model with 6-month follow-up

Echocardiography-guided percutaneous intramyocardial septal radiofrequency ablation (PIMSRA, Liwen procedure) is a novel treatment option for hypertrophic obstructive cardiomyopathy (HOCM). The safety and feasibility of using this procedure for cryoablation are unknown. We aimed to investigate the feasibility and safety of echocardiography-guided percutaneous intramyocardial septal cryoablation (PIMSCA) for septal thickness reduction in a canine model. Eight canines underwent PIMSCA, and had electrocardiography, echocardiography(ECG), myocardial contrast echocardiography (MCE), serological and pathological examinations during the preoperative, immediate postoperative, and 6-month follow-up. All eight canines underwent successful cryoablation and continued to be in sinus rhythm during ablation and without malignant arrhythmias. MCE showed that the ablation area had decreased myocardial perfusion after the procedure. Troponin I levels were significantly elevated [0.010 (0.005, 0.297) ng/mL vs. 3.122 (1.152, 7.990) ng/mL, p < 0.05)]. At 6-month follow-up after the procedure, all animals were alive, with thinning of the interventricular septum (7.26 ± 0.52 mm vs. 3.86 ± 0.29 mm, p < 0.05). Echocardiography showed no significant decrease in the left ventricular ejection fractions (LVEF) (54.32 ± 2.93 % vs. 54.70 ± 2.47 %, p > 0.05) or changes by pulse-wave Doppler E/A (1.17 ± 0.43 vs. 1.07 ± 0.43, p > 0.05), E/e' (8.09 ± 1.49 vs. 10.05 ± 2.68, p > 0.05). Pathological findings proved the effectiveness of cryoablation in myocardial tissues. We observed pericardial effusions and premature ventricular complexes (PVCs) associated with the procedure. Our findings provided preliminary evidence of the safety and feasibility of PIMSCA in reducing interventricular septum, which provides a potentially new treatment option for HOCM.

Signal generator framework-based variation expansion-matching method for polynomial systems

This paper proposes an interpolatory model reduction method rooted in the signal generator framework for polynomial systems. This innovative approach not only surpasses the constraints on input spaces found in moment-matching methods, such as u(t)∈L2([0,+∞]) recommended in H2 optimal interpolation method, but also eliminates redundant reduced bases through variations of solution in the time domain. Our contributions can be delineated into three key aspects. Firstly, we establish the closure under addition, multiplication, and composition for the transformations from all elementary functions and rational functions to a signal generator-driven system. This validates the applicability of the signal generator framework to general inputs. Secondly, we enhance the numerical stability of two-sided reduced bases for polynomial systems through a variation expansion. Thirdly, we employ tensor decomposition techniques to streamline the complexity of the model reduction process. The simulation results have verified the effectiveness and applicability of the proposed method, which requires smaller computational complexity while maintaining the same level of accuracy.

Prediction of diabetic retinopathy among type 2 diabetic patients in University of Gondar Comprehensive Specialized Hospital, 2006–2021: A prognostic model

BackgroundThere has been a paucity of evidence for the development of a prediction model for diabetic retinopathy (DR) in Ethiopia. Predicting the risk of developing DR based on the patient’s demographic, clinical, and behavioral data is helpful in resource-limited areas where regular screening for DR is not available and to guide practitioners estimate the future risk of their patients. MethodsA retrospective follow-up study was conducted at the University of Gondar (UoG) Comprehensive Specialized Hospital from January 2006 to May 2021 among 856 patients with type 2 diabetes (T2DM). Variables were selected using the Least Absolute Shrinkage and Selection Operator (LASSO) regression. The data were validated by 10-fold cross-validation. Four ML techniques (naïve Bayes, K-nearest neighbor, decision tree, and logistic regression) were employed. The performance of each algorithm was measured, and logistic regression was a well-performing algorithm. After multivariable logistic regression and model reduction, a nomogram was developed to predict the individual risk of DR. ResultsLogistic regression was the best algorithm for predicting DR with an area under the curve of 92%, sensitivity of 87%, specificity of 83%, precision of 84%, F1-score of 85%, and accuracy of 85%. The logistic regression model selected seven predictors: total cholesterol, duration of diabetes, glycemic control, adherence to anti-diabetic medications, other microvascular complications of diabetes, sex, and hypertension. A nomogram was developed and deployed as a web-based application. A decision curve analysis showed that the model was useful in clinical practice and was better than treating all or none of the patients. ConclusionsThe model has excellent performance and a better net benefit to be utilized in clinical practice to show the future probability of having DR. Identifying those with a higher risk of DR helps in the early identification and intervention of DR.

Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers

Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers

3D modelling of ground magnetic-source airborne TEM data for anisotropic media based on the rational Krylov method

With the rapid development of the transient electromagnetic method (TEM), the numerical simulations of three-dimensional anisotropic media have garnered attention. This study used the rational Krylov method to realise the three-dimensional modelling of the ground magnetic-source airborne transient electromagnetic data for arbitrary anisotropic media. First, based on Maxwell's equations, a time-domain electromagnetic equation was derived for anisotropic media. The finite difference method was employed to discretise the equation, and the electromagnetic field was then expressed as the product of an exponential matrix and a vector related to the source. Subsequently, the rational Arnoldi algorithm was adopted to construct the rational Krylov subspace orthogonal basis vectors, and the electromagnetic fields at different times were obtained based on a model reduction strategy. The accuracy of the proposed algorithm was verified through the comparisons of its results with those of other software packages. We calculated the response for different anisotropic parameters of the typical three-dimensional anisotropic models, and analysed the influence of the anisotropic media on the ground magnetic-source airborne transient electromagnetic data. The proposed three-dimensional forward TEM algorithm provides important technical support for identifying the anisotropic geological bodies and lays the foundation for further inversion research.

Order reduction of special valve models based on pod method

Abstract The opening size of the disc-type special regulating valve (referred to as the special valve) can be used to regulate the fluid pressure and flow inside the cabin, and is an important equipment in the intake system of the high-altitude simulation test bench (referred to as the high-altitude bench). Due to the high dimensionality and complex solution of the special valve model, the calculation time is relatively long. In this paper, the Proper Orthogonal Decomposition (POD) model reduction method is applied to the mathematical model of the special valve, and the POD basis matrix is constructed through snapshots, thus obtaining a low-dimensional model and improving computational efficiency. The research results indicate that, while ensuring sufficient accuracy, this method can reduce the dimensionality of the special valve model, improve the computational speed of the simulator in calculating this case, and verify the effectiveness of the POD method.

Non-intrusive reduced-order model for time-dependent stochastic partial differential equations utilizing dynamic mode decomposition and polynomial chaos expansion

In this study, we present a novel non-intrusive reduced-order model (ROM) for solving time-dependent stochastic partial differential equations (SPDEs). Utilizing proper orthogonal decomposition (POD), we extract spatial modes from high-fidelity solutions. A dynamic mode decomposition (DMD) method is then applied to vertically stacked matrices of projection coefficients for future prediction of coefficient fields. Polynomial chaos expansion (PCE) is employed to construct a mapping from random parameter inputs to the DMD-predicted coefficient field. These lead to the POD–DMD–PCE method. The innovation lies in vertically stacking projection coefficients, ensuring time-dimensional consistency in the coefficient matrix for DMD and facilitating parameter integration for PCE analysis. This method combines the model reduction of POD with the time extrapolation strengths of DMD, effectively recovering field solutions both within and beyond the training time interval. The efficiency and time extrapolation capabilities of the proposed method are validated through various nonlinear SPDEs. These include a reaction–diffusion equation with 19 parameters, a two-dimensional heat equation with two parameters, and a one-dimensional Burgers equation with three parameters.

Model order reduction by convex displacement interpolation

We present a nonlinear interpolation technique for parametric fields that exploits optimal transportation of coherent structures of the solution to achieve accurate performance. The approach generalizes the nonlinear interpolation procedure introduced in [Iollo, Taddei, J. Comput. Phys., 2022] to multi-dimensional parameter domains and to datasets of several snapshots. Given a library of high-fidelity simulations, we rely on a scalar testing function and on a point set registration method to identify coherent structures of the solution field in the form of sorted point clouds. Given a new parameter value, we exploit a regression method to predict the new point cloud; then, we resort to a boundary-aware registration technique to define bijective mappings that deform the new point cloud into the point clouds of the neighboring elements of the dataset, while preserving the boundary of the domain; finally, we define the estimate as a weighted combination of modes obtained by composing the neighboring snapshots with the previously-built mappings. We present several numerical examples for compressible and incompressible, viscous and inviscid flows to demonstrate the accuracy of the method. Furthermore, we employ the nonlinear interpolation procedure to augment the dataset of simulations for linear-subspace projection-based model reduction: our data augmentation procedure is designed to reduce offline costs — which are dominated by snapshot generation — of model reduction techniques for nonlinear advection-dominated problems.

Nonautonomous spectral submanifolds for model reduction of nonlinear mechanical systems under parametric resonance.

We use the recent theory of spectral submanifolds (SSMs) for model reduction of nonlinear mechanical systems subject to parametric excitations. Specifically, we develop expressions for higher-order nonautonomous terms in the parameterization of SSMs and their reduced dynamics. We provide these results for both general first-order and second-order mechanical systems under periodic and quasiperiodic excitation using a multi-index based approach, thereby optimizing memory requirements and the computational procedure. We further provide theoretical results that simplify the SSM parametrization for general second-order dynamical systems. More practically, we show how the reduced dynamics on the SSM can be used to extract the resonance tongues and the forced response around the principal resonances in parametrically excited systems. In the case of two-dimensional SSMs, we formulate explicit expressions for computing the steady-state response as the zero-level set of a two-dimensional function for systems that are subject to external as well as parametric excitation. This allows us to parallelize the computation of the forced response over the range of excitation frequencies. We demonstrate our results on several examples of varying complexity, including finite-element-type examples of mechanical systems. Furthermore, we provide an open-source implementation of all these results in the software package SSMTool.