Proper Orthogonal Decomposition Basis Functions Research Articles

A reduced-order model for nonlinear radiative transfer problems based on moment equations and POD-Petrov-Galerkin projection of the normalized Boltzmann transport equation

A data-driven projection-based reduced-order model (ROM) for nonlinear thermal radiative transfer (TRT) problems is presented. The TRT ROM is formulated by (i) a hierarchy of low-order quasidiffusion (aka variable Eddington factor) equations for moments of the radiation intensity and (ii) the normalized Boltzmann transport equation (BTE). The multilevel system of moment equations is derived by projection of the BTE onto a sequence of subspaces which represent elements of the phase space of the problem. Exact closure for the moment equations is provided by the Eddington tensor. A Petrov-Galerkin (PG) projection of the normalized BTE is formulated using a proper orthogonal decomposition (POD) basis representing the normalized radiation intensity over the whole phase space and time. The Eddington tensor linearly depends on the solution of the normalized BTE. By linear superposition of the POD basis functions, a low-rank expansion of the Eddington tensor is constructed with coefficients defined by the PG projected normalized BTE. The material energy balance (MEB) equation is coupled with the effective gray low-order equations which exist on the same dimensional scale as the MEB equation. The resulting TRT ROM is structure and asymptotic preserving. A detailed analysis of the ROM is performed on the classical Fleck-Cummings (F-C) TRT multigroup test problem in 2D geometry. Numerical results are presented to demonstrate the ROM's effectiveness in the simulation of radiation wave phenomena. The ROM is shown to produce solutions with sufficiently high accuracy while using low-rank approximation of the normalized BTE solution. Essential physical characteristics of supersonic radiation wave are preserved in the ROM solutions.

Reduced-order model-based variational inference with normalizing flows for Bayesian elliptic inverse problems

We propose a reduced-order model-based variational inference method with normalizing flows for Bayesian elliptic inverse problems. The coefficient of the elliptic PDE is represented by a finite number of parameters. We aim to estimate the posterior distributions of these parameters in the framework of variational inference, and the approximation of the posterior distribution is constructed through a normalizing flow. Moreover, as data of inputs and outputs of the forward problem are accumulated during the training of the normalizing flow, we can naturally exploit the low-dimensional intrinsic structure of the forward elliptic PDE using reduced-order models based on these data, in which information of the true posterior is incorporated. We construct a low-dimensional set of data-driven basis functions in the solution space using proper orthogonal decomposition (POD) and train a neural network that maps the parameters to the coefficients of these data-driven basis functions. The surrogate forward map, which is the combination of the reduced-order model and the parameter-to-coefficient neural network, is then applied to the Bayesian elliptic inverse problem, where in each iteration the forward problem is evaluated in the space spanned by the POD basis functions using the surrogate forward map. Thus the inversion will be significantly accelerated as the surrogate forward map is essentially a low-dimensional model. We present numerical examples to show the accuracy and efficiency of the proposed method.

A numerical comparison of simplified Galerkin and machine learning reduced order models for vaginal deformations

High-fidelity computer simulations of childbirth remain prohibitively expensive and time consuming, making them impractical for guiding decision-making during obstetric emergencies. Cheap computer simulations that preserve the accuracy of high-fidelity models can be developed using surrogate modeling. Two common approaches to surrogate modeling are physics-based reduced order modeling (ROM) and machine learning (ML), with the latter gaining popularity as the scientific computing community seeks to leverage advances from other, mostly non-physics-based, computational strategies. Although ROM and ML have been compared for various problems, to our knowledge, such a comparison for simulations of vaginal deformations is currently missing. This study provides a baseline numerical comparison between methods from these two fundamentally different approaches. Since there are many methods falling into each modeling approach, to provide a fair and natural comparison, we select a basic model from each category, with each allowing (i) a straightforward implementation in commercial software packages, and (ii) use by practitioners with limited experience in the field. As a benchmark for the numerical comparison of the ROM and ML approaches, we use the finite element (FE) modeling of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. From the ROM strategies, we consider a simplified Galerkin ROM (G-ROM) that is based on the linearization of the underlying nonlinear equations. From the ML strategies, we select a feed-forward neural network to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the proper orthogonal decomposition (POD) coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical investigation of G-ROM, ML, and POD-ML takes place in the reconstructive regime. The numerical results show that the G-ROM outperforms the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The G-ROM achieves superior error performance to the best ML model with 11 POD basis functions. With higher-dimensional POD bases, the G-ROM achieves a relative error 3 orders of magnitude lower than that of the best ML model with an online CPU time still on the same order of magnitude as the best ML model. The POD-ML model improves on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model does not improve on the error performance of the ML and is still outperformed by the G-ROM for POD bases of size greater than 11. This baseline numerical investigation serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, and material properties of the human vagina, as well as their changes during labor.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

<p style='text-indent:20px;'>We present a non-intrusive model order reduction (NIMOR) approach with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level <inline-formula><tex-math id="M1">\begin{document}$ L $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ L\geq1 $\end{document}</tex-math></inline-formula>) over a target parameterized space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level <inline-formula><tex-math id="M3">\begin{document}$ L+1 $\end{document}</tex-math></inline-formula>, which includes the sparse grids from approximation level <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula>. A nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of the NIMOR method.</p>

Non-Intrusive Reduced Order Modelling for the fast simulation of gas reservoirs

Non-Intrusive Reduced Order Modelling (NIROM) is evaluated for the prediction of gas reservoir performance, including the time evolution of the pressure distribution within the reservoir. The NIROM projects simulation solutions into hyperspace using proper orthogonal decomposition (POD), then radial basis function (RBF) interpolation is used to describe the dynamics of the fluid flow in hyperspace. POD-RBF NIROMs have been used previously for the estimation of saturation distributions over time in oil reservoirs during pressure maintenance (e.g. by gas flooding) but have not been tested for gas reservoir depletion problems. In this work, the accuracy and efficiency of the NIROM were evaluated by comparing its outputs to those obtained from conventional simulations for two test cases: a homogeneous gas reservoir model with gas rate or bottom hole pressure production controls, and a realistic heterogeneous reservoir model with a similar geological description to the Norne field. We show that the NIROM can provide accurate estimates of pressure and saturation distributions as well as production data and it runs between 4 and 40 times faster. This is despite having been implemented using an interpreted language. We show that a 99.9% energy criterion can be used for determining the number of POD basis functions for pressure and saturation distributions, but more basis functions are needed for estimating production data. In all cases, the linear radial basis function gives the best estimation.

An iterative algorithm for POD basis adaptation in solving parametric convection–diffusion equations

In this paper, we develop an iterative algorithm for proper orthogonal decomposition (POD) basis adaptation in solving linear parametric PDEs. Specifically, we consider the convection–diffusion equations with diffusivity as a parameter. To construct POD basis functions for the convection–diffusion equation with a small diffusivity, we need a fine-grid solver to obtain accurate solution snapshots, which leads to a large amount of computation and memory costs. Meanwhile, a coarse-grid solver is sufficient for obtaining accurate solution snapshots for the convection–diffusion equation with a large diffusivity. We aim to adapt the POD basis functions extracted from the solution snapshots of a large diffusivity for the construction of a reduced-order model at a small diffusivity without resorting to a fine-grid solver. Our POD basis adaptation method exploits the implicit dependence of solutions on diffusivity. The POD basis functions are adapted through an iterative algorithm, where the full-order model simulation at a large diffusivity and the POD-based reduced-order model simulation at a small diffusivity are implemented, alternatively. We also provide convergence analysis for our POD basis adaptation method. The algorithm and convergence analysis can be generalized to other types of linear parametric PDEs without any difficulty. Finally, we present numerical results to demonstrate the performance and accuracy of the proposed method.

A Data-Driven Model Reduction Method for Parabolic Inverse Source Problems and its Convergence Analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract time- and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time- and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of POD-CSI method.

A discontinuous and adaptive reduced order model for the angular discretization of the Boltzmann transport equation

SummaryThis article presents a new adaptive reduced order model for resolving the angular direction of the Boltzmann transport equation, based on proper orthogonal decomposition (POD) and the method of snapshots. It builds upon previous methods of applying POD to the angular dimension, with modifications to increase accuracy and solver stability. Previous methods used continuous global functions spanning the whole sphere. The new approach, discontinuous POD (DPOD), partitions the surface of the sphere into angular regions, each with an independent set of POD basis functions. Combined, these can approximate flux distributions which span the sphere using optimized basis functions for each angular region. In addition, a novel implementation of adaptive angular resolution known as adaptive discontinuous POD (ADPOD) is presented, which allows the number of DPOD basis functions to vary by angular octant and spatial element. DPOD and ADPOD are applied to two problems in order to demonstrate their benefits compared with POD. Both are shown to reduce the number of solver iterations required to find a solution and decrease the error in the angular flux.

Parametric hull shape variations by Reduced Order Model based geometric transformation

The representation and the variation of hull shapes are two challenging problems in naval architecture due to the complexity of the geometry and to the need to ensure the fairness of the surfaces. Conventional CAD techniques are widely used to accurately describe the hull shape. However, they are rather complex to be easily used to generate hull shape variations due to the great number of variables involved. We propose a study to highlight the pro and cons of the application of space reduction techniques, usually referred to as Reduced Order Models (ROMs), to create a parametric model for both global and local hull shape variations. A geometric transformation relying on the Proper Orthogonal Decomposition (POD) method is applied on top of a combined subdivision surface — Free Form Deformation (FFD) approach conceived for modeling and variation of hull shapes. The analysis focuses on highlighting the geometric meaning of the new POD basis functions and on how this model affects the variability of the explored design space. Two studies are developed in order to show possible applications of this technique. In the first one a new set of selection criteria for the POD modes based on geometric considerations are proposed to use it on uncorrelated geometric domains. In the latter the large-dimensional space of the so-called Design Velocities (DV), related to the shape sensitivity, is reduced by using the POD approach to create a new transformation of the hull shape.

Learning patient‐specific parameters for a diffuse interface glioblastoma model from neuroimaging data

Parameters in mathematical models for glioblastoma multiforme (GBM) tumour growth are highly patient specific. Here, we aim to estimate parameters in a Cahn–Hilliard type diffuse interface model in an optimised way using model order reduction (MOR) based on proper orthogonal decomposition (POD). Based on snapshots derived from finite element simulations for the full‐order model (FOM), we use POD for dimension reduction and solve the parameter estimation for the reduced‐order model (ROM). Neuroimaging data are used to define the highly inhomogeneous diffusion tensors as well as to define a target functional in a patient‐specific manner. The ROM heavily relies on the discrete empirical interpolation method, which has to be appropriately adapted in order to deal with the highly nonlinear and degenerate parabolic partial differential equations. A feature of the approach is that we iterate between full order solvers with new parameters to compute a POD basis function and sensitivity‐based parameter estimation for the ROM problems. The algorithm is applied using neuroimaging data for two clinical test cases, and we can demonstrate that the reduced‐order approach drastically decreases the computational effort.

Model-order reduction technique for temperature prediction and sensor placement in cylindrical steam reformer for HT–PEMFC

A model-order reduction technique employing proper orthogonal decomposition (POD) with a robust point interpolation method (RPIM) is proposed and validated herein; a proposed method has been used for temperature prediction and sensor placement for a cylindrical steam reformer. The uncertainty associated with a set of obtained observations typically leads to an infeasible prediction for designing and operating systems. To address this problem, a novel interpolation method is introduced for reducing the uncertainty via optimal sensor placement. The objective function of the proposed method is the norm of the linear combination of the POD basis function extracted from a set of error-free observations and a mask matrix determined based on the sensor placement. The error bound of the proposed POD–RPIM method is analyzed and compared with that of well-known conventional interpolation methods. It is demonstrated that the POD–RPIM is a promising method that can address the shortcomings of the conventional methods, with regard to the uncertainty involved.

A Galerkin‐free/equation‐free model reduction method for single‐phase flow in fractured porous media

AbstractUsing traditional high‐fidelity numerical simulation to simulate fluid flow in fractured porous media in a real field remains challenging. It involves a large number of degrees of freedom when matrix and fracture equations are solved. To address this challenge, we propose a Galerkin‐free framework to construct a reduced‐order model (ROM) based on the proper orthogonal decomposition (POD). Compared with the typical POD‐based modeling process commonly used in previous studies, the POD‐ROM can be built without performing the Galerkin projection of flow equations onto the low‐dimensional space spanned by the POD basis functions. The numerical integration method was incorporated to obtain the POD time coefficients based on the flow equations solved by the conventional finite volume method. Two complex fracture cases reflecting high‐contrast porous media in a two‐dimensional domain were designed to verify the accuracy and efficiency of the established Galerkin‐free POD‐ROM. Sensitivity analysis of parameters was conducted to examine the adaptability of the ROM. The results illustrate that, compared with the fine‐scale model, the ROM can significantly reduce the CPU time without compromising the quality of the numerical solutions.

Proper Orthogonal Decomposition Method to Nonlinear Filtering Problems in Medium-High Dimension

In this paper, we investigate the proper orthogonal decomposition (POD) method to numerically solve the forward Kolmogorov equation (FKE). Our method aims to explore the low-dimensional structures in the solution space of the FKE and to develop efficient numerical methods. As an important application and our primary motivation to study the POD method to FKE, we solve the nonlinear filtering (NLF) problems with a real-time algorithm proposed by Yau and Yau combined with the POD method. This algorithm is referred as POD algorithm in this paper. Our POD algorithm consists of offline and online stages. In the offline stage, we construct a small number of POD basis functions that capture the dynamics of the system and compute propagation of the POD basis functions under the FKE operator. In the online stage, we synchronize the coming observations in a real-time manner. Its convergence analysis has also been discussed. Some numerical experiments of the NLF problems are performed to illustrate the feasibility of our algorithm and to verify the convergence rate. Our numerical results show that the POD algorithm provides considerable computational savings over existing numerical methods.

Proper Orthogonal Decomposition-Based Method for Predicting Flow and Heat Transfer of Oil and Water in Reservoir.

Calculation process of some reservoir engineering problems involves several passes of full-order numerical reservoir simulations, and this makes it a time-consuming process. In this study, a fast method based on proper orthogonal decomposition (POD) was developed to predict flow and heat transfer of oil and water in a reservoir. The reduced order model for flow and heat transfer of oil and water in the hot water-drive reservoir was generated. Then, POD was used to extract a reduced set of POD basis functions from a series of "snapshots" obtained by a finite difference method (FDM), and these POD basis functions most efficiently represent the dynamic characteristics of the original physical system. After injection and production parameters are changed constantly, the POD basis functions combined with the reduced order model were used to predict the new physical fields. The POD-based method was approved on a two-dimensional hot water-drive reservoir model. For the example of this paper, compared with FDM, the prediction error of water saturation and temperature fields were less than 1.3% and 1.5%, respectively; what is more, it was quite fast, where the increase in calculation speed was more than 70 times.

An angular reduced order model for radiative transfer in non grey media

This paper investigates a reduced order model for the angular discretisation of the radiative transfer equation (RTE) when considering non grey participating gases. The key idea is to use a global model for the gas radiative properties and to derive an angular reduced order model, based on the Proper Orthogonal Decomposition (POD) method, for each absorption coefficient class independently. Angular POD basis functions are extracted from high order SN reference solutions. A finite element approach is used to discretised the RTE in space and angle and the POD angular matrices of the reduced system are easily constructed from the SN angular matrices of the reference solutions. The angular POD basis sets are truncated at different levels depending on the absorption coefficient class in order to optimally compute the total radiative power. The method is applied to solve the radiation field associated to an air/H2O mixture flowing in a square differentially heated cavity, with black isothermal walls and diffuse reflecting adiabatic walls. Results show that the POD model is very accurate and efficient for treating the thick classes but it suffers from a low convergence rate for the thin classes. For computing the radiative power, the reduced order model allows to reduce the averaged number of angular basis functions of an order of magnitude and to reduce the CPU time by a factor 2 to 3 to reach a given level of accuracy, compared to a standard SN method.

Machine learning-based rapid response tools for regional air pollution modelling

A parameterised non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) and machine learning methods has been firstly developed for model reduction of pollutant transport equations. Our motivation is to provide rapid response urban air pollution predictions and controls. The varying parameters in the P-NIROM are pollutant sources. The training data sets are obtained from the high fidelity modelling solutions (called snapshots) for selected parameters (pollutant sources, here) over the parameter space RP. From these training data sets, the machine learning method is used to generate the relationship between the reduced solutions and inputs (pollutant sources) over RP. Furthermore a set of hyper-surface functions associated with each POD basis function is constructed for representing the fluid dynamics over the reduced space. The accuracy of the P-NIROM is highly dependent on the quality of the training set, here obtained from the high fidelity model. Over existing machine learning methods, the P-NIROM algorithm proposed here has the advantages that (1) it is combined with NIROM, thus providing rapid and reasonably accurate solutions; and (2) it is a robust and efficient approach for representation of any parametrised partial differential equations as the model parameters/inputs vary. In this study, we demonstrate the way how to implement the P-NIROM for the pollutant transport equation (but not limited to due to its robustness). Its predictive capability is illustrated in a three-dimensional (3-D) simulation of power plant plumes over a large region in China, where the varying parameters are the emission intensity at three locations. Results indicate that in comparison to the high fidelity model, the CPU cost is reduced by factor up to five orders of magnitude while reasonable accuracy remains.

Robust non‐linear control design for systems governed by Burgers' equation subjected to parameter variation

In this study, a robust sliding mode controller is proposed for dynamic systems governed by Burgers' equation with Neumann boundary conditions in the presence of parameter variations. The main objective is to design a reduced-order model based controller at a nominal value of the system parameter that stabilises the full-order model while being robust with respect to model uncertainties in the obtained reduced-order model. The model uncertainties resulted from the variation of the parameter ν are discussed under two categories; first, the error arising from the change in the states of the full-order model, and second, the error associated with the estimated proper orthogonal decomposition basis functions at the nominal value of ν 0 . In this work, the boundedness of the error functions is studied and an estimation of these bounds is obtained in terms of the reduced-order model and matrices of the full-order system that are known a priori. Next, the bounds on the error functions are used to design a reduced-order sliding mode controller that guarantees the stability of the full-order model obtained via a finite element approximation of the Burgers' equation for a trajectory tracking problem.

Extrapolating POD reduced-order model based on temperature gradient for unsteady heat conduction

A new proper orthogonal decomposition (POD) reduced-order model based on temperature gradient is established for heat conduction differential equation. In the most of present studies of heat conduction POD reduced-order model, the POD basis functions are extracted from temperature fields by adopting proper orthogonal decomposition or singular value decomposition (SVD). Different from conventional POD reduced-order method, this paper optimize the gradient of POD solutions by extracting the optimal orthogonal basis from temperature gradient. After getting POD basis, the POD reduced-order model is solved by using Galerkin projection. POD-Galerkin reduced-order model of two dimensional, constant property and non-steady heat conduction differential equations is established, and the accuracy of the reduced-order model under different boundary conditions is further studied. The results suggests that the relative error of POD solution is less than 0.1% while the average computing time reduces from 1.64 s to 0.02 s.