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

In 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.     

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

Model order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 John Wiley & Sons, Ltd.     

Development and application of reduced‐order modeling procedures for subsurface flow simulation

The optimization of subsurface flow processes is important for many applications, including oil field operations and the geological storage of carbon dioxide. These optimizations are very demanding computationally due to the large number of flow simulations that must be performed and the typically large dimension of the simulation models. In this work, reduced‐order modeling (ROM) techniques are applied to reduce the simulation time of complex large‐scale subsurface flow models. The procedures all entail proper orthogonal decomposition (POD), in which a high‐fidelity training simulation is run, solution snapshots are stored, and an eigen‐decomposition (SVD) is performed on the resulting data matrix. Additional recently developed ROM techniques are also implemented, including a snapshot clustering procedure and a missing point estimation technique to eliminate rows from the POD basis matrix. The implementation of the ROM procedures into a general‐purpose research simulator is described. Extensive flow simulations involving water injection into a geologically complex 3D oil reservoir model containing 60 000 grid blocks are presented. The various ROM techniques are assessed in terms of their ability to reproduce high‐fidelity simulation results for different well schedules and also in terms of the computational speedups they provide. The numerical solutions demonstrate that the ROM procedures can accurately reproduce the reference simulations and can provide speedups of up to an order of magnitude when compared with a high‐fidelity model simulated using an optimized solver. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

This report presents a numerical study of reduced‐order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced‐order representations combine ideas of approximation for nonlinear terms, of local bases, and of least‐squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user‐specified tolerance. An adaptive time‐integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced‐order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 John Wiley & Sons, Ltd.     

结构物理参数识别的贝叶斯估计马尔可夫蒙特卡罗方法

从结构动力特征方程出发,以结构主模态参数为观测量,推得结构物理参数线性回归模型。对该模型应用贝叶斯估计理论得到物理参数后验联合分布,再结合马尔可夫蒙特卡罗抽样方法给出各个物理参数的边缘概率分布和最优估计值,而提出了基于结构主模态参数的结构物理参数识别贝叶斯估计马尔可夫蒙特卡罗方法。对五层剪切型结构的数值研究表明,此方法能够利用少数主模态参数给出结构质量和刚度参数的概率分布和最优识别值,而且在主模态参数较准确时识别误差很小。     

Adaptive reduced order modeling for nonlinear dynamical systems through a new a posteriori error estimator: Application to uncertainty quantification

Multiquery problems such as uncertainty quantification (UQ), optimization of a dynamical system require solving a differential equation at multiple parameter values. Therefore, for large systems, the computational cost becomes prohibitive. This issue can be addressed by using a cheaper reduced order model (ROM) instead. However, the ROM entails error in the solution due to approximation in a lower dimensional subspace. Moreover, the ROM lacks robustness over a wide range of parameter values. To address these issues, first, an upper bound on the norm of the state transition matrix is derived. This bound, along with the residual in the governing equation, are then used to develop an error estimator for general nonlinear dynamical systems. Furthermore, this error estimator is used in conjunction with the modified greedy search algorithm proposed by Hossain and Ghosh (Int J Numer Methods Eng, 2018;116(12-13): 741-758) to adaptively construct a robust proper orthogonal decomposition-based ROM. This adaptive ROM is subsequently deployed for UQ by invoking it in a statistical simulation. Two numerical studies: (i) viscous Burgers' equation and (ii) beam on nonlinear Winkler foundation, showed an improved accuracy of the error estimator compared to the current literature. A significant computational speed-up in UQ is achieved.     

Limited‐memory adaptive snapshot selection for proper orthogonal decomposition

Reduced order models are useful for accelerating simulations in many‐query contexts, such as optimization, uncertainty quantification, and sensitivity analysis. However, offline training of reduced order models (ROMs) can have prohibitively expensive memory and floating‐point operation costs in high‐performance computing applications, where memory per core is limited. To overcome this limitation for proper orthogonal decomposition, we propose a novel adaptive selection method for snapshots in time that limits offline training costs by selecting snapshots according an error control mechanism similar to that found in adaptive time‐stepping ordinary differential equation solvers. The error estimator used in this work is related to theory bounding the approximation error in time of proper orthogonal decomposition‐based ROMs, and memory usage is minimized by computing the singular value decomposition using a single‐pass incremental algorithm. Results for a viscous Burgers' test problem demonstrate convergence in the limit as the algorithm error tolerances go to zero; in this limit, the full‐order model is recovered to within discretization error. A parallel version of the resulting method can be used on supercomputers to generate proper orthogonal decomposition‐based ROMs, or as a subroutine within hyperreduction algorithms that require taking snapshots in time, or within greedy algorithms for sampling parameter space. Copyright © 2016 John Wiley & Sons, Ltd.     

Multivariate predictions of local reduced‐order‐model errors and dimensions

This paper introduces multivariate input‐output models to predict the errors and bases dimensions of local parametric Proper Orthogonal Decomposition reduced‐order models. We refer to these mappings as the multivariate predictions of local reduced‐order model characteristics (MP‐LROM) models. We use Gaussian processes and artificial neural networks to construct approximations of these multivariate mappings. Numerical results with a viscous Burgers model illustrate the performance and potential of the machine learning‐based regression MP‐LROM models to approximate the characteristics of parametric local reduced‐order models. The predicted reduced‐order models errors are compared against the multifidelity correction and reduced‐order model error surrogates methods predictions, whereas the predicted reduced‐order dimensions are tested against the standard method based on the spectrum of snapshots matrix. Since the MP‐LROM models incorporate more features and elements to construct the probabilistic mappings, they achieve more accurate results. However, for high‐dimensional parametric spaces, the MP‐LROM models might suffer from the curse of dimensionality. Scalability challenges of MP‐LROM models and the feasible ways of addressing them are also discussed in this study.     

Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits

Various reduced basis element methods are compared for performing transient thermal simulations of integrated circuits. The reduced basis element method is a type of reduced order modeling that takes advantage of repeated geometrical features. It uses a reduced set of basis functions to approximate the solution of a PDE on subdomains (blocks); then these blocks are coupled together to perform a simulation on an entire domain. As the simulations are transient, a proper orthogonal decomposition basis is used, and the proper orthogonal decomposition eigenvalues from each block are used to derive error bounds for the entire simulation. These bounds are used to examine choices of block decompositions for a simplified integrated circuit structure. A decomposition that uses a single block for each transistor device is compared with a decomposition that uses one block for multiple devices. It was found that larger blocks are more computationally efficient; however, the advantage decreases if the devices within a block receive independent signals. Continuous and discontinuous methods of coupling the blocks were also compared. The coupling methods lend themselves to different solution approaches such as static condensation (continuous coupling) and block‐based inversion (discontinuous). Static condensation yielded the best convergence rate, accuracy, and operation count. Copyright © 2017 John Wiley & Sons, Ltd.     

A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

A nonparametric probabilistic approach for modeling uncertainties in projection‐based, nonlinear, reduced‐order models is presented. When experimental data are available, this approach can also quantify uncertainties in the associated high‐dimensional models. The main underlying idea is twofold. First, to substitute the deterministic reduced‐order basis (ROB) with a stochastic counterpart. Second, to construct the probability measure of the stochastic reduced‐order basis (SROB) on a subset of a compact Stiefel manifold in order to preserve some important properties of a ROB. The stochastic modeling is performed so that the probability distribution of the constructed SROB depends on a small number of hyperparameters. These are determined by solving a reduced‐order statistical inverse problem. The mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations. Its potential is demonstrated through several example problems from computational structural dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

A sequential constant‐stress accelerated life testing scheme and its Bayesian inference

In the analysis of accelerated life testing (ALT) data, some stress‐life model is typically used to relate results obtained at stressed conditions to those at use condition. For example, the Arrhenius model has been widely used for accelerated testing involving high temperature. Motivated by the fact that some prior knowledge of particular model parameters is usually available, this paper proposes a sequential constant‐stress ALT scheme and its Bayesian inference. Under this scheme, test at the highest stress is firstly conducted to quickly generate failures. Then, using the proposed Bayesian inference method, information obtained at the highest stress is used to construct prior distributions for data analysis at lower stress levels. In this paper, two frameworks of the Bayesian inference method are presented, namely, the all‐at‐one prior distribution construction and the full sequential prior distribution construction. Assuming Weibull failure times, we (1) derive the closed‐form expression for estimating the smallest extreme value location parameter at each stress level, (2) compare the performance of the proposed Bayesian inference with that of MLE by simulations, and (3) assess the risk of including empirical engineering knowledge into ALT data analysis under the proposed framework. Step‐by‐step illustrations of both frameworks are presented using a real‐life ALT data set. Copyright © 2008 John Wiley & Sons, Ltd.     

Proper Generalized Decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems

In this paper, the proper generalized decomposition (PGD) is used for model reduction in the solution of an inverse heat conduction problem within the Bayesian framework. Two PGD reduced order models are proposed and the approximation Error model (AEM) is applied to account for the errors between the complete and the reduced models. For the first PGD model, the direct problem solution is computed considering a separate representation of each coordinate of the problem during the process of solving the inverse problem. On the other hand, the second PGD model is based on a generalized solution integrating the unknown parameter as one of the coordinates of the decomposition. For the second PGD model, the reduced solution of the direct problem is computed before the inverse problem within the parameter space provided by the prior information about the parameters, which is required to be proper. These two reduced models are evaluated in terms of accuracy and reduction of the computational time on a transient three-dimensional two region inverse heat transfer problem. In fact, both reduced models result on substantial reduction of the computational time required for the solution of the inverse problem, and provide accurate estimates for the unknown parameter due to the application of the approximation error model approach.     

Bayesian transformed models for small area estimation

Sample surveys are usually designed and analyzed to produce estimates for larger areas. However, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To overcome such difficulties,borrowing strength from related small areas via modeling becomes an appropriate approach. In line with this, we propose hierarchical models with power transformations for improving the precision of small area predictions. The proposed methods are applied to satellite data in conjunction with survey data to estimate mean acreage under a specified crop for counties in Iowa.     

Optimal input design for online parameter estimation for aircraft with multiple control surfaces

An optimal input design method is proposed for online parameter estimation for aircraft with multiple control surfaces. The optimal input is designed considering the input and output constraints. These constraints are constructed based on a military standard, MIL-STD-8785C, viz., the flying qualities of a piloted aircraft. The accuracy of parameter estimation using the optimal input is compared with that of parameter estimation using conventional doublet/3211 inputs. Two online parameter estimation schemes are also considered to evaluate the performance of the designed optimal input: a Bayesian method based on the time domain and an equation-error method based on the frequency domain. The recursive form of the Bayesian method is also derived. Numerical simulations are performed, and the performance, convergence, and accuracy of two online parameter estimation schemes are compared.     

Sequential Bayesian technique: An alternative approach for software reliability estimation

This paper proposes a sequential Bayesian approach similar to Kalman filter for estimating reliability growth or decay of software. The main advantage of proposed method is that it shows the variation of the parameter over a time, as new failure data become available. The usefulness of the method is demonstrated with some real life data.     

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

This 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.     

Parameter estimation in a thermoelastic composite problem via adjoint formulation and model reduction

Advances in nondestructive material characterization are providing a wealth of information that could be exploited to gain insight into general aspects of material performance and, in particular, discover relationships between microstructure and thermo‐mechanical properties in polycrystalline and other complex composite materials. In order to facilitate the integration of such measurements into existing models, as well as inform new physics‐based predictions, we developed a C++/MPI computational framework for sensitivity analysis and parameter estimation. The framework utilizes a micro‐mechanical modeling based on fast Fourier transforms, direct and adjoint formulations, and Markov chain Monte Carlo sampling techniques. We illustrate the characteristics of this framework and demonstrate its utility by computing the residual stresses arising from thermal expansion of an elastic composite and using data from simulated experiments. We show that the availability of nondestructive 3‐D measurements is crucial to reduce the uncertainty in predictions, emphasizing the importance of an integrated experimental/modeling/data analysis approach for improved material characterization and design. Copyright © 2017 John Wiley & Sons, Ltd.     

Bayesian state and parameter estimation of uncertain dynamical systems

The focus of this paper is Bayesian state and parameter estimation using nonlinear models. A recently developed method, the particle filter, is studied that is based on stochastic simulation. Unlike the well-known extended Kalman filter, the particle filter is applicable to highly nonlinear models with non-Gaussian uncertainties. Recently developed techniques that improve the convergence of the particle filter simulations are introduced and discussed. Comparisons between the particle filter and the extended Kalman filter are made using several numerical examples of nonlinear systems. The results indicate that the particle filter provides consistent state and parameter estimates for highly nonlinear models, while the extended Kalman filter does not.     

Transport Map sampling with PGD model reduction for fast dynamical Bayesian data assimilation

The motivation of this work is to address real-time sequential inference of parameters with a full Bayesian formulation. First, the proper generalized decomposition (PGD) is used to reduce the computational evaluation of the posterior density in the online phase. Second, Transport Map sampling is used to build a deterministic coupling between a reference measure and the posterior measure. The determination of the transport maps involves the solution of a minimization problem. As the PGD model is quasi-analytical and under a variable separation form, the use of gradient and Hessian information speeds up the minimization algorithm. Eventually, uncertainty quantification on outputs of interest of the model can be easily performed due to the global feature of the PGD solution over all coordinate domains. Numerical examples highlight the performance of the method.