Local proper generalized decomposition

One of the main difficulties that a reduced‐order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (proper orthogonal decomposition, reduced basis) and a priori [proper generalized decomposition (PGD)] model order reduction. Early approaches to solve it include the construction of local reduced‐order models in the framework of POD. We present here an extension of local models in a PGD—and thus, a priori—context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced‐order models, in contrast to most proper orthogonal decomposition local approximations. The resulting method can be seen as a sort of a priori manifold learning or nonlinear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.     

Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

We present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.     

Fast Bayesian approach for parameter estimation

This paper presents two techniques, i.e. the proper orthogonal decomposition (POD) and the stochastic collocation method (SCM), for constructing surrogate models to accelerate the Bayesian inference approach for parameter estimation problems associated with partial differential equations. POD is a model reduction technique that derives reduced‐order models using an optimal problem‐adapted basis to effect significant reduction of the problem size and hence computational cost. SCM is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations. The utility of the techniques is assessed on the non‐linear inverse problem of probabilistically calibrating scalar Robin coefficients from boundary measurements arising in the quenching process and non‐destructive evaluation. A hierarchical Bayesian model that handles flexibly the regularization parameter and the noise level is employed, and the posterior state space is explored by the Markov chain Monte Carlo. The numerical results indicate that significant computational gains can be realized without sacrificing the accuracy. Copyright © 2008 John Wiley & Sons, Ltd.     

Galerkin reduced‐order modeling scheme for time‐dependent randomly parametrized linear partial differential equations

In this paper, we consider the problem of constructing reduced‐order models of a class of time‐dependent randomly parametrized linear partial differential equations. Our objective is to efficiently construct a reduced basis approximation of the solution as a function of the spatial coordinates, parameter space, and time. The proposed approach involves decomposing the solution in terms of undetermined spatial and parametrized temporal basis functions. The unknown basis functions in the decomposition are estimated using an alternating iterative Galerkin projection scheme. Numerical studies on the time‐dependent randomly parametrized diffusion equation are presented to demonstrate that the proposed approach provides good accuracy at significantly lower computational cost compared with polynomial chaos‐based Galerkin projection schemes. Comparison studies are also made against Nouy's generalized spectral decomposition scheme to demonstrate that the proposed approach provides a number of computational advantages. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems

The identification of the geological structure from seismic data is formulated as an inverse problem. The properties and the shape of the rock formations in the subsoil are described by material and geometric parameters, which are taken as input data for a predictive model. Here, the model is based on the Helmholtz equation, describing the acoustic response of the system for a given wave length. Thus, the inverse problem consists in identifying the values of these parameters such that the output of the model agrees the best with observations. This optimization algorithm requires multiple queries to the model with different values of the parameters. Reduced order models are especially well suited to significantly reduce the computational overhead of the multiple evaluations of the model. In particular, the proper generalized decomposition produces a solution explicitly stating the parametric dependence, where the parameters play the same role as the physical coordinates. A proper generalized decomposition solver is devised to inexpensively explore the parametric space along the iterative process. This exploration of the parametric space is in fact seen as a post‐process of the generalized solution. The approach adopted demonstrates its viability when tested in two illustrative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Proper generalized decomposition of a geometrically parametrized heat problem with geophysical applications

The solution of a steady thermal multiphase problem is assumed to be dependent on a set of parameters describing the geometry of the domain, the internal interfaces and the material properties. These parameters are considered as new independent variables. The problem is therefore stated in a multidimensional setup. The proper generalized decomposition (PGD) provides an approximation scheme especially well suited to preclude dramatically increasing the computational complexity with the number of dimensions. The PGD strategy is reviewed for the standard case dealing only with material parameters. Then, the ideas presented in [Ammar et al., “Parametric solutions involving geometry: A step towards efficient shape optimization.” Comput. Methods Appl. Mech. Eng., 2014; 268 :178–193] to deal with parameters describing the domain geometry are adapted to a more general case including parametrization of the location of internal interfaces. Finally, the formulation is extended to combine the two types of parameters. The proposed strategy is used to solve a problem in applied geophysics studying the temperature field in a cross section of the Earth crust subsurface. The resulting problem is in a 10-dimensional space, but the PGD solution provides a fairly accurate approximation (error ≤1%) using less that 150 terms in the PGD expansion. Copyright © 2015 John Wiley & Sons, Ltd.     

Randomized residual-based error estimators for the proper generalized decomposition approximation of parametrized problems

This article introduces a novel error estimator for the proper generalized decomposition (PGD) approximation of parametrized equations. The estimator is intrinsically random: it builds on concentration inequalities of Gaussian maps and an adjoint problem with random right-hand side, which we approximate using the PGD. The effectivity of this randomized error estimator can be arbitrarily close to unity with high probability, allowing the estimation of the error with respect to any user-defined norm as well as the error in some quantity of interest. The performance of the error estimator is demonstrated and compared with some existing error estimators for the PGD for a parametrized time-harmonic elastodynamics problem and the parametrized equations of linear elasticity with a high-dimensional parameter space.     

The proper generalized decomposition for the simulation of delamination using cohesive zone model

The use of cohesive zone models is an efficient way to treat the damage especially when the crack path is known a priori. It is the case in the modeling of delamination in composite laminates. However, the simulations using cohesive zone models are expensive in a computational point of view. When using implicit time integration or when solving static problems, the non‐linearity related to the cohesive model requires many iteration before reaching convergence. In explicit approaches, an important number of iterations are also needed because of the time step stability condition. In this article, a new approach based on a separated representation of the solution is proposed. The proper generalized decomposition is used to build the solution. This technique coupled with a cohesive zone model allows a significant reduction of the computational cost. The results approximated with the proper generalized decomposition are very close the ones obtained using the classical finite element approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Real‐time direct integration of reduced solid dynamics equations

A new method is developed here for the real‐time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black‐box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so‐called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.     

A new bi-fidelity model reduction method for Bayesian inverse problems

This work presents a new bi-fidelity model reduction approach to the inverse problem under the framework of Bayesian inference. A low-rank approximation is introduced to the solution of the corresponding forward problem and admits a variable-separation form in terms of stochastic basis functions and physical basis functions. The calculation of stochastic basis functions is computationally predominant for the low-rank expression. To significantly improve the efficiency of constructing the low-rank approximation, we propose a bi-fidelity model reduction based on a novel variable-separation method, where a low-fidelity model is used to compute the stochastic basis functions and a high-fidelity model is used to compute the physical basis functions. The low-fidelity model has lower accuracy but efficient to evaluate compared with the high-fidelity model; it accelerates the derivative of recursive formulation for the stochastic basis functions. The high-fidelity model is computed in parallel for a few samples scattered in the stochastic space when we construct the high-fidelity physical basis functions. The required number of forward model simulations in constructing the basis functions is very limited. The bi-fidelity model can be constructed efficiently while retaining good accuracy simultaneously. In the proposed approach, both the stochastic basis functions and physical basis functions are calculated using the model information. This implies that a few basis functions may accurately represent the model solution in high-dimensional stochastic spaces. The bi-fidelity model reduction is applied to Bayesian inverse problems to accelerate posterior exploration. A few numerical examples in time-fractional derivative diffusion models are carried out to identify the smooth field and channel-structured field in porous media in the framework of Bayesian inverse problems.     

Non-incremental boundary element discretization of parabolic models based on the use of the proper generalized decompositions

There are many ways to solve space–time linear parabolic partial differential equations by using the boundary element method (BEM). In general, standard techniques make use of an incremental strategy. In this paper we propose a novel alternative of efficient non-incremental solution strategy for that kind of models. The proposed technique combines the use of the BEM with a proper generalized decomposition (PGD) that allows a space–time separated representation of the unknown field within a non-incremental integration scheme.     

Hyper‐reduction of mechanical models involving internal variables

We propose to improve the efficiency of the computation of the reduced‐state variables related to a given reduced basis. This basis is supposed to be built by using the snapshot proper orthogonal decomposition (POD) model reduction method. In the framework of non‐linear mechanical problems involving internal variables, the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model. This drawback is due to the fact that, using a Galerkin formulation, the size of the reduced basis has no effect on the complexity of the constitutive equations. In this paper we show how a reduced‐basis approximation and a Petrov–Galerkin formulation enable to reduce the computational effort related to the internal variables. The key concept is a reduced integration domain where the integration of the constitutive equations is performed. The local computations being not made over the entire domain, we extrapolate the computed internal variable over the full domain using POD vectors related to the internal variables. This paper shows the improvement of the computational saving obtained by the hyper‐reduction of the elasto‐plastic model of a simple structure. Copyright © 2008 John Wiley & Sons, Ltd.     

Certified real-time shape optimization using isogeometric analysis,PGD model reduction,and a posteriori error estimation

In this paper, we address the effective and accurate solution of problems with parameterized geometry. Considering the attractive framework of isogeometric analysis, which enables a natural and flexible link between computer-aided design and simulation tools, the parameterization of the geometry is defined on the mapping from the isogeometric analysis parametric space to the physical space. From the subsequent multidimensional problem, model reduction based on the proper generalized decomposition technique with off-line/online steps is introduced in order to describe the resulting manifold of parametric solutions with reduced CPU cost. Eventually, a posteriori estimation of various error sources inheriting from discretization and model reduction is performed in order to control the quality of the approximate solution, for any geometry, and feed a robust adaptive algorithm that optimizes the computational effort for prescribed accuracy. The overall approach thus constitutes an effective and reliable numerical tool for shape optimization analyses. Its performance is illustrated on several two- and three-dimensional numerical experiments.     

Parametric uncertainty quantification using proper generalized decomposition applied to neutron diffusion

In this paper, a proper generalized decomposition (PGD) approach is employed for uncertainty quantification purposes. The neutron diffusion equation with external sources, a diffusion-reaction problem, is used as the parametric model. The uncertainty parameters include the zone-wise constant material diffusion and reaction coefficients as well as the source strengths, yielding a large uncertain space in highly heterogeneous geometries. The PGD solution, parameterized in all uncertain variables, can then be used to compute mean, variance, and more generally probability distributions of various quantities of interest. In addition to parameterized properties, parameterized geometrical variations of three-dimensional models are also considered in this paper. To achieve and analyze a parametric PGD solution, algorithms are developed to decompose the model's parametric space and semianalytically integrate solutions for evaluating statistical moments. Varying dimensional problems are evaluated to showcase PGD's ability to solve high-dimensional problems and analyze its convergence.     

A Computational Inverse Technique to Determine the Dynamic Constitutive Model Parameters of Concrete

In this paper, a computational inverse technique is presented to determine the constitutive parameters of concrete based on the penetration experiments. In this method, the parameter identification problem is formulated as an inverse problem, in which the parameters of the constitutive model can be characterized through minimizing error functions of the penetration depth measured in experiments and that computed by forward solver LS-DYNA. To reduce the time for forward calculation during the inverse procedure, radial basis function approximate model is used to replace the actual computational model. In order to improve the accuracy of approximation model, a local-densifying method combined with RBF approximation model is adopted. The intergeneration projection genetic algorithm is employed as the inverse solver. Through the application of this method, the parameters of HJC constitutive model are determined. Results show that the identified constitutive parameters' computational penetration depth and projectile deceleration-time curves are closely in accordance with experimental data. The proposed inverse approach is a potentially useful tool to effectively help identify material parameters.     

Galerkin lumped parameter methods for transient problems

In this paper we propose a general methodology to obtain lumped parameter models for systems governed by parabolic partial differential equations which we call Galerkin lumped parameter methods. The idea consists of decomposing the computational domain into several subdomains connected through so‐called ports. Then a time‐independent adapted reduced basis is introduced by numerically solving several elliptic problems in each subdomain. The proposed lumped parameter model is the Galerkin approximation of the original problem in the space spanned by this basis. The relationship of these methods with classical lumped parameter models is analyzed. Numerical results are shown as well as a comparison of the solution obtained with the lumped model and the ‘exact’ one computed by standard finite element procedures. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient mold cooling optimization by using model reduction

Optimization and inverse identification are two procedures usually encountered in many industrial processes reputed gourmand for the computing time view point. In fact, optimization implies to propose a trial solution whose accuracy is then evaluated, and if needed it must be updated in order to minimize a certain cost function. In the case of mold cooling optimization the evaluation of the solution quality needs the solution of a thermal model, in the whole domain and during the thermal history. Thus, the optimization process needs several iterations and then the computational cost can become enormous. In this work we propose the use of model reduction for accomplishing this kind of simulations. Thus, only one thermal model is solved using the standard discretization technique. After that, the most important modes defining the temperature evolution are extracted by invoking the proper orthogonal decomposition, and all the other thermal model solutions are performed by using the reduced order approximation basis just extracted. The CPU time savings can be impressive.