General Decomposition Framework Research Articles

Separated representation of the finite element solution of nonlinear magnetostatic problem based on non-intrusive Proper Generalized Decomposition

The Proper Generalized Decomposition has shown its efficiency to solve parameterized problems in nonlinear system events when it is combined with the Discrete Empirical Interpolation Method. Nevertheless, the solution of finite element model with the Proper Generalized Decomposition framework requires to have access to matrices and vectors of the discretized problem, which makes the method highly intrusive. In this context, based on a set of finite element solutions for a set of input parameters, a surrogate model can be developed applying a non-intrusive Proper Generalized Decomposition approach. The proposed non-intrusive approach is based on a canonical decomposition of the finite element solutions combined with an interpolation method. We then obtain a surrogate model approximating the finite element solutions for a wide range of parameters. The surrogate model, given its evaluation speed, can be used for real-time applications. In this paper, the proposed non-intrusive Proper Generalized Decomposition approach is employed to approximate a nonlinear magnetostatic problem and is applied to a single phase standard transformer and to a three-phase inductance.

A separated representation involving multiple time scales within the Proper Generalized Decomposition framework

Solutions of partial differential equations can exhibit multiple time scales. Standard discretization techniques are constrained to capture the finest scale to accurately predict the response of the system. In this paper, we provide an alternative route to circumvent prohibitive meshes arising from the necessity of capturing fine-scale behaviors. The proposed methodology is based on a time-separated representation within the standard Proper Generalized Decomposition, where the time coordinate is transformed into a multi-dimensional time through new separated coordinates, each representing one scale, while continuity is ensured in the scale coupling. For instance, when considering two different time scales, the governing Partial Differential Equation is commuted into a nonlinear system that iterates between the so-called microtime and macrotime, so that the time coordinate can be viewed as a 2D time. The macroscale effects are taken into account by means of a finite element-based macro-discretization, whereas the microscale effects are handled with unidimensional parent spaces that are replicated throughout the time domain. The resulting separated representation allows us a very fine time discretization without impacting the computational efficiency. The proposed formulation is explored and numerically verified on thermal and elastodynamic problems.

A Generalized Decomposition Framework for Large-Scale Transmission Expansion Planning

In this paper, we propose a scalable and configurable decomposition framework for solving large-scale transmission capacity expansion planning with security constraints under uncertainties. This framework is capable of using both progressive hedging and Benders decomposition algorithms to decompose and parallelize a large-scale problem both vertically and horizontally. A scenario bundling method is also developed to create bundles through three steps, i.e., classification, clustering, and grouping with the objective of maximizing similarity between bundles. This bundling method can improve both quality of results (decreasing optimality gap) and performance (reducing computational time) of the proposed framework. To verify capabilities of the proposed method, it is applied to a reduced ERCOT system with 3179 buses, 4458 branches, and 10 scenarios. The numerical result for this case study shows that the proposed framework can make solving large-scale problems tractable, and provides high quality results (with less than $1\%$ optimality gap) in a reasonable time (around 2.8 days).

On the Use of Generalized Volume Scattering Models for the Improvement of General Polarimetric Model-Based Decomposition

Recently, a general polarimetric model-based decomposition framework was proposed by Chen et al., which addresses several well-known limitations in previous decomposition methods and implements a simultaneous full-parameter inversion by using complete polarimetric information. However, it only employs four typical models to characterize the volume scattering component, which limits the parameter inversion performance. To overcome this issue, this paper presents two general polarimetric model-based decomposition methods by incorporating the generalized volume scattering model (GVSM) or simplified adaptive volume scattering model, (SAVSM) proposed by Antropov et al. and Huang et al., respectively, into the general decomposition framework proposed by Chen et al. By doing so, the final volume coherency matrix structure is selected from a wide range of volume scattering models within a continuous interval according to the data itself without adding unknowns. Moreover, the new approaches rely on one nonlinear optimization stage instead of four as in the previous method proposed by Chen et al. In addition, the parameter inversion procedure adopts the modified algorithm proposed by Xie et al. which leads to higher accuracy and more physically reliable output parameters. A number of Monte Carlo simulations of polarimetric synthetic aperture radar (PolSAR) data are carried out and show that the proposed method with GVSM yields an overall improvement in the final accuracy of estimated parameters and outperforms both the version using SAVSM and the original approach. In addition, C-band Radarsat-2 and L-band AIRSAR fully polarimetric images over the San Francisco region are also used for testing purposes. A detailed comparison and analysis of decomposition results over different land-cover types are conducted. According to this study, the use of general decomposition models leads to a more accurate quantitative retrieval of target parameters. However, there exists a trade-off between parameter accuracy and model complexity which constrains the physical validity of solutions and must be further investigated.

Approximation of constrained problems using the PGD method with application to pure Neumann problems

In this paper we introduce, analyze, and compare several approaches designed to incorporate a linear (or affine) constraint within the Proper Generalized Decomposition framework. We apply the considered methods and numerical strategies to two classes of problems: the pure Neumann case where the role of the constraint is to recover unicity of the solution; and the Robin case, where the constraint forces the solution to move away from the already existing unique global minimizer of the energy functional.

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to poly-tope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max-min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.

Efficient PGD-based dynamic calculation of non-linear soil behavior

Abstract Non-linear behavior of soils during a seismic event has a predominant role in current site response analysis. Soil response analysis consistently indicates that the stress–strain relationship of soils is non-linear and shows hysteresis. When focusing on forced response simulations, time integrations based on modal analysis are widely considered; however, parametric analysis, non-linear behavior and complex damping functions make difficult the online use of standard discretization strategies, e.g., those based on the use of finite element. In this paper, we propose a new harmonic analysis formulation, able to address forced response simulation of soils exhibiting their characteristic non-linear behavior. The solution can be evaluated in real-time from the offline construction of a parametric solution to the associated linearized problem within the Proper Generalized Decomposition framework.

Path‐space Motion Estimation and Decomposition for Robust Animation Filtering

AbstractRenderings of animation sequences with physics‐based Monte Carlo light transport simulations are exceedingly costly to generate frame‐by‐frame, yet much of this computation is highly redundant due to the strong coherence in space, time and among samples. A promising approach pursued in prior work entails subsampling the sequence in space, time, and number of samples, followed by image‐based spatio‐temporal upsampling and denoising. These methods can provide significant performance gains, though major issues remain: firstly, in a multiple scattering simulation, the final pixel color is the composite of many different light transport phenomena, and this conflicting information causes artifacts in image‐based methods. Secondly, motion vectors are needed to establish correspondence between the pixels in different frames, but it is unclear how to obtain them for most kinds of light paths (e.g. an object seen through a curved glass panel).To reduce these ambiguities, we propose a general decomposition framework, where the final pixel color is separated into components corresponding to disjoint subsets of the space of light paths. Each component is accompanied by motion vectors and other auxiliary features such as reflectance and surface normals. The motion vectors of specular paths are computed using a temporal extension of manifold exploration and the remaining components use a specialized variant of optical flow. Our experiments show that this decomposition leads to significant improvements in three image‐based applications: denoising, spatial upsampling, and temporal interpolation.

High-resolution thermal analysis at thermoplastic pre-impregnated acomposite interfaces

In situ consolidation of pre-impregnated thermoplastic composite requires as much as possible intimate contact as well as an appropriate molecular diffusion across the interface, where the intimate contact takes place. The last depends on the temperature that influences the molecular mobility. Thus, in order to evaluate the consolidation, first, we must describe accurately the composite surfaces and then, when placed in contact for manufacturing the laminate, a high-resolution simulation of the resulting thermal model must be accomplished to have access to the thermal history at the rough contact surface. In this paper, we proceed in three steps: (i) first the pre-preg surface is assimilated to a random process, a fractional Brownian motion, that can be characterized by two main descriptors, the topothesy and the fractal dimension; then, (ii) assuming the rough surface is represented by a Cantor set, the different parameters describing the fractal are determined from the statistical surface description; and finally, (iii) the resulting Cantor set representation will be considered as the geometrical support of a high-resolution thermal analysis accomplished within the proper generalized decomposition framework that allows exploring scales never until now attained.

Kinetic Theory Modeling and Efficient Numerical Simulation of Gene Regulatory Networks Based on Qualitative Descriptions

In this work, we begin by considering the qualitative modeling of biological regulatory systems using process hitting, from which we define its probabilistic counterpart by considering the chemical master equation within a kinetic theory framework. The last equation is efficiently solved by considering a separated representation within the proper generalized decomposition framework that allows circumventing the so-called curse of dimensionality. Finally, model parameters can be added as extra-coordinates in order to obtain a parametric solution of the model.

A separated representation of an error indicator for the mesh refinement process under the proper generalized decomposition framework

Today industries do not only require fast simulation techniques but also verification techniques for the simulations. The proper generalized decomposition (PGD) has been situated as a suitable tool for fast simulation for many physical phenomena. However, so far, verification tools for the PGD are under development. The PGD approximation error mainly comes from two different sources. The first one is related with the truncation of the PGD approximation and the second one is related with the discretization error of the underlying numerical technique. In this work we propose a fast error indicator technique based on recovery techniques, for the discretization error of the numerical technique used by the PGD technique, for refinement purposes.

General Polarimetric Model-Based Decomposition for Coherency Matrix

Orientation angle compensation was incorporated into model-based decomposition to cure overestimation of the volume scattering contribution for interpretation of polarimetric synthetic aperture radar (PolSAR) data. The compensation is based on rotating the coherency matrix to minimize the cross-polarization term. However, this processing cannot always guarantee that the double- and odd-bounce scattering components will be rotated back to zero orientation angle and left with zero cross-polarization power. As a result, built-up patches with large orientation angles may still suffer from the scattering mechanism ambiguity. In this paper, double- and odd-bounce scattering models were generalized to fit the cross-polarization and off-diagonal terms, by separating their independent orientation angles. A general decomposition framework is proposed that utilizes all elements of a coherency matrix. The residual minimization criterion is used for model inversion. All the model parameters are simultaneously obtained using a nonlinear least squares optimization technique. The manual intervention, branch conditions, and negative power issues are avoided. The performance and advantages of this approach are demonstrated and evaluated with spaceborne L-band ALOS/PALSAR and airborne X-band Pi-SAR PolSAR data sets. Comparison studies are also carried out and demonstrate that further improved decomposition performance is achieved by the proposed method, especially in oriented built-up areas.

On the convergence of inexact block coordinate descent methods for constrained optimization

We consider the problem of minimizing a smooth function over a feasible set defined as the Cartesian product of convex compact sets. We assume that the dimension of each factor set is huge, so we are interested in studying inexact block coordinate descent methods (possibly combined with column generation strategies). We define a general decomposition framework where different line search based methods can be embedded, and we state global convergence results. Specific decomposition methods based on gradient projection and Frank–Wolfe algorithms are derived from the proposed framework. The numerical results of computational experiments performed on network assignment problems are reported.

A General Primal-Dual Envelope Method for Convex Programming Problems

A general decomposition framework for solving large-scale convex programming problems is described. New algorithms are obtained and several known techniques are recovered as special cases, including Dantzig--Wolfe column generation, the finite envelope method of Rockafellar, and Zhu's primal-dual steepest descent method.

On augmented Lagrangian decomposition methods for multistage stochastic programs

A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented.