Affine Decomposition Research Articles

Parametric Model-Order Reduction for Radiation Transport Simulations Based on an Affine Decomposition of the Operators

This work presents a data-driven, projection-based parametric reduced-order model (ROM) for the neutral particle radiation transport (linear Boltzmann transport) equation. The ROM utilizes the method of snapshots with proper orthogonal decomposition. The novelty of the work is in the detailed proposal to exploit the parametrically affine transport operators to intrusively, yet efficiently, build the reduced transport operators in real time in a matrix-free manner compatible with sweep-based transport solvers. This affine-based ROM is applied to one-dimensional (1-D), two-dimensional (2-D), and 2-D multigroup transport benchmarks and is found to significantly outperform less intrusive ROMs in terms of speed for a desired accuracy level. The ROM has an 18.2 to 89.4 speedup with an error range of 0.0002% to 0.01% for the 1-D benchmark, a 1120× to 4870× speedup with an error range of 0.0009% to 0.01% for the 2-D benchmark, and a 54 600× to 399 800× speedup with an error range of 0.00022% to 0.01% for the multigroup 2-D benchmark. Even higher speedups are expected for three-dimensional multigroup transport problems.

A DEIM driven reduced basis method for the diffuse Stokes/Darcy model coupled at parametric phase-field interfaces

In this article, we develop a reduced basis method for efficiently solving the coupled Stokes/Darcy equations with parametric internal geometry. To accommodate possible changes in topology, we define the Stokes and Darcy domains implicitly via a phase-field indicator function. In our reduced order model, we approximate the parameter-dependent phase-field function with a discrete empirical interpolation method (DEIM) that enables affine decomposition of the associated linear and bilinear forms. In addition, we introduce a modification of DEIM that leads to non-negativity preserving approximations, thus guaranteeing positive-semidefiniteness of the system matrix. We also present a strategy for determining the required number of DEIM modes for a given number of reduced basis functions. We couple reduced basis functions on neighboring patches to enable the efficient simulation of large-scale problems that consist of repetitive subdomains. We apply our reduced basis framework to efficiently solve the inverse problem of characterizing the subsurface damage state of a complete in-situ leach mining site.

A discrete-to-continuum model of protein complexes

On the basis of a tensor representation of protein shape, obtained by an affine decomposition of residue velocity, we show how to identify actions at continuum scale for both single proteins and their complexes in terms of power equivalence. The approach constructs and justifies a continuum modeling of protein complexes, which avoids a direct, atomistic-based, simulation of the whole complex, rather it focuses (in a statistical sense) on a single protein and its interactions with the neighbors. In the resulting setting we also prove the existence of equilibrium configurations (native states) under large strains.

Approximate Piecewise Affine Decomposition of Neural Networks

Abstract Neural networks are increasingly being used for nonlinear system identification due to their flexibility and the constantly growing ecosystem of available tools and methods. However, there are concerns about the interpretability of these models. With the aim of regaining interpretability, while retaining the generalization capabilities of these model structures, this work presents a method to compute low complexity piecewise affine (PWA) functions that closely approximate neural networks, and demonstrates it on a nonlinear system identification benchmark based on data gathered from a F-16 jet. The approach enables researchers to take advantage of both the applicability of machine learning methods to large, high-dimensional datasets, and the numerous controller synthesis techniques available for PWA systems.

Efficient Algorithm for Finding Roots of Error-Locator Polynomials

A novel method for finding roots of polynomials over finite fields has been proposed. This method is based on the cyclotomic discrete Fourier transform algorithm. The improvement is achieved by using the normalized cyclic convolutions, which have a small complexity and allow matrix decomposition, as well as methods of adapting the truncated normalized cyclic convolutions calculation. For small values of degree of the error-locator polynomial the novel method has not only the smallest multiplicative complexity, but the full computational complexity of this method is also less than with other methods. Thus, the multiplicative complexity of the novel method in comparison to the method of affine decomposition (the Fedorenko-Trifonov method) is up to ten times less, although the additive complexity is approximately 10-15% more. The novel method has matrix representation convenient for implementation.

Fast solution of neutron diffusion problem with movement of control rods

Accurate and fast-running reduced order neutronics models are expected in many challenging applications such as multi-physics coupling analysis and data assimilations of reactors. The emphasis of the present study is to investigate an appropriate order reduction method for treating the movement of control rods. The empirical interpolation method is adopted to perform an affine decomposition to the spatial distribution of the cross sections with respect to the positions of control rods. The method can well recover the abrupt change of the cross sections near the end of control rods. With incorporation of the decomposed components, accurate and very efficient reduced order models can be constructed for the neutron diffusion problem with one-step or two-step movements of control rods.

Certified Reduced Basis VMS-Smagorinsky model for natural convection flow in a cavity with variable height

In this work we present a Reduced Basis VMS-Smagorinsky Boussinesq model, applied to natural convection problems in a variable height cavity, in which the buoyancy forces are involved. We take into account in this problem both physical and geometrical parametrizations, considering the Rayleigh number as a parameter, so as the height of the cavity. We perform an Empirical Interpolation Method to approximate the sub-grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters. We construct an a posteriori error estimator, based upon the Brezzi–Rappaz–Raviart theory, used in the greedy algorithm for the selection of the basis functions. Finally we present several numerical tests for different parameter configuration.

Data driven approximation of parametrized PDEs by reduced basis and neural networks

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial differential equation, but that the value of the physical parameters is unknown or difficult to be directly measured. Our method allows to estimate fields of interest, for instance temperature of a sample of material or velocity of a fluid, given data at a handful of points in the domain. We propose to accomplish this task with a neural network embedding a reduced basis solver as exotic activation function in the last layer. The reduced basis solver accounts for the underlying physical phenomenon and it is constructed from snapshots obtained from randomly selected values of the physical parameters during an expensive offline phase. The same full order solutions are then employed for the training of the neural network. As a matter of fact, the chosen architecture resembles an asymmetric autoencoder in which the decoder is the reduced basis solver and as such it does not contain trainable parameters. The resulting latent space of our autoencoder includes parameter-dependent quantities feeding the reduced basis solver, which – depending on the considered partial differential equation – are the values of the physical parameters themselves or the affine decomposition coefficients of the differential operators.

Robust Optimization of a Permanent-Magnet Synchronous Machine Considering Uncertain Driving Cycles

This work focuses on the robust optimization of a permanent magnet (PM) synchronous machine while considering a driving cycle. The robustification is obtained by considering uncertainties of different origins. Firstly, there are geometrical uncertainties caused by manufacturing inaccuracies. Secondly, there are uncertainties linked to different driving styles. The final set of uncertainties is linked to ambient parameters such as traffic and weather conditions. The optimization goal is to minimize the PM's volume while maintaining a desired machine performance measured by the energy efficiency over the driving cycle and the machine's maximal torque. The magnetic behavior of the machine is described by a partial differential equation (PDE) and is simulated by the finite-element method employing an affine decomposition to avoid reassembling of the system of equations due to the changing PM geometry. The Sequential Quadratic Programming algorithm is used for the optimization. Stochastic collocation is applied to compute moments of stochastic quantities. The robustness of the optimized configurations is validated by a Monte Carlo sampling. It is found that the uncertainties in driving style and road conditions have significant influence on the optimal PM configuration.

Non-intrusive proper generalized decomposition involving space and parameters: application to the mechanical modeling of 3D woven fabrics

In our former works we proposed different Model Order Reduction strategies for alleviating the complexity of computational simulations. In fact we proved that separated representations are specially appealing for addressing many issues, in particular, the treatment of 3D models defined in degenerated domains (those involving very different characteristic dimensions, like beams, plate and shells) as well as the solution of parametrized models for calculating their parametric solutions. However it was proved that the efficiency of solvers based on the construction of such separated representations strongly depends on the affine decompositions (separability) of operators, parameters and geometry. Even if our works proved that different techniques exists for performing such beneficial separation prior of applying the separated representation constructor, the complexity of the solver increases in certain circumstances too much, as the one involving the space separation of complex microstructures concerned by 3D woven fabrics. In this paper we explore an alternative route that allows circumventing the just referred difficulties. Thus, instead of following the standard procedure that consists of introducing the separated representation of the unknown field prior to discretize the models, the strategy here proposed consists of proceeding inversely: first the model is discretized and then the separated representation of the discrete unknown field is enforced. Such a procedure enables the consideration of very complex and non separable features, like complex domains, boundary conditions and microstructures as the ones concerned by homogenized models of complex and rich 3D woven fabrics. It will be proved that such a procedure can be also easily coupled with a non-intrusive treatment of the parametric dimensions by using a sparse hierarchical collocation technique.

From linear to nonlinear PGD-based parametric structural dynamics

The present paper analyzes different integration schemes of solid dynamics in the frequency domain involving the so-called Proper Generalized Decomposition – PGD. The last framework assumes for the solution a parametric dependency with respect to frequency. This procedure allowed introducing other parametric dependences related to loading, geometry, and material properties. However, in these cases, affine decompositions are required for an efficient computation of separated representations. A possibility for circumventing such difficulty consists in combining modal and harmonic analysis for defining an hybrid integration scheme. Moreover, such a procedure, as proved in the present work, can be easily generalized to address nonlinear parametric dynamics, as well as to solve problems with non-symmetric stiffness matrices, always operating in the domain of low frequencies.

Robust optimisation formulations for the design of an electric machine

In this study, two formulations for the robust optimisation of the size of the permanent magnet in a synchronous machine are discussed. The optimisation is constrained by a partial differential equation to describe the electromagnetic behaviour of the machine. The need for a robust optimisation procedure originates from the fact that optimisation parameters have deviations. The first approach, i.e. worst-case optimisation, makes use of local sensitivities. The second approach takes into account expectation values and standard deviations. The latter are associated with global sensitivities. The geometry parametrisation is elegantly handled thanks to the introduction of an affine decomposition. Since the stochastic quantities are determined by tools from uncertainty quantification (UQ) and thus require a lot of finite element evaluations, model order reduction is used in order to increase the efficiency of the procedure. It is shown that both approaches are equivalent if a linearisation is carried out. This finding is supported by the application on an electric machine. The optimisation algorithms used are sequential quadratic programming, particle swarm optimisation and genetic algorithm. While both formulations reduce the size of the magnets, the UQ-based optimisation approach is less pessimistic with respect to deviations and yields smaller magnets.

Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver

Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.

A three-scale offline–online numerical method for fluid flow in porous media

A new multiscale method combined with model order reduction is proposed for flow problems in a three-scale porous material. We derive an effective three-scale model that couples a macroscopic Darcy equation, a mesoscopic Stokes–Brinkman equation, and a microscopic Stokes equation. A corresponding three-scale numerical method is then derived using the finite element discretization with numerical quadrature, where the macroscopic and mesoscopic permeability is upscaled at quadrature points from mesoscopic and microscopic problems, respectively. The computational cost of solving numerous mesoscopic and microscopic flow problems is further reduced by applying a Petrov–Galerkin reduced basis method at the mesoscopic and microscopic scales. As there is no natural way to obtain an affine decomposition of the mesoscopic problems, which is instrumental for the efficiency of the model order reduction, we derive a mesoscopic solver that makes use of empirical interpolation techniques. A priori and a posteriori error estimates are derived for the new method that is also tested numerically to corroborate the theoretical convergence rates and illustrate its efficiency.

A reduced basis for a local high definition wind model

In this paper we present an application of the reduced basis method to a local high definition adjusted wind model. The model provides a precise description of the wind in 3D and takes into account topography and thermal gradients on the surface by solving only 2D linear equations; the buoyancy forces, slope effects, and mass conservation are also considered. The wind field is adjusted to the point measurements through an optimal control problem in which the wind flux acts as a control on the boundary. In order to use a reduced basis method, we consider an affine decomposition in terms of the parameter related to the friction coefficient and the wind measures at some given observation points. We also design an a posteriori error estimator that is needed to conduct our reduced basis process. Finally, two numerical examples are presented: a test problem and a real-data scenario, we corroborate the correct behavior of the method in both cases.

An Effective Approach of Teeth Segmentation within the 3D Cone Beam Computed Tomography Image Based on Deformable Surface Model

In order to extract the pixels of teeth from 3D Cone Beam Computed Tomography (CBCT) image, in this paper, a novel 3D segmentation approach based on deformable surface mode is developed for 3D tooth model reconstruction. Different forces are formulated to handle the segmentation problem by using different strategies. First, the proposed method estimates the deformation force of vertex model by simulating the deformation process of a bubble under the action of internal pressure and external force field. To handle the blurry boundary, a “braking force” is proposed deriving from the 3D gradient information calculated by transforming the Sobel operator into three-dimension representation. In addition, a “border reinforcement” strategy is developed for handling the cases with complicate structures. Moreover, the proposed method combines affine cell image decomposition (ACID) grid reparameterization technique to handle the unstable changes of topological structure and deformability during the deformation process. The proposed method was performed on 510 CBCT images. To validate the performance, the results were compared with those of two other well-studied methods. Experimental results show that the proposed approach had a good performance in handling the cases with complicate structures and blurry boundaries well, is effective to converge, and can successfully achieve the reconstruction task of various types of teeth in oral cavity.

Simultaneous empirical interpolation and reduced basis method for non-linear problems

In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parameterized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [9,8]. To deal with this issue, it is now standard to apply the EIM methodology [1,3] before deploying the Reduced Basis (RB) methodology. However, the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [3,2]. We propose a simultaneous EIM reduced basis algorithm, named SER, which provides a huge computational gain and requires as little as N+1 finite-element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [3].

Efficient model reduction of parametrized systems by matrix discrete empirical interpolation

In this work, we apply a Matrix version of the so-called Discrete Empirical Interpolation (MDEIM) for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations. Dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced-order models (ROMs). However, in many cases such an affine decomposition is not readily available, and must be recovered through (often) intrusive procedures, such as the empirical interpolation method (EIM) and its discrete variant DEIM. In this paper we show that MDEIM represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non-intrusive, efficient and purely algebraic way. We propose different strategies to combine MDEIM with a state approximation resulting either from a reduced basis greedy approach or Proper Orthogonal Decomposition. A posteriori error estimates accounting for the MDEIM error are also developed in the case of parametrized elliptic and parabolic equations. Finally, the capability of MDEIM to generate accurate and efficient ROMs is demonstrated on the solution of two computationally-intensive classes of problems occurring in engineering contexts, namely PDE-constrained shape optimization and parametrized coupled problems.

Implicit partitioning methods for unknown parameter sets

The key condition for the application of the Reduced Basis Method (RBM) to Parametrized Partial Differential Equations (PPDEs) is the availability of affine decompositions of the equations in parameter and space. The efficiency of the RBM depends on both the number of reduced basis functions and the number of affine terms. A possible way to reduce the costs is a partitioning of the parameter domain. One creates separate RB spaces (Haasdonk et al., Math. Comput.Model. Dyn. Syst. 17(4), 423---442, 2011) and affine decompositions (Eftang and Stamm, Int. J. Numer. Methods Eng. 90(4), 412---428, 2012) on each subdomain. Since the solutions are supposed to be smooth in parameter, the variation of the solutions on a subdomain becomes small and only few basis functions and affine terms are needed. Based upon the Empirical Interpolation Method (EIM) (Barrault et al., C. R. Math. Acad. Sci. Paris 339(9), 667---672, 2004 and Tonn, 2012), we generalize the existing partitioning concepts to arbitrary input functions with possibly unknown, high-dimensional, or even without direct parameter dependencies. No a-priori information about the input is necessary. We create affine decomposition and partitions without the knowledge of either an explicit description of the parameter domain or of the form of the partitions. The main idea is to perform several EIMs such that different parts of the family of possible input functions are covered. For a new input in the online stage, the coefficients of the resulting affine decomposition are used for the assignment to the appropriate EIM. An application includes PPDEs with stochastic influences (Chen et al., Math. Model. Numer. Anal. 48(4), 943---953, 2014, Haasdonk et al., SIAM/ASA J. Uncertain. Quantif. 1, 79---105, 2013 and Wieland, 2013). For some probability space, the parameter domain is now associated with the set of possible random outcomes. Hence, the elements of the parameter set are not parameters in classical sense and there often is no explicit description of the class of outcomes.

Affine Approximation for Moment Generating Function of Log-Normal Stochastic Volatility Model

While empirical studies have established that the log-normal stochastic volatility (SV) model is superior to its alternatives, the model does not allow for the analytical solutions available for affine models. To circumvent this, we show that the joint moment generating function (MGF) of the log-price and the quadratic variance (QV) under the log-normal SV model can be decomposed into a leading term, which is given by an exponential-affine form, and a residual term, whose estimate depends on the higher order moments of the volatility process. We prove that the second-order leading term is theoretically consistent with the expected values and covariance matrix of the log-price and the quadratic variance. We further extend this approach to the log-normal SV model with jumps. We use Fourier inversion techniques to value vanilla options on the equity and the QV and, by comparison to Monte Carlo simulations, we show that the second-order leading term is precise for the valuation of vanilla options. We generalize the affine decomposition to other non-affine stochastic volatility models with polynomial drift and volatility functions, and with jumps in the volatility process.