Maximum entropy snapshot sampling for reduced basis modelling

Purpose The maximum entropy snapshot sampling (MESS) method aims to reduce the computational cost required for obtaining the reduced basis for the purpose of model reduction. Hence, it can significantly reduce the original system dimension whilst maintaining an adequate level of accuracy. The purpose of this paper is to show how these beneficial results are obtained. Design/methodology/approach The so-called MESS method is used for reducing two nonlinear circuit models. The MESS directly reduces the number of snapshots by recursively identifying and selecting the snapshots that strictly increase an estimate of the correlation entropy of the considered systems. Reduced bases are then obtained with the orthogonal-triangular decomposition. Findings Two case studies have been used for validating the reduction performance of the MESS. These numerical experiments verify the performance of the advocated approach, in terms of computational costs and accuracy, relative to gappy proper orthogonal decomposition. Originality/value The novel MESS has been successfully used for reducing two nonlinear circuits: in particular, a diode chain model and a thermal-electric coupled system. In both cases, the MESS removed unnecessary data, and hence, it reduced the snapshot matrix, before calling the QR basis generation routine. As a result, the QR-decomposition has been called on a reduced snapshot matrix, and the offline stage has been significantly scaled down, in terms of central processing unit time.

Combining spectral and POD modes to improve error estimation of numerical model reduction for porous media

Numerical model reduction (NMR) is used to solve the microscale problem that arises from computational homogenization of a model problem of porous media with displacement and pressure as unknown fields. The reduction technique and an associated error estimator for the NMR error have been presented in prior work, where both spectral decomposition (SD) and proper orthogonal decomposition (POD) were used to construct the reduced basis. It was shown that the POD basis performs better w.r.t. minimizing the residual, but the SD basis has some advantageous properties for the estimator. Since it is the estimated error that will govern the error control, the most efficient procedure is the one that results in the lowest error bound. The main contribution of this paper is further development of the previous work with a proposed combined basis constructed using both SD and POD modes together with an adaptive mode selection strategy. The performance of the combined basis is compared to (i) the pure SD basis and (ii) the pure POD basis via numerical examples. The examples show that it is possible to find a combination of SD/POD modes which is improved, i.e. it yields a smaller estimate, compared to the cases of pure SD or pure POD.

How biased are our models? – a case study of the alpine region

Geophysical process simulations play a crucial role in the understanding of the subsurface. This understanding is required to provide, for instance, clean energy sources such as geothermal energy. However, the calibration and validation of the physical models heavily rely on state measurements such as temperature. In this work, we demonstrate that focusing analyses purely on measurements introduces a high bias. This is illustrated through global sensitivity studies. The extensive exploration of the parameter space becomes feasible through the construction of suitable surrogate models via the reduced basis method, where the bias is found to result from very unequal data distribution. We propose schemes to compensate for parts of this bias. However, the bias cannot be entirely compensated. Therefore, we demonstrate the consequences of this bias with the example of a model calibration.

On Rational Krylov and Reduced Basis Methods for Fractional Diffusion

We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function 𝑧 −𝑠 with 𝑧 ranging over the spectral interval of the spatial discretization matrix. We prove convergence rates for this method and demonstrate numerically that it is competitive with or superior to many methods from the reduced basis, rational Krylov, and direct rational approximation classes. We provide numerical tests for some elliptic fractional diffusion model problems.

A Training Set Subsampling Strategy for the Reduced Basis Method

We present a subsampling strategy for the offline stage of the Reduced Basis Method. The approach is aimed at bringing down the considerable offline costs associated with using a finely-sampled training set. The proposed algorithm exploits the potential of the pivoted QR decomposition and the discrete empirical interpolation method to identify important parameter samples. It consists of two stages. In the first stage, we construct a low-fidelity approximation to the solution manifold over a fine training set. Then, for the available low-fidelity snapshots of the output variable, we apply the pivoted QR decomposition or the discrete empirical interpolation method to identify a set of sparse sampling locations in the parameter domain. These points reveal the structure of the parametric dependence of the output variable. The second stage proceeds with a subsampled training set containing a by far smaller number of parameters than the initial training set. Different subsampling strategies inspired from recent variants of the empirical interpolation method are also considered. Tests on benchmark examples justify the new approach and show its potential to substantially speed up the offline stage of the Reduced Basis Method, while generating reliable reduced-order models.

Spatialized Epidemiological Forecasting applied to Covid-19 Pandemic at Departmental Scale in France

In this paper, we present a spatialized extension of a SIR model that accounts for undetected infections and recoveries as well as the load on hospital services. The spatialized compartmental model we introduce is governed by a set of partial differential equations (PDEs) defined on a spatial domain with complex boundary. We propose to solve the set of PDEs defining our model by using a meshless numerical method based on a finite difference scheme in which the spatial operators are approximated by using radial basis functions. Such an approach is reputed as flexible for solving problems on complex domains. Then we calibrate our model on the French department of Isère during the first period of lockdown, using daily reports of hospital occupancy in France. Our methodology allows to simulate the spread of Covid-19 pandemic at a departmental level, and for each compartment. However, the simulation cost prevents from online short-term forecast. Therefore, we propose to rely on reduced order modeling tools to compute short-term forecasts of infection number. The strategy consists in learning a time-dependent reduced order model with few compartments from a collection of evaluations of our spatialized detailed model, varying initial conditions and parameter values. A set of reduced bases is learnt in an offline phase while the projection on each reduced basis and the selection of the best projection is performed online, allowing short-term forecast of the global number of infected individuals in the department.

A new entropy-variable-based discretization method for minimum entropy moment approximations of linear kinetic equations

In this contribution we derive and analyze a new numerical method for kinetic equations based on a variable transformation of the moment approximation. Classical minimum-entropy moment closures are a class of reduced models for kinetic equations that conserve many of the fundamental physical properties of solutions. However, their practical use is limited by their high computational cost, as an optimization problem has to be solved for every cell in the space-time grid. In addition, implementation of numerical solvers for these models is hampered by the fact that the optimization problems are only well-defined if the moment vectors stay within the realizable set. For the same reason, further reducing these models by, e.g., reduced-basis methods is not a simple task. Our new method overcomes these disadvantages of classical approaches. The transformation is performed on the semi-discretized level which makes them applicable to a wide range of kinetic schemes and replaces the nonlinear optimization problems by inversion of the positive-definite Hessian matrix. As a result, the new scheme gets rid of the realizability-related problems. Moreover, a discrete entropy law can be enforced by modifying the time stepping scheme. Our numerical experiments demonstrate that our new method is often several times faster than the standard optimization-based scheme.