Perfectly Matched Layers equations for 3D acoustic wave propagation in heterogeneous media

International audienceThis work is dedicated to the analysis of Berenger PML method applied to the 3D linearized Euler equations without advection terms, with variable wave velocity and acoustic impedance. It is an extension of a previous work presented in a 2D context [8]. The 3D linearized Euler equations are used to simulate propagation of acoustic waves beneath the subsurface. We propose an analysis of these equations in a general heterogeneous context, based on a priori error estimates. Following the method introduced by M ́ tral and Vacus [9], we derive an augmented system from the original one, involving the primitive unknowns and their first order spatial derivatives. We define a symetrizer for this augmented system. This allows to compute energy estimates in the three following cases: the Cauchy problem, the half-space problem with a non homogeneous Dirichlet boundary condition and finally the transmission problem between two half-spaces separated by an impedance discontinuity

Applying Gauss-Newton and Exact Newton method to Full Waveform Inversion

International audienceFull Waveform Inversion (FWI) applications classically rely on efficient first-order optimization schemes, as the steepest descent or the nonlinear conjugate gradient optimization. However, second-order information provided by the Hessian matrix is proven to give a useful help in the scaling of the FWI problem and in the speed-up of the optimization. In this study, we propose an efficient matrix-free Hessian-vector formalism, that should allow to tackle Gauss-Newton (GN) and Exact-Newton (EN) optimization for large and realistic FWI targets. Our method relies on general second order adjoint formulas, based on a Lagrangian formalism. These formulas yield the possibility of computing Hessian-vector products at the cost of 2 forward simulations per shot. In this context, the computational cost (per shot) of one GN or one EN nonlinear iteration amounts to the resolution of 2 forward simulations for the computation of the gradient plus 2 forward simulations per inner linear conjugate gradient iteration. A numerical test is provided, emphasizing the possible improvement of the resolution when accounting for the exact Hessian in the inversion algorithm

Optimization for engine calibration

International audienceNowadays, automotive manufacturers are submitted to strong constraints in engine calibration such as: low fuel consumption, emission-control legislation and driver requests for driving comfort and performances. These constraints lead to an increasing complexity of the engines and thus an increasing number of parameters to be tuned, making the empirical engine calibration by a scan of parameter values impossible at engine test-bench. New methodologies in automated engine calibration based on statistics and optimization have emerged in order to limit the number of experimental tests to be run. The optimization problem of engine calibration consists in the determination of engine tuning parameters that minimize the cumulated fuel consumption and pollutant emissions on a driving cycle generally associated with legislation norms. This cycle is decomposed in a set of stationary operating points of the engine characterized by its speed and its torque (the transient behaviors of the engine are not taken into account in the stabilized calibration). Then, the optimal tuning parameters of the engine should be defined for each operating points, the functions defining these parameters on the whole engine operating domain are called the engine maps. These two-dimensional optimal engine maps are then integrated in the engine control unit in the vehicle. We illustrate the difficulties associated with this application and propose adapted optimization methodologies: LoLiMoT models for engine map parameterization in order to handle intrinsic constraints on the map regularity, multi-objective optimization method based on CMA-ES approach. Finally , application on real dataset obtained at IFP automated test-bench for a diesel engine are presented. 2. Keywords: Engine calibration, LoLiMoT, Multi-objective optimization, Evolutionary algorithm 3. Introduction Engine calibration consists in fulfilling the engine tuning maps that are used in engine controls of the vehicle, i.e. in defining the optimal tuning of parameters used by engine control strategies. Due to the highly increased number of these parameters (especially for diesel engines but spark ignition engines are following the same trend) and the reduction of the development schedule available for the calibration process, manual tuning of engine parameters is now replaced by mathematically assisted calibration process. Such a process is based on the design of experiments with associated modeling methods, in order to reduce the number of tests used to build engine response models depending on engine control parameters, and optimization techniques to determine the optimal settings within the model definition domain. In order to perform the tests in a more productive way, these mathematical techniques are generally associated with test automation, requiring well controlled measurement methods and reliable test equipments. This paper describes the optimization methods developed for this application and illustrates their effectiveness on a real case of a common rail diesel Engine. The first section introduces the classical steps of the calibration process and discusses the associated difficulties. In the second section, we propose the Multi-Objective Covariance-Adaptation Evolutionary Strategy method for solving the optimization problem associated with a given engine operating point defined by the engine speed and the engine load. In the third part, an integrated approach is proposed in order to directly optimize the engine maps on the whole driving cycle (associated with legislation norms) instead of the individual optimization of each engine operating point. 4. Engine calibration 4.1. Sketch of the engine calibration process The emission calibration workflow is classically divided into four steps: 1. a preliminary phase consisting in choosing a sample of operating points (referred to as OP in the

Engine calibration: multi-objective constrained optimization of engine maps

International audienceWe present two new approaches to address the optimization problem associated with engine calibration. In this area, the tuning parameters are traditionally determined in a local way, i.e., at each engine operating point, via a single-objective minimization problem. To overcome these restrictions, the first method we propose is able to cope with several objective functions simultaneously in the local formulation. The second method we put forward relies on a global formulation, which allows the whole driving cycle to be taken into account while remaining single-objective. At the practical level, the two methods are implemented by combining various existing techniques such as the LoLiMoT (Local Linear Model Tree) parameterization and the MO-CMA-ES (Multi-Objective Covariance Matrix Adaptation Evolution Strategy) algorithm. A better compromise appears to be achieved on real case applications. Keywords Engine calibration · Response surface · LoLiMoT · Multi-objective optimization · Evolutionary algorithm Nomenclature Abbreviations (by alphabetical order

A review of the use of optimal transport distances for high resolution seismic imaging based on the full waveform

We consider the high-resolution seismic imaging method called full-waveform inversion (FWI). FWI is a data fitting method aimed at inverting for subsurface mechanical parameters. Despite the large adoption of FWI by the academic and industrial communities, and many successful results, FWI still suffers from severe limitations. From a mathematical standpoint, FWI is a large scale PDE-constrained optimization problem. The misfit function that is used, which measures the discrepancy between observed seismic data and data calculated through the solution of a wave propagation problem, is non-convex. After discretization, the size of the FWI problem requires the use of local optimization solvers, which are prone to converge towards local minima. Thus the success of FWI strongly depends on the choice of the initial model to ensure the convergence towards the global minimum of the misfit function. This limitation has been the motivation for a large variety of strategies. Among the different methods that have been investigated, the use of optimal transport (OT) distances-based misfit functions has been recently promoted. The leading idea is to benefit from the inherent convexity of OT distances with respect to dilation and translation to render the FWI problem more convex. However, the application of OT distances in the framework of FWI is not straightforward, as seismic data is signed, while OT has been developed for the comparison of probability measures. The purpose of this study is to review two methods that were developed to overcome this difficulty. Both have been successfully applied to field data in an industrial framework. Both make it possible to better exploit the seismic data, alleviating the sensitivity to the initial model and to various conventional workflow steps, and reducing the uncertainty attached to the subsurface mechanical parameters inversion.Comment: 18 figure

Inner product preconditioned trust-region methods for frequency-domain full waveform inversion

Full waveform inversion is a seismic imaging method which requires to solve a large-scale minimization problem, typically through local optimization techniques. Most local optimization methods can basically be built up from two choices: the update direction and the strategy to control its length. In the context of full waveform inversion, this strategy is very often a line search. We here propose to use instead a trust-region method, in combination with non-standard inner products which act as preconditioners. More specifically, a line search and several trust-region variants of the steepest descent, the limited memory BFGS algorithm and the inexact Newton method are presented and compared. A strong emphasis is given to the inner product choice. For example, its link with preconditioning the update direction and its implication in the trust-region constraint are highlighted. A first numerical test is performed on a 2D synthetic model then a second configuration, containing two close reflectors, is studied. The latter configuration is known to be challenging because of multiple reflections. Based on these two case studies, the importance of an appropriate inner product choice is highlighted and the best trust-region method is selected and compared to the line search method. In particular we were able to demonstrate that using an appropriate inner product greatly improves the convergence of all the presented methods and that inexact Newton methods should be combined with trust-region methods to increase their convergence speed

A trust-region Newton method for frequency-domain full waveform inversion

editorial reviewedExploiting Hessian information greatly enhances the convergence of full waveform inversion. A theoretically simple way to incorporate these second-order derivatives is to minimize the misfit using Newton methods. In practice however the pure Newton method is too computationally intensive to implement, because it requires inverting the Hessian operator. In addition, the misfit is not necessarily quadratic, thus the exact Newton direction is not necessarily appropriate. Consequently, it is natural to turn to inexact Newton methods, where the search direction is constructed iteratively to approximate the pure Newton direction. The bottleneck of these methods lies in the compromise to find between a direction built in few iterations, but which hardly takes the Hessian into account and a nearly exact direction which is very expensive to compute. In this work we present an inexact Newton method based on a particular trust-region algorithm, in the context of frequency-domain full waveform inversion. A numerical test is performed on the Marmousi model to compare convergence speeds with a line search based inexact Newton algorithm. This illustrates that the trust-region method is more robust and provides faster convergence for an adequate choice of trust-region parameters

Accounting robustly for instantaneous chemical equilibriums in reactive transport: A numerical method and its application to liquid-liquid extraction modeling

International audienceReactive transport equations are used in numerous application fields: CO2 or nuclear waste storage monitoring, separation process in chemical engineering. We present a general method to account robustly for instantaneous equilibriums in reactive transport. This method is adapted to all kind of hydraulic transport models including 1D to 3D convection-diffusion equations. This leads to the resolution of a bound constrained system of Differential Algebraic Equations (DAE). The algebraic constraints come from the adjunction of mass action laws related to the equilibriums, whereas the bounds account for the positivity of the computed quantities. In order to solve the numerical system associated with our method, we use an adaptation of the DASSL solver, CDASSL, that can handle the resolution of bound constrained DAE systems. We present an application of this method to liquid-liquid extraction modeling. Numerical experiments demonstrate the interest of using the CDASSL solver to ensure the bound constraints are satisfied