A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems

We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box. The method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme [14] in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad-hoc fixes, such as covariance localization, are implemented. We provide a numerical investigation of the conditions under which the proposed method inherits the regularizing properties of the LM scheme of [14]. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method. The numerical investigation is carried out with synthetic experiments on two model inverse problems: (i) identification of conductivity on a Darcy flow model and (ii) electrical impedance tomography with the complete electrode model. We further demonstrate the potential application of the method in solving shape identification problems by means of a level-set approach for the parameterization of unknown geometries

Iterative regularization for ensemble data assimilation in reservoir models

We propose the application of iterative regularization for the development of ensemble methods for solving Bayesian inverse problems. In concrete, we construct (i) a variational iterative regularizing ensemble Levenberg-Marquardt method (IR-enLM) and (ii) a derivative-free iterative ensemble Kalman smoother (IR-ES). The aim of these methods is to provide a robust ensemble approximation of the Bayesian posterior. The proposed methods are based on fundamental ideas from iterative regularization methods that have been widely used for the solution of deterministic inverse problems [21]. In this work we are interested in the application of the proposed ensemble methods for the solution of Bayesian inverse problems that arise in reservoir modeling applications. The proposed ensemble methods use key aspects of the regularizing Levenberg-Marquardt scheme developed by Hanke [16] and that we recently applied for history matching in [18]. In the case where the forward operator is linear and the prior is Gaussian, we show that the proposed IR-enLM and IR-ES coincide with standard randomized maximum likelihood (RML) and the ensemble smoother (ES) respectively. For the general nonlinear case, we develop a numerical framework to assess the performance of the proposed ensemble methods at capturing the posterior. This framework consists of using a state-of-the art MCMC method for resolving the Bayesian posterior from synthetic experiments. The resolved posterior via MCMC then provides a gold standard against to which compare the proposed IR-enLM and IR-ES. We show that for the careful selection of regularization parameters, robust approximations of the posterior can be accomplished in terms of mean and variance. Our numerical experiments showcase the advantage of using iterative regularization for obtaining more robust and stable approximation of the posterior than standard unregularized methods

Groupoid symmetry and constraints in general relativity

When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold M\Sigma\ of riemannian metrics on a Cauchy hypersurface \Sigma. As in every lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection.Comment: 22 pages, major revisio

Filter Based Methods For Statistical Linear Inverse Problems

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory

Adaptive regularisation for ensemble Kalman inversion

We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the early stopping of the scheme. In contrast to existing approaches, our parameter choice does not rely on additional tuning parameters which often have severe effects on the efficiency of EKI. We motivate our approach using the interpretation of EKI as a Gaussian approximation in the Bayesian tempering setting for inverse problems. We show that our parameter choice controls the symmetrised Kulback-Leibler divergence between consecutive tempering measures. We further motivate our choice using a heuristic statistical discrepancy principle. We test our framework using electrical impedance tomography with the complete electrode model. Parameterisations of the unknown conductivity are employed which enable us to characterise both smooth or a discontinuous (piecewise-constant) fields. We show numerically that the proposed regu-larisation of EKI can produce efficient, robust and accurate estimates, even for the discontinuous case which tends to require larger ensembles and more iterations to converge. We compare the proposed technique with a standard method of choice and demonstrate that the proposed method is a viable choice to address computational efficiency of EKI in practical/operational settings

Well-posed Bayesian geometric inverse problems arising in subsurface flow

In this paper, we consider the inverse problem of determining the permeability of the subsurface from hydraulic head measurements, within the framework of a steady Darcy model of groundwater flow. We study geometrically defined prior permeability fields, which admit layered, fault and channel structures, in order to mimic realistic subsurface features; within each layer we adopt either constant or continuous function representation of the permeability. This prior model leads to a parameter identification problem for a finite number of unknown parameters determining the geometry, together with either a finite number of permeability values (in the constant case) or a finite number of fields (in the continuous function case). We adopt a Bayesian framework showing existence and well-posedness of the posterior distribution. We also introduce novel Markov Chain-Monte Carlo (MCMC) methods, which exploit the different character of the geometric and permeability parameters, and build on recent advances in function space MCMC. These algorithms provide rigorous estimates of the permeability, as well as the uncertainty associated with it, and only require forward model evaluations. No adjoint solvers are required and hence the methodology is applicable to black-box forward models. We then use these methods to explore the posterior and to illustrate the methodology with numerical experiments

Ensemble-marginalized Kalman filter for linear time-dependent PDEs with noisy boundary conditions: application to heat transfer in building walls

In this work, we present the ensemble-marginalized Kalman filter (EnMKF), a sequential algorithm analo- gous to our previously proposed approach [1, 2], for estimating the state and parameters of linear parabolic partial differential equations in initial-boundary value problems when the boundary data are noisy. We apply EnMKF to infer the thermal properties of building walls and to estimate the corresponding heat flux from real and synthetic data. Compared with a modified Ensemble Kalman Filter (EnKF) that is not marginalized, EnMKF reduces the bias error, avoids the collapse of the ensemble without needing to add in- flation, and converges to the mean field posterior using 50% or less of the ensemble size required by EnKF. According to our results, the marginalization technique in EnMKF is key to performance improvement with smaller ensembles at any fixed time

Tuning equation ford dynamic matrix control in siso loops

El Control por Matriz Dinámica (DMC) es una de las estrategias de control avanzado que más aplicaciones industriales tiene en la actualidad. Sin embargo, la literatura presenta pocas opciones para el cálculo del parámetro de sintonización que gobierna la agresividad del controlador. Esta investigación propone una nueva ecuación de sintonización para calcular este parámetro de sintonización. Se presentan los análisis estadísticos realizados para formular la ecuación de sintonización. Para probar la eficacia de la ecuación propuesta, se presenta pruebas de rendimiento del controlador usando diferentes métodos de sintonización. Estas pruebas incluyen tanto sistemas lineales como no lineales./Dynamic Matrix Control (DMC) is one of the most used advanced control strategies used in industrial environments. However, the available literature does not present many alternatives to calculate the controller tuning parameter (also called suppression factor). This research proposes a new tuning equation to calculate this parameter. The statistical analysis and regression used to develop the equation, as well as the tests used to validate it are shown. Linear and nonlinear systems were used to compare different tuning methods

Primer registro de la cojolita (Penelope purpurascens) en el estado de Guanajuato, México

-Aquí se informa sobre el registro de la cojolita en la Reserva de Biosfera Sierra Gorda de Guanajuato, el cual es el registro más central para esa latitud y el primero para el estado. Esta especie fue registrada fotográfícamente en un bosque de encino. La presencia de esta especie enfatiza la necesidad de continuar realizando inventarios biológicos en esta Reserva de la Biosfer