Ensemble-based implicit sampling for Bayesian inverse problems with non-Gaussian priors

In the paper, we develop an ensemble-based implicit sampling method for Bayesian inverse problems. For Bayesian inference, the iterative ensemble smoother (IES) and implicit sampling are integrated to obtain importance ensemble samples, which build an importance density. The proposed method shares a similar idea to importance sampling. IES is used to approximate mean and covariance of a posterior distribution. This provides the MAP point and the inverse of Hessian matrix, which are necessary to construct the implicit map in implicit sampling. The importance samples are generated by the implicit map and the corresponding weights are the ratio between the importance density and posterior density. In the proposed method, we use the ensemble samples of IES to find the optimization solution of likelihood function and the inverse of Hessian matrix. This approach avoids the explicit computation for Jacobian matrix and Hessian matrix, which are very computationally expensive in high dimension spaces. To treat non-Gaussian models, discrete cosine transform and Gaussian mixture model are used to characterize the non-Gaussian priors. The ensemble-based implicit sampling method is extended to the non-Gaussian priors for exploring the posterior of unknowns in inverse problems. The proposed method is used for each individual Gaussian model in the Gaussian mixture model. The proposed approach substantially improves the applicability of implicit sampling method. A few numerical examples are presented to demonstrate the efficacy of the proposed method with applications of inverse problems for subsurface flow problems and anomalous diffusion models in porous media.Comment: 27 page

Multi-sensor large scale land surface data assimilation using ensemble approaches

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2006.Includes bibliographical references (p. 223-234).One of the ensemble Kalman filter's (EnKF) attractive features in land surface applications is its ability to provide distributional information. The EnKF relies on normality approximations that improve its efficiency but can also compromise the accuracy of its distributional estimates. The effects of these approximations are evaluated by comparing the conditional marginal distributions and moments estimated by the EnKF to those obtained from an SIR particle filter, which gives exact solutions for large ensemble sizes. The results show that overall the EnKF appears to provide a good approximation for nonlinear, non-normal land surface problems. A difficulty in land data assimilation problems results from the high dimensionality of states created by spatial discretization over large computational grids. The high dimensionality can be reduced by exploiting the fact that soil moisture field may have significant spatial correlation structure especially after extensive rainfall while it may have local structure determined by soil and vegetation variability after prolonged drydown. This is confirmed by SVD of the replicate matrix produced in an ensemble forecasting experiment. Local EnKF's are suitable for problems during dry periods but give less accurate results after rainfall.(cont.) The most promising option is to develop a generalized method that reflects structural changes in the ensemble. A highly efficient ensemble multiscale filter (EnMSF) is then proposed to solve large scale nonlinear estimation problems with arbitrary uncertainties. At each prediction step realizations of the state variables are propagated. At update times, joint Gaussian distribution of states and measurements are assumed and the Predictive Efficiency method is used to identify a multiscale tree to approximate statistics of the propagated ensemble. Then a two-sweep update is performed to estimate the state variables using all the data. By controlling the tree parameters, the EnMSF can reduce sampling error while keep long range correlation in the ensemble. Applications of the EnMSF to Navier-Stokes equation and a nonlinear diffusion problem are demonstrated. Finally, the EnMSF is successfully applied to soil moisture and surface fluxes estimation over the Great Plains using synthetic multiresolution L-band passive and active microwave soil moisture measurements following HYDROS specifications.by Yuhua Zhou.Ph.D

Electromechanical large scale computational models of the ventricular myocardium

Els models computacionals del cor són una eina important que pot donar als investigadors biomèdics una font addicional d’informació per entendre el funcionament del miocardi. Els models numèrics poden ajudar a interpretar dades experimentals i proporcionar informació complementària sobre mecanismes cardíacs que no poden ser determinats amb precisió mitjançant dispositius clínics clàssics. En aquesta tesi, s’apliquen tècniques de computació a gran escala per construir una eina computacional capaç d’executar-se en paral•lel en milers de processadors, permetent simulacions d’alta fidelitat en malles fines. Per simular el bombeig del cor, s’utilitza un esquema d’acoblament explícit entre les equacions electrofisiològiques en tres dimensions i la formulació en mecànica de sòlids. Per trobar la solució numèrica, s’utilitza el mètode d’elements finits. A més, s’implementen tècniques en assimilació de dades per a l’estimació efectiva dels paràmetres electrofisiològics i mecànics rellevants que apareixen a les equacions, la qual cosa ´es un pas crucial cap a un model cardíac sensible a cada pacient. El codi computacional s’aplica per simular problemes físics reals. S’estudia la propagació electromecànica en una geometria de conill, on es prova la sensibilitat del model a les variacions d’entrada. En particular, l’eina de càlcul s’utilitza per avaluar la influència del camp de fibres cardíaques en la contracció del teixit. Per desenvolupar una simulació cardíaca útil per a fins clínics, el model requereix la integració i combinació de la mecànica computacional i les tècniques de processament d’imatge més recents. El model resultant pot ser la base d’estudis teòrics sobre mecanismes de patologies, oferint als investigadors i cardiòlegs pistes addicionals per comprendre el funcionament del cor. Pot ajudar a la planificació de cirurgia i modelització, com és la predicció dels efectes de compostos farmacològics en el ritme cardíac o l’estudi de l’efecte de medicaments. Aquest projecte només és possible en un equip multidisciplinar, on grups especialitzats uneixen les seves forces en les respectives disciplines: cardiòlegs, investigadors imatge, bioenginyers i científics de la computació. El present model computacional del cor és un pas més cap a la creació d’un laboratori cardíac virtual.A cardiac computational model is a relevant tool that can give biomedical researchers an additional source of information to understand how the heart works. Numerical models can help to interpret experimental data and provide information about cardiac mechanisms that can not be determined accurately by classical clinical devices. In this thesis, High Performance Computing (HPC) techniques are used to build a cardiac computational tool, which is capable of running in parallel in thousands of processors, bioengineers and computational scientists. The present cardiac computational model is one further step towards the creation of a virtual lab, allowing high fidelity simulations on fine meshes. To simulate the pumping heart, an explicit coupling scheme between the three-dimensional electrophysiological equations and the solid mechanics formulation is used, solving the governing equations with finite element methods. Also, data assimilation techniques are implemented for the effective estimation of some relevant electrophysiological parameters, which is a crucial step towards the patient-sensitive cardiac model. The data assimilation techniques are assessed on synthetic data generated by the model. Finally, the computational code is applied to simulate real physical problems. The electromechanical propagation in a rabbit geometry is studied to test the sensitivity of the framework to input variations. Particularly, the computational tool is used to evaluate the influence of the fiber field in the contraction of the tissue. To develop a cardiac simulation useful for clinical purposes, the integrative model requires combining computational mechanics and image processing techniques via data assimilation methods. Coupled with the most advanced image processing analysis, the framework can be the base of theoretical studies into the mechanisms of cardiac pathologies. It can help surgery planning and cardiac modeling, such as the prediction of the impact of pharmacological compounds on the heart’s rhythm or to improve the knowledge of drug study, giving medical researchers additional hints to understand the heart. This realization is only possible in a multidisciplinary team, where specialized groups join forces in their respective disciplines: cardiologists, image researchers, bioengineers and computational scientists. The present cardiac computational model is one further step towards the creation of a virtual la