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

Living in an Irrational Society: Wealth Distribution with Correlations between Risk and Expected Profits

Different models to study the wealth distribution in an artificial society have considered a transactional dynamics as the driving force. Those models include a risk aversion factor, but also a finite probability of favoring the poorer agent in a transaction. Here we study the case where the partners in the transaction have a previous knowledge of the winning probability and adjust their risk aversion taking this information into consideration. The results indicate that a relatively equalitarian society is obtained when the agents risk in direct proportion to their winning probabilities. However, it is the opposite case that delivers wealth distribution curves and Gini indices closer to empirical data. This indicates that, at least for this very simple model, either agents have no knowledge of their winning probabilities, either they exhibit an ``irrational'' behavior risking more than reasonable.Comment: 7 pages, 8 figure

Modelling exchange bias in core/shell nanoparticles

We present an atomistic model of a single nanoparticle with core/shell structure that takes into account its lattice strucutre and spherical geometry, and in which the values of microscopic parameters such as anisotropy and exchange constants can be tuned in the core, shell and interfacial regions. By means of Monte Carlo simulations of the hysteresis loops based on this model, we have determined the range of microscopic parameters for which loop shifts after field cooling can be observed. The study of the magnetic order of the interfacial spins for different particles sizes and values of the interfacial exchange coupling have allowed us to correlate the appearance of loop asymmetries and vertical displacements to the existence of a fraction of uncompensated spins at the shell interface that remain pinned during field cycling, offering new insight on the microscopic origin of the experimental phenomenology.Comment: 7 pages, 3 figures. Contribution presented at HMM 2007 held at Napoli 4-6 June 2007. To be published in J. Phys. Condens. Matte

Convective regularization for optical flow

We argue that the time derivative in a fixed coordinate frame may not be the most appropriate measure of time regularity of an optical flow field. Instead, for a given velocity field $v$ we consider the convective acceleration $v_t + \nabla v v$ which describes the acceleration of objects moving according to $v$. Consequently we investigate the suitability of the nonconvex functional $\|v_t + \nabla v v\|^2_{L^2}$ as a regularization term for optical flow. We demonstrate that this term acts as both a spatial and a temporal regularizer and has an intrinsic edge-preserving property. We incorporate it into a contrast invariant and time-regularized variant of the Horn-Schunck functional, prove existence of minimizers and verify experimentally that it addresses some of the problems of basic quadratic models. For the minimization we use an iterative scheme that approximates the original nonlinear problem with a sequence of linear ones. We believe that the convective acceleration may be gainfully introduced in a variety of optical flow models