Stabilized variational formulation for direct solution of inverse problems in heat conduction and elasticity with discontinuities

We consider the design of finite element methods for inverse problems with full-field data governed by elliptic forward operators. Such problems arise in applications in inverse heat conduction, in mechanical property characterization, and in medical imaging. For this class of problems, novel finite element methods have been proposed (Barbone et al., 2010) that give good performance, provided the solutions are in the H^1(Ω) function space. The material property distributions being estimated can be discontinuous, however, and therefore it is desirable to have formulations that can accommodate discontinuities in both data and solution. Toward this end, we present a mixed variational formulation for this class of problems that handles discontinuities well. We motivate the mixed formulation by examining the possibility of discretizing using a discontinuous discretization in an irreducible finite element method, and discuss the limitations of that approach. We then derive a new mixed formulation based on a least-square error in the constitutive equation. We prove that the continuous variational formulations are well-posed for applications in both inverse heat conduction and plane stress elasticity. We derive a priori error bounds for discretization error, valid in the limit of mesh refinement. We demonstrate convergence of the method with mesh refinement in cases with both continuous and discontinuous solutions. Finally we apply the formulation to measured data to estimate the elastic shear modulus distributions in both tissue mimicking phantoms and in breast masses from data collected in vivo

Characterizing and Extending Answer Set Semantics using Possibility Theory

Answer Set Programming (ASP) is a popular framework for modeling combinatorial problems. However, ASP cannot easily be used for reasoning about uncertain information. Possibilistic ASP (PASP) is an extension of ASP that combines possibilistic logic and ASP. In PASP a weight is associated with each rule, where this weight is interpreted as the certainty with which the conclusion can be established when the body is known to hold. As such, it allows us to model and reason about uncertain information in an intuitive way. In this paper we present new semantics for PASP, in which rules are interpreted as constraints on possibility distributions. Special models of these constraints are then identified as possibilistic answer sets. In addition, since ASP is a special case of PASP in which all the rules are entirely certain, we obtain a new characterization of ASP in terms of constraints on possibility distributions. This allows us to uncover a new form of disjunction, called weak disjunction, that has not been previously considered in the literature. In addition to introducing and motivating the semantics of weak disjunction, we also pinpoint its computational complexity. In particular, while the complexity of most reasoning tasks coincides with standard disjunctive ASP, we find that brave reasoning for programs with weak disjunctions is easier.Comment: 39 pages and 16 pages appendix with proofs. This article has been accepted for publication in Theory and Practice of Logic Programming, Copyright Cambridge University Pres

Relations Between Stochastic Orderings and generalized Stochastic Precedence

The concept of "stochastic precedence" between two real-valued random variables has often emerged in different applied frameworks. In this paper we consider a slightly more general, and completely natural, concept of stochastic precedence and analyze its relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas. Motivations for our study can arise from different fields. In particular we consider the frame of Target-Based Approach in decisions under risk. This approach has been mainly developed under the assumption of stochastic independence between "Prospects" and "Targets". Our analysis concerns the case of stochastic dependence.Comment: 13 pages, 6 figure