Minisymposia Abstracts

Speaker: Luther White
University of Oklahoma
Title: Modeling, data, validation, and inversion for a girder
Abstract: A beam model with shearing having spatially dependent elastic properties is determined for the purpose of modeling a prestressed concrete mended girder. Data has been obtained from a collection of experiments in which the girder is loaded and the displacements at points along the girder are observed. This data is used for a validation procedure. Different elastic parameterizations are considered. Numerical studies and presented. Finally, issues involving the inversion of data and applications for health monitoring are discussed.

Speaker: Kurt M. Bryan
Rose-Hulman Institute of Technology
Title: Efficient computational methods for thermal imaging of small cracks in plates
Abstract: We present some fast computational methods for identifying cracks in a twodimensional plate using input flux and temperature measurements, based on the “small volume” asymptotic expansions of Ammari, Moskow, Vogelius, et. al. For a given input heat flux the temperature of the plate is measured using an infrared image of the entire surface of the plate (so unlike problems in which measurements are restricted to the one-dimensional boundary of the object, we have access to the “interior”of the object.) The novelty here is that the cracks we seek to image are very small, below the single-pixel resolution of the infrared camera.

Speaker: Herb Kunze
University of Guelph
Title: Using the ideas and philosophy of fractal-based analysis to solve differential equations inverse problems
Abstract: In fractal imaging, one seeks to approximate a target image by the fixed point of a fractal transform, but the true error in terms of parameters is not available to be minimized. One instead shifts to a different minimization problem via the Collage Theorem. Broadly speaking, a similar situation occurs when considering inverse problems for differential equations: the true error in terms of parameters is not available. Motivated by a fractal-based approach, in this talk, we shift to a different minimization problem. We develop and discuss the theoretical machinery for treating both ODEs and PDEs inverse problems, in deterministic and random settings, with applications.

Speaker: Assad Oberai
Rensselaer Polytechnic Institute
Title: Direct computation of inverse problems of incompressible elasticity
Abstract: Solving inverse problems for the distribution of mechanical properties is typically based on constrained optimization procedures, requiring numerous forward solutions. Current image processing techniques often provide full-field kinematic measurements. In such cases, the inverse problem may be approached directly. For linear elasticity this leads to system of advection-type equations for the sought moduli, with coefficients that depend on the known strains. A novel variational formulation related to the adjoint operator (AWE) leads to a boundary value problem that is well posed under mild conditions, allowing for the presence of measurement error, and lends itself well to Galerkin discretization. In this talk we consider the application of this formulation to different problems in linear elasticity imaging including, those of qausi-static plane stress and strain and time-harmonic plane stress.

Speaker: Paul Barbone
Boston University
Title: Improved inverse problem solutions using improved forward solvers
Abstract: A large class of inverse problems may be described as trying to infer medium properties in a domain based upon measurement of a field within or on the boundary of that domain. The ability to do this depends upon the measured field being sensitive to changes in the material properties. This sensitivity may be expressed in terms of an inequality which may be derived directly from the governing partial differential equations. More precisely, there exists a constant C ≤ 1 such that:
                 ||δμ||μ ≤ ||μ||.                       (1)
Here, δμ is an infinitesimal variation in material properties, δu is the corresponding change in the relevant field, C is a constant. The norms in equation (1) are application dependent. Equation (1) indicates that any variation in material properties δμ results in a measurable variation in the field, δμ. When equation (1) is satisfied for a particular problem, then we may hope to uniquely infer the properties from the given measurements. When (1) is violated, we cannot hope to infer the properties from the measurements without some external assumptions. We consider a class of partial differential equations (PDEs) that govern inverse scalar and vector potential problems. For these (PDEs), we can show that equation (1) is satisfied. We then consider traditional FEM discretization of the PDEs, and examine the discrete counterpart to equation (1). We find in many cases that, though (1) is satisfied in the continuous case, it is not satisfied in the discrete case. We then derive improved FEM discretization methods that satisfy (1), even in the discrete case. Numerical examples shall also be presented.

Speaker: Ian Knowles
University of Alabama at Birmingham
Title: Well-posedness for the inverse groundwater problem
Abstract: The inverse problem of determining subsurface parameters, such as hydraulic conductivity, storativity, and recharge from well measurements in an aquifer system has a long history. W n this problem showing that the recovered parameters depend continuously on the well data. We also discuss the related problem of practical error estimation in aquifer modelling.

Speaker: Fabio Raciti
University of Catania
Title: On an inverse problem related to the Lamé system
Abstract: We consider an inverse problem for an elastic body occupying a bounded domain A. Let B be a subdomain of A such that the Lamé parameters have different constant values in B and A\B. We study the problem of localizing B with boundary measurements under the assumption that B has the shape of a polyhedron.

Speaker: Akhtar A. Khan
Rochester Institute of Technology
Title: Parameter identification in variational and quasivariational inequalities
Abstract: In recent years new optimality conditions, by means of multiplier rules, were obtained for abstract optimization problems in function spaces where the associated ordering cone of has a nonempty interior. However, it turns out that the multipliers for these problems belong to non-regular spaces of measures. One of the essential requirement of these studies is the validity of a Slater’s type constraint qualification. It is known that Slater’s type constraint qualification is a stringent condition and it does not hold for many important cases of interest. In this talk, we will discuss a new conical regularization technique that gives optimality conditions without requiring any Slater’s type constraint qualification. The Henig dilating cones is the basic technical tool for this study. Finite element discretization of the dilating cone will be discussed and numerical examples will be presented.

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