Conferencias Plenarias V MACI 2015

Estimation of Distributed Parameters in Permittivity Models of
Composite Dielectric Materials Using Reflectance

H.T. Banks,
Center for Research in Scientific Computation
North Carolina State University

Abstract.
We investigate the feasibility of quantifying properties of a composite dielectric
material through the reflectance, where the permittivity is described by the
Lorentz model in which an unknown probability measure is placed on the model
parameters. We summarize the computational and theoretical framework (the
Prohorov Metric Framework) developed by our group in the past two decades for
nonparametric estimation of probability measures using a least-squares method,
and point out the limitation of the existing computational algorithms for this
particular application. We then improve the algorithms, and demonstrate the
feasibility of our proposed methods by numerical results obtained for both
simulated data and experimental data for inorganic glass when considering the
resonance wavenumber as a distributed parameter. Finally, in the case where the
distributed parameter is taken as the relaxation time, we show using simulated
data how the addition of derivative measurements improves the accuracy of the
method.

Non-modular Losses in Structured Prediction with Application to Computer Vision

Matthew B. Blaschko
Chargé de Recherche (Inria) and Associate Professor
Center for Visual Computing
École Centrale Paris
Francia

Abstract.

Statistical learning has been a driving force in the development of computer vision applications.  While statistical estimation is very well developed for binary classification and regression, it is less so for prediction of structured outputs, such as segmentation of an image, or prediction of object location and interrelated parts.  Typically, modular losses such as Hamming loss are applied for such prediction problems.  We investigate the use of non-modular loss functions, which may be isomorphic to submodular or supermodular set functions, and explore convex surrogate loss functions in these cases.  An important question is the computational complexity of regularized risk minimization using the resulting convex surrogate, and we show polynomial time convex surrogates for both submodular and supermodular loss functions.

A TWO-STRAIN SPATIO-TEMPORAL MATHEMATICAL MODEL OF TUMOR GROWTH

Prof. Fabio Augusto Milner, STEM Education School of Mathematical and Statistical Sciences. Division of Educational Leadership and Innovation Honors Faculty, Barrett’s Honors College, Arizona State University.

Abstract.  Cancer in most organs develops in the form of solid tumors that begin as spheroids consisting entirely of cancerous (parenchyma) cells. When the tumor reaches the size of approximately 1 mm, it needs blood vessels (vascularization) to develop in it in order to be able to grow further. A model of vascularized tumor growth with competing parenchyma cells of two dierent strains is presented, consisting of a free boundary value problem for a system of nonlinear parabolic PDEs describing the dynamics of the parenchyma cells and vascular endothelial cells (VECs), coupled with an ODE describing the dynamics of vascular density and an algebraic equation describing a variable used as proxy for local resource availability. The tumor is assumed to have radial symmetry and the mathematical domain is then taken to be a sphere with moving boundary, with the boundary moving at the unique physically determined velocity that changes local tumor volume exactly to maintain density. Some theoretical analytical properties of the model are presented, and results from several simulations are shown and biological interpretations provided, including how selection for increased proliferation or for increased angiogenesis may lead to tumors that are structurally like integrated tissues or like segregated ecological niches, as well as to hypertumors.

Numerical Solutions of Fractional Stefan Problems

Prof. Vaughan R. Voller, Department of Civil, Environmental, and Geo- Engineering, University of Minnesota,

Abstract. The Stefan problem, involving the tracking of a phase change interface, provides the frame work for studying moving boundary problems in a variety of physical situations. The solution of the classical one-dimensional Stefan problem predicts that in time t the phase interface goes as s(t) ∼ t can be observed where the time exponent n write down the governing equations of the Stefan problem in terms of fractional order time (1 ≥ β > 0) and space (1 ≥ α > 0) derivatives.

In this talk we start with providing a physical justification for using fractional Stefan problems and introduce a limit case problem related to the horizontal moisture movement in a porous media. Two model forms, which admit closed form solutions are presented; one assuming a sharp interface between the advancing wet and dry domains, the other assuming a diffuse interface, of finite thickness, across which a liquid fraction changes smoothly from a value of one to a value of zero. Numerical solutions based on fixed grid (phase field like) and deforming grid methods are presented and tested. Results indicate, counter to those seen in integer derivative Stefan problems, that in the fractional time case (β < 1), a solution of the fractional diffuse interface model in the sharp interface limit does not coincide with the solution of the sharp interface counterpart.

A virtual power principle for RVE-based multiscale models

Pablo Javier Blanco
HemoLab LNCC – Brasil

Abstract:
In this talk a unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element (RVE). The entire theory lies on three fundamental principles: (i) kinematical admissibility, whereby the macro- and micro-scale kinematics are defined and linked so as to ensure their magnitudes are in some sense preserved in the micro-macro transition; (ii) duality, through which the force- and stress-like quantities are uniquely identified as the dual objects of the adopted kinematical variables; and (iii) the principle of multiscale virtual power, a generalization of the well-known Hill-Mandel principle of macrohomogeneity, from which equilibrium equations and homogenization relations for the force- and stress-like quantities are unequivocally derived by straightforward variational arguments.
The overall theory is presented within a clear, logically structured multiscale framework that provides a rational justification of classical formulations and facilitates
the rigorous development of new multiscale models in systematic, well-defined steps. In addition, due to its variational basis, the format in which resulting models are presented is naturally well suited for discretization by finite element methods or any other numerical approximation. Multiscale models already published in the literature can be cast straightforwardly inside the proposed general theory including coupled multiphysics, high order strains effects, dynamics, localization of deformations at macro and micro level and material failure, among others. Several of the above applications are also discussed.