Brown University Center for Statistical Sciences Seminar
Abstract: Three types of data are now available to test for changes in brain shape: 3D binary masks, 2D triangulated surfaces, and trivariate 3D vector displacement data from the non-linear deformations required to align the structure with an atlas standard. We use the Euler characteristic of the excursion set of a random field as a tool to test for localized shape changes. We extend these ideas to scale space, where the scale of the smoothing kernel is added as an extra dimension to the random field. Extending this further still, we look at fields of correlations between all pairs of voxels, which can be used to assess brain connectivity. Shape data is highly non-isotropic, that is, the effective smoothness is not constant across the image, so the usual random field theory does not apply. We propose a solution that warps the data to isotropy using local multidimensional scaling. We then show that the subsequent corrections to the random field theory can be done without actually doing the warping - a result guaranteed in part by the famous Nash Embedding Theorem. This has recently been formalized by Jonathan Taylor who has extended Robert Adler's random field theory to arbitrary manifolds.
Brown University, Joint Materials/Solid Mechanics Seminar Series
Abstract: Current research efforts in structural health monitoring are motivated by the need to move past alarm-based diagnostics to "gray scale" damage tracking, which is essential for the development of a true prognostic capability. In recent years we have developed a theoretical framework and methods based on ideas from dynamical systems theory that provide tracking of incipient damage and make possible accurate predictions of mechanical failures. Damage identification is accomplished by using a novel mathematical concept of phase space warping that tracks nonstationarity in dynamical systems. This concept describes distortions in a dynamical system's phase space due to the drifts in its parameters. It is a measure of change of global phase space structures. The damage tracking metrics developed in this framework are inherently capable of handling both linear and nonlinear systems. These methods dramatically improve upon available techniques both theoretically and experimentally. In laboratory and computer experiments, we have been able to accurately track damage evolution and predict failures well in advance of actual breakdowns in systems with drastically different notions of "damage." Furthermore, this framework allows for identification of dimensionality and dynamics of damage processes. In systems with more than one failure mode, we have been able to reliably identify active damage modes and predict which particular mode will fail and when.
Brown Analysis Seminar
PDE Seminar
Brown University, Joint Materials/Solid Mechanics Seminar Series
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