Special PDE & Lefschetz Center for Dynamical Systems Seminar
Please Note -- Change in Room and Location for Today Only |
Brown University Center for Statistical Sciences Seminar
MD Anderson Cancer Center, The University of Texas | |
Abstract: Spatial heterogeneity plays an important role in a diverse set of applications. For example, in ecology, heterogeneous environments promote camouflaged prey species and disruptive selection; in economics, local characteristics determine regional policies; in health sciences, histopathological heterogeneity characterizes certain cancer tissues within and among tumor types.
Recent Bayesian modeling of univariate spatial data has considered mixed effect models, where a residual stationary (homogeneous) Gaussian effect is assumed. Arguably, one might prefer the flexibility of a nonstationary, non-Gaussian specification. In a nonparametric setting, this can be accommodated by mixture of Dirichlet process (DP) models. The DP is an example of a species sampling prior, which are typically used to describe diversity of different ecological groups of species under different environmental conditions.
However, a limitation of the mixture of DP models is that the latent factor driving species sampling is globally defined and may fail to account for spatial heterogeneity. In this work, we introduce a novel class of prior distributions, the hybrid Dirichlet Processes (hDP), which generalize the DP and overcome this limitation. In a spatial setting, the hDP are defined as mixtures of Gaussian random fields with spatially varying weights. A crucial feature of this specification is the possibility to model local speciation and hybrid clustering. We illustrate the procedure by means of a simulated example and an application to the analysis of hippocampal atrophy in brains of patients affected by Alzheimer's disease.
This is joint work with Alan Gelfand (Duke University, USA) and Sonia Petrone (Bocconi University, Italy).
Candidate for Assistant Professor in the Biostatistics Section of the Program in Public Health
Joint Scientific Computing and Probability Seminar
School for Computational Science and Department of Mathematics, Florida State University, Tallahassee, FL, 32306, email: webster@scs.fsu.edu | |
Abstract: This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms (input data of the model). Here, we especially address the situation where the input data are assumed to depend on a moderately large number of random variables, where in general the curse of dimensionality is encountered.
This method can be viewed as an extension of the Sparse Grid Stochastic Collocation method proposed in [Nobile-Tempone-Webster, Technical report #85, MOX, Dipartimento di Matematica, 2006] which consists of a Galerkin approximation in space and a collocation, in probability space, at the zeros of sparse tensor product spaces utilizing either Clenshaw-Curtis or Gaussian interpolants. As a consequence of the collocation approach our techniques naturally lead to the solution of uncoupled deterministic problems as in the Monte Carlo method.
Our previous sparse collocation procedure is very effective for problems whose input data depend on a moderate number of random variables, which "weigh equally" in the solution. For such an isotropic situation the displayed convergence is faster than standard collocation techniques built upon full tensor product spaces. On the other hand, the convergence rate deteriorates when we attempt to solve highly anisotropic problems, such as those appearing when the input random variables come e.g. from Karhunen-Lo`eve -type truncations of "smooth" random fields. In such cases, a full anisotropic tensor product approximation may still be more effective for a small or modest number of random variables. However, if the number of random variables is large, the construction of the full tensor product spaces becomes infeasible, since the dimension of the approximating space grows exponentially fast in the number of random variables.
Instead, this work proposes the use of anisotropic sparse tensor product spaces constructed from the Smolyak algorithm utilizing suitable abscissas. This approach is particularly attractive in the case of truncated expansions of random fields, since the anisotropy can be tuned to the decay properties of the expansion. We will present a priori and a posteriori procedures for choosing the anisotropy of the sparse grids which are extremely effective for the problems under study.
This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates: (sub)-exponential convergence of the "probability error" in the asymptotic regime and algebraic convergence of the "probability error" in the pre-asymptotic regime, with respect to the total number of collocation points. Numerical examples exemplify the theoretical results and are used to compare this approach with several others, including standard Monte Carlo. In particular, for moderately large dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.
Scientific Computing Seminar
Validations and Applications | |
Special Analysis/PDE Seminar
Abstract: The multidimensional L2 conjecture and delocalization for random decaying potentials will be discussed. The main steps of the proofs will be provided if time permits.
Department of Mathematics Colloquium
Abstract: We will start with classical results in approximation theory and then will quickly move to related problems in PDE (scattering problems for multidimensional Schrodinger and Dirac operators). The talk will be targeted to the general audience.
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