Adaptive stochastic Galerkin FEM for lognormal coefficients in hierarchical tensor representations

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with lognormal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm

Articulating Local Politics and Market Forces for Economic Development: A Case Study of Ethanol Development in Upstate New York

CaRDI Reports Issue 8; Community & Energy: Economics of Energy; Community & Energy: Renewable Energy Sources & Sustainabilit

Immigrants and the Community

First in a series based on the research project "Integrating the Needs of Immigrant Workers and Rural Communities," which attempts to inform New York communities about the nature and consequences of increasing immigrant settlement.Many upstate New York communities have experienced population loss and decline in the last decade. Increasing numbers of immigrants have settled in many of these communities, which poses possible community development challenges and opportunities. Because each community must address these issues in its own way, this report is intended to make communities aware of changes in their populations and highlight issues they may choose to address.USDA Fund for Rural America (grant #2001-36201-11283) and Cornell University Agricultural Experiment Station (grant #33452

Immigrants and the Community: Farmworkers with Families

Second in a series based on the research project ?Integrating the Needs of Immigrant Workers and Rural Communities,? which attempts to inform New York communities about the nature and consequences of increasing immigrant settlement.America's hired farm workforce has changed considerably in the last decade. The most apparent change has been its "latinization" during the past two decades. This is largely a consequence of large numbers of Mexicans coming to the United States to work. Although Mexican immigrants work in numerous industries across the American landscape, they are especially important in agriculture. There has been a growing tendency of farmworkers to settle in rural communities together with their immediate family. But how and to what extent does community integration occur? How do foreigners who have little familiarity with American culture become integrated into the community? Answers to these questions have practical importance to farmers interested in retaining their workforce, service providers working to improve farmworker well-being and communities interested in helping the new residents contribute to community development. To help us understand the factors that both promote and limit the integration of immigrants into rural communities, we chose for study five New York agricultural communities in different economic and social contexts that have relied heavily on hired farm labor. Each community has a minority population of some significance and a history of immigrant farmworkers settling there

Learning Paths from Signature Tensors

Matrix congruence extends naturally to the setting of tensors. We apply methods from tensor decomposition, algebraic geometry and numerical optimization to this group action. Given a tensor in the orbit of another tensor, we compute a matrix which transforms one to the other. Our primary application is an inverse problem from stochastic analysis: the recovery of paths from their third order signature tensors. We establish identifiability results, both exact and numerical, for piecewise linear paths, polynomial paths, and generic dictionaries. Numerical optimization is applied for recovery from inexact data. We also compute the shortest path with a given signature tensor.Comment: 22 pages, 3 figure

Adaptive stochastic Galerkin FEM with hierarchical tensor representations

The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated

Riemannian thresholding methods for row-sparse and low-rank matrix recovery

In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular, a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity

Adaptive stochastic Galerkin FEM for log-normal coefficients in hierarchical tensor representations

Stochastic Galerkin methods for non-affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view. In this work, an adaptive Galerkin FE method for linear parametric PDEs with log-normal coefficients discretized in Hermite chaos polynomials is derived. It employs problem-adapted function spaces to ensure solvability of the variational formulation. The inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations. For this, a new tensor train format of the lognormal coefficient is derived and verified numerically. The central novelty is the derivation of a reliable residual-based a posteriori error estimator. This can be regarded as a unique feature of stochastic Galerkin methods. It allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic Wiener chaos polynomial degrees. For the evaluation of the error estimator to become feasible, a numerically efficient tensor format discretization is developed. Benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed Galerkin discretization and the fully adaptive algorithm