Scientific Computing Group Report - Detail view

Identifier: 2007-21

Author(s): Y. Cheng and C.-W. Shu

Title: Superconvergence and Time Evolution of Discontinuous Galerkin Finite Element Solutions

Page count: 26 pp.

Date: 2007-01-01

Abstract:

This proposal and quite substantial proposed effort relates to the development, analysis, and implementation of efficient and accurate methods for uncertainty quantification in problems described by PDE -- the emphasis appears to be on elliptic problems although there are also some discussions of applications to Navier-Stokes and large scale structutal problems. The basic approach for the uncertainty quantification is the polynomial chaos expansion with an emphasis on sparse grid collocaton methods The proposed effort is focused around 4 pillars of work, all interlocked in various ways .. a careful theoretical and computational study of collocation vs Galerkin polynomial chaos methods .. further development and analysis of sparse grid methods, including error estimation techniques to enable adaptivity .. the combination of reduced basis methods (POD style) in space with efficient sparse grid methods for the uncertainty modeling .. extension of the developed methods toward parameter estimation and optimal control for uncertain/stochastic problems The extensive and very detailed proposal spends the majority of the discussion on an extensive litterature overview, putting the work into context as well as substantial details of past related work, in particular in the uncertainty modeling and sparse grid methods. Unfortunately, the details are sparse on how the proposers suggest to address the many open questions and proposed research efforts -- if one simply adds up the content of the 4 paragraphs where the 4 projects are proposed, it amounts to less than 1 page out of a proposal of 24 pages. This makes it difficult at best to judge what is being proposed and the likeliness of success. The exception is a few proposed problesm which have, to a large extend, be addressed in the litteratur already. A few examples .. the first project focuses on a detailed study of Galerkin vs collocation methods. This is a rich area and with a rich litterature in area of spectral methods for solving PDE where this exact problem has been studied for decades. The exception is with the use of sparse grid methods where there is less theory but this approach does not have a direct Galerkin based analogy. .. the use of sparse grids have been proposed by several authors during the last decade and this is certainty well worth further exploring, in particular as it relates to error estimation in propability space. For 'simple' elliptic problems there are several possibilities and the proposers have had quite some success with this. The situation for time-dependent problems is very different and with applications like Navier-Stokes in mind, a discussion of even simple time-dependent problems would have been appropriate. It also seems that the proposed adaptive domain decomposition approach is very similar to the multielement approach of Wan and Karniadakis -- I am not aware of an analysis of this but I am not sure there is any new computational approach being proposed here. .. the combination of the reduced basis methods with the sparse grid methods is, in my view, perhaps the most interesting component of the proposal although I fail to see how this can be a practical tool without an error-estimator associated with POD basis -- this estimator must guarantee that the POD basis is rich enough to cover the dynamics over the range of the stochastic parameter variation. There is no discussion of this but the general area of fast solvers combined with the uncertainty quantification is certainly interesting. .. the description of the fourth project is so sparse and general that it is hard to judge the proposed work -- this effort, it appears, is entirely computational and builds directly on the 3 previous efforts to enable paramater estimation and optimal control in complex nonlinear PDE's -- it remains unclear to me just how the proposers will make the leap from the relatively simple cases studied in the other efforts to these much harder problems. These are general comments but reflects the superficial nature of the discussion of what is being proposed -- even simple 'proofs-of-concept' would have been a strength here. This sparse discussions are in stark contrast to very detailed discussion of the general area of numerical/analytic methods for uncertainty quantification and the past work of the proposers. The proposing team consists of a very senior PI with a terrific track record and two junior people, both with strong records. The junior people are a recent student of the PI and a recent postdoc of PI. i.e., one can expect the collaborative team to be well integrated. Given the above concerns and, in particular, the lack of details of proposed work and the very substrantial fundign level being requested, I am hesitant to recommend funding of this effort. A severely reduced effort based on the second and third effort may be worth considering with a main emphasis on the second topic unless the concern of errors in the POD approach be addressed.

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