Browsing by Author "Carden, Russell L."

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Automatic Discrete Empirical Interpolation for Nonlinear Model Reduction
    (2012-08) Carden, Russell L.; Sorensen, Danny C.
    The Discrete Empirical Interpolation Method (DEIM) is a technique for model reduction of non-linear dynamical systems. It is based upon a modification to proper orthogonal decomposition which is designed to reduce the computational complexity for evaluating the reduced order nonlinear term. The DEIM approach is based upon an interpolatory projection and only requires evaluation of a few selected components of the original nonlinear term. Thus, implementation of the reduced order nonlinear term requires a new code to be derived from the original code for evaluating the nonlinearity. This work describes a methodology for automatically deriving a code for the reduced order nonlinearity directly from the original nonlinear code. The methodology is derived from standard techniques of automatic differentiation. This capability removes the possibly tedious and error prone task of producing such a code by hand and hence can facilitate the use of DEIM by non-experts.
  • Thumbnail Image
    Item
    Ritz Value for Non-Hermitian Matrices
    (Society for Industrial and Applied Mathematics, 2012) Carden, Russell L.; Embree, Mark
    Rayleigh-Ritz eigenvalue estimates for Hermitian matrices obey Cauchy interlacing, which has helpful implications for theory, applications, and algorithms. In contrast, few results about the Ritz values of non-Hermitian matrices are known, beyond their containment within the numerical range. To show that such Ritz values enjoy considerable structure, we establish regions within the numerical range in which certain Ritz values of general matrices must be contained. To demonstrate that localization occurs even for extreme examples, we carefully analyze possible Ritz value combinations for a three-dimensional Jordan block.