Christopher A. Beattie

Professor of Mathematics

Ph.D. (Mathematical Sciences), The Johns Hopkins University, 1982

M.S. (Mechanical Engineering), Duke University, 1977

B.S.E. (Biomedical Engineering/Computer Science), Duke University, 1975

Visiting Fellow, Einstein Foundation - Berlin, 2015 - 2017

 
 

Spectral Estimation for Linear Operators
The analysis of physical phenomena often depends on the ability to closely approximate eigenvalues and eigenfunctions of differential operators. Examples range from the prediction of resonant frequencies, vibrational mode shapes, and buckling loads of elastic structures; through the determination of bound state energy levels and associated electronic configurations for atoms and molecules; to the computation of critical values of Reynolds numbers in viscous fluid flows.  Eigen problems for differential operators are often fundamentally more difficult than matrix eigen problems due to their infinite dimensional (often noncompact) character.  Both theoretical and practical issues are involved in constructing computational methods for approaching problems such as these successfully.


Selected papers:

    1. 1.Methods for computing lower bounds to eigenvalues of self-adjoint operators (with F. Goerisch). Numerische Mathematik, 72, pp. 143-172 (1995).

    2. 2.Convergence theorems for intermediate problems. II.(with W. M. Greenlee) Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132.05 : pp. 1057-1072. (2002)

    3. 3.Galerkin Eigenvector Approximations, Mathematics of Computation. 69, pp.1409- 1434. (2000)

    4. 3.Location of essential spectra of intermediate Hamiltonians restricted to symmetry subspaces (with M. B. Ruskai). Journal of Mathematical Physics, 29 (10), pp. 2236-2240. (1988)

 

Research Areas: model reduction, computational linear algebra,

     spectral estimation, numerical analysis, scientific computing.

 
 

Computational Linear Algebra

Computational linear algebra is a core enabling technology of modern science and engineering - developments in this area are crucial to the development of accurate and efficient computational strategies and not surprisingly, these methods often are at the heart of major computer codes.  The development and refinement of these methods to make them fast, accurate, and reliable depends on careful, detailed analysis of their behavior.


Selected papers:

    1. 1.Matrix Factorizations and Direct Solution of Linear Systems, Chapter 51 in Handbook of Linear Algebra (edited by L. Hogben), Chapman & Hall, 2013

    2. 2.Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computations, (with M. Embree, and D. Sorensen). SIAM Review, 47 (3), pp. 492 – 515, (2005)

    3. 3.Convergence of restarted Krylov subspaces to invariant subspaces, (with M. Embree and J. Rossi), SIAM J. Matrix Analysis and Applications, 25 (4), pp.1074-1109, (2004)

    4. 4.A note on shifted Hessenberg systems and frequency response computation (with Z. Drmac, and S. Gugercin) ACM Transactions on Mathematical Software, 38(2), pp. 12:1-12:16 2011

    5. 5. Harmonic Ritz and Lehmann Bounds, Electronic Transactions of Numerical Analysis (ETNA), 7, pp. 18-39, (1998).

    6. 6. Localization criteria and containment for Rayleigh quotient iteration (with D. W. Fox). SIAM Journal of Matrix Analysis and Applications, 10 (1), pp. 80-93. (1989)

 

Model reduction

Direct numerical simulation of dynamical systems can play a crucial role in studying a wide variety of complex physical phenomena in areas ranging from ocean circulation to microchip design. Many such phenomena involve heterogeneous mixtures of physical processes that evolve on fine time and length scales while system behaviors of interest occupy much coarser time and length scales.  As models are refined at increasingly fine time and length scales in order to attain high accuracy, dynamical systems of enormous scale and complexity are often produced, leading to overwhelming demands made on computational resources.   One may address this problem in some cases through methods that encode the fine scale dynamic structure of complex systems into compactly-represented high-fidelity reduced models that may then serve as efficient surrogates for the original systems.

    While there are a variety of approaches that can accomplish this, my principal focus has been on those associated originally with Krylov subspace projection (sometimes called moment-matching), which now is more descriptively termed interpolatory model reduction.  This class of approaches is numerically reliable, widely applicable,  and can be realized effectively either with direct methods or inexact iterative methods for problem dimensions exceeding 106. 

 

Selected papers:

    1. 1.Interpolatory model reduction of large-scale systems (with A. C. Antoulas and S. Gugercin) [Survey Paper], in Efficient Modeling and Control of Large-Scale Systems (J. Mohammadpour and K. Grigoriadis, editors), Springer, 2009.

    2. 2.H2 model reduction of large-scale dynamical systems. (with A. C. Antoulas and S. Gugercin), SIAM Journal on Matrix Analysis and Applications, 30(2), pp. 609-638 (2008).

    3. 3.Interpolatory projection methods for structure-preserving model reduction (with S. Gugercin), Systems and Control Letters, 58(3), pp. 225-232. (2009)

    4. 4.Structure-preserving model reduction for nonlinear port-Hamiltonian systems (with S. Chaturantabut and S. Gugercin), SIAM J. on Scientific Computing, 38(5), pp. B837-B865 (2016)

    5. 5.Inexact Solves in Interpolatory Model Reduction.(with S. Gugercin, and S. Wyatt). Linear Algebra and its Applications, Vol.436(8), 2012, pp. 2916–2943 (Special Issue dedicated to Danny Sorensen's 65th birthday)

  3. 6.Interpolatory H-infinity Model Reduction. (with G. Flagg and S. Gugercin), Systems and Control Letters, 62(7), pp.567-574, (2013), 2013


    •  

                Cover art from:

    C.A. Beattie, M. Embree and D.C. Sorensen,

    Convergence of Polynomial Restart Krylov

        Methods for Eigenvalue Computation,

    SIAM Review, 47(3), pp. 492-515, (2005).


    Compact computational representations of phenomena that involve significant transport, convection, or propagation delays can create vexing tradeoffs that defy acceptable balance between fidelity and complexity. View perspectives on this problem offered by an international group of experts at an opening project workshop sponsored by the Einstein Foundation - Berlin, and TU - Berlin with the collaboration of the European Model Reduction Network, EU-MORNET
    May 19-20, 2015