Publications of Oliver Ernst

1. Efficient Iterative Solvers for Stochastic Galerkin Discretizations of log-Transformed Random Diffusion Problems
   H. C. Elman, O. G. Ernst, and E. Ullmann
   submitted        (2011)
   
   We show how to reformulate a diffusion problem with lognormal random field as a convection-diffusion problem with a random convection term, turning a stochastically nonlinear problem into one which is stochastically linear.
2. Expansion of random field gradients using hierarchical matrices
   I. Busch, O. G. Ernst, and E. Ullmann
   PAMM    (to appear)    (2011)
   
   A short note describing how to compute the KL expansion of the gradient of a random field.
3. Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods
   O. G. Ernst and M. J. Gander
   in I. Graham, T. Hou, O. Lakkis, and R. Scheichl, editors, Numerical Analysis of Multiscale Problems , volume 83, page 325--361. Springer-Verlag, Berlin Heidelberg, 2011.
   A survey article on iterative solvers for discrete Helmholtz problems, written in connection with the 2010 London Mathematical Society Symposium in Durham (UK).
4. In the Convergence of Generalized Polynomial Chaos Expansions
   O. G. Ernst, A. Mugler, H.-J. Starkloff, and E. Ullmann
   Mathematical Modelling and Numerical Analysis    46 (2)  317--339  (2012)
   
   Answers the question when second-order random variables can be expanded in mean-square convergent generalized polynomial chaos expansions.
5. Deflated Restarting for Matrix Functions
   M. Eiermann, O. G. Ernst, and S. Güttel
   SIAM Journal on Matrix Analysis and Applications    32  621--641  (2011)
   
   Here we show how Ron Morgan-style deflation for accelerated restarting with Krylov subspace methods can be generalized from linear systems to general matrix functions.
6. Stochastic Galerkin Matrices
   O. G. Ernst and E. Ullmann
   SIAM Journal on Matrix Analysis and Applications    31  1848--1872  (2010)
   
   This is a collection of results on the properties of matrices arising in stochastic Galerkin discretization.
7. Efficient Solvers for a Linear Stochastic Galerkin Mixed Formulation of Diffusion Problems with Random Data
   O. G. Ernst, C. E. Powell, D. Silvester, and E. Ullmann
   SIAM Journal on Scientific Computing    31  1424--1447  (2009)
   
   Here we discuss mean-based preconditioning for the mixed discretization of second-order elliptic problems with random coefficients in the stochatically linear case.
8. A Generalization of the Steepest Descent Method for Matrix Functions
   M. Afanasjew, M. Eiermann, O. G. Ernst, and S. Güttel
   Electronic Transactions on Numerical Analysis    28  206--222  (2008)
   
   An analysis of our restarted Krylov subspace approximation for matrix functions in the special case of restart length one.
9. Implementation of a Restarted Krylov Subspace Method for the Evaluation of Matrix Functions
   M. Afanasjew, M. Eiermann, O. G. Ernst, and S. Güttel
   Linear Algebra and its Applications    429  2293--2314  (2008)
   
   Presents a stable implementation of our restarted Krylov subspace method for matrix function which now has constant work per restart cycle. Also includes an a posteriori error estimator and applications to time dependent problems from heat conduction, the quasi-static Maxwell equations and convection diffusion.
10. Fast 3D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection
    R.-U. Börner, O. G. Ernst, and K. Spitzer
    Geophysical Journal International    173  766--780  (2008)
    
    Krylov subspace model reduction applied to a geophysical electromagnetic diffusion problem based on Nédélec element discretization.
11. Computational Aspects of the Stochastic Finite Element Method
    M. Eiermann, O. G. Ernst, and E. Ullmann
    Computing and Visualization in Science    10  3--15  (2007)
    
    My invited presentation for the Algoritmy 2005 conference.
12. A Restarted Krylov Subspace Method for the Evaluation of Matrix Functions
    M. Eiermann and O. G. Ernst
    SIAM Journal on Numerical Analysis    44  2481--2504  (2006)
    
    We present a method of restarting Krylov subspace approximations to functions of a matrix times a vector.
13. Efficient Iterative Algorithms for the Stochastic Finite Element Method with Applications to Acoustic Scattering
    H. C. Elman, O. G. Ernst, D. P. O'Leary, and M. Stewart
    Computer Methods in Applied Mechanics and Engineering    194  1037--1055  (2005)
    
    This paper discusses efficient solution strategies for the linear systems of equations arising from the stochastic finite element method applied to acoustic scattering as well as some post processing for involving the stochastic properties of the solution.
14. A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations
    H. C. Elman, O. G. Ernst, and D. P. O'Leary
    SIAM Journal on Scientific Computing    23  1290--1314  (2001)
    
    This paper describes a new multigrid method which uses GMRES as a smoother on meshes too coarse to resolve the waves as well as an outer FGMRES acceleration.
15. Geometric Aspects of the Theory of Krylov Subspace Methods
    M. Eiermann and O. G. Ernst
    Acta Numerica    10  251--312  (2001)
    
    A sort of 'grand-unifying theory' for Krylov subspace algorithms based on an abstract orthogonal and oblique projection in a Hilbert space and involving the angles between certain subspaces.
16. Minimal and Orthogonal Residual Methods and their Generalizations for Solving Linear Operator Equations
    O. G. Ernst
      Habilitation Thesis, TU Bergakademie Freiberg      (2001)
    
    My habilitation thesis. This presents a framework for MR and OR methods beginning with abstract projections, then applying these to equation-solving and finally to Krylov subspace methods.
17. Numerical Experiences with a Krylov-enhanced Multigrid Solver for Exterior Helmholtz Problems
    H. C. Elman and O. G. Ernst
    in A. Bermúdez, D. Gómez, C. Hazard, P. Joly, and J. E. Roberts, editors, Proceedings of the Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation, Waves 2000 (Santiago de Compostela) . SIAM, Philadelphia, 2000.
    Our contribution to the Waves 2000 Meeting in Santiago de Compostela, Spain, July 2000. This contains some further numerical results obtained with our muiltigrid enhanced Krylov solver applied to acoustic scattering by obstacles and an inhomogeneous medium.
18. Equivalent Iterative Methods for p-Cyclic Matrices
    O. G. Ernst
    Numerical Algorithms    25  161--180  (2000)
    
    My contribution to the meeting in honor of Richard Varga's 70th birthday in Kent, March 1999. Contains an overview of how various iterative methods simplify when applied to p-cyclic matrices by resorting to the reduced system.
19. Analysis of Acceleration Strategies for Restarted Minimal Residual Methods
    M. Eiermann, O. G. Ernst, and O. Schneider
    Journal of Computational and Applied Mathematics    123  262--292  (2000)
    
    Another theoretical paper on Krylov subspace methods and their generalizations to arbitrary correction spaces, this time focussing on augmented correction spaces, preconditioners that invert on a subspace and de Sturler's optimal truncation scheme. Also contains new simple derivations of how the implicitly restarted Arnoldi method can be used for augmented Krylov subspace methods.
20. Residual-Minimizing Krylov Subspace Methods for Stabilized Discretizations of Convection-Diffusion Equations
    O. G. Ernst
    SIAM Journal on Matrix Analysis and Applications    21  1079--1101  (2000)
    
    This paper discusses the behavior of (unpreconditioned) GMRES on stabilized discrete convection-diffusion problems.
21. A Very Short Finite Element Tutorial
    M. Eiermann, O. G. Ernst, and W. Queck
    in Peter Klimanek and Wolfgang Pantleon, editors, Simulationstechniken in der Materialwissenschaft , volume B 279, page 17--40. TU Bergakademie Freiberg, 1996.
22. A Finite Element Capacitance Matrix Method for Exterior Helmholtz Problems
    O. G. Ernst
    Numerische Mathematik    75  175--204  (1996)
    
    This is more or less one chapter of my thesis: the finite element iterative imbedding algorithm.
23. Fast Numerical Solution of Exterior Helmholtz Problems with Radiation Boundary Condition by Imbedding
    O. G. Ernst
      PhD thesis, Stanford University      (1994)
    
    My dissertation.
24. A Domain Decomposition Approach to Solving the Helmholtz Equation with a Radiation Boundary Condition
    O. G. Ernst and G. H. Golub
    Contemporary Mathematics    157    (1992)