Abstracts
Artan Borici: Overlap Fermions and Arnoldi unitary process
Computing quark propagators with overlap fermions requires the solution of a shifted unitary linear system. Jagels and Reichel have shown that for such systems it is possible to construct a minimal residual algorithm by short recurrences. Arnold et. al. have found this algorithm to be the fastest among overlap solvers. In this talk we present a three-term recurrence for the Arnoldi unitary process. Using the new recurrence we construct a minimal residual solver which is the fastest among all Krylov subspace algorithms considered so far for the overlap inversion.
Markus Hegland: Penalised least squares, the combination technique and backfitting
Statistical smoothing, machine learning with kernels and radial basis function regression all lead to penalised least squares problems. While sparse in low dimensions, the system matrix of these penalised least squares problems become increasingly dense in higher dimensions. Consequently, the computational complexity of numerical solvers for these problems scales at least quadratically with problem size. As the problem size can be very large, this is an important computational challenge.
In this talk I will first discuss joint work with J. Garcke on the application of the combination technique and its limitations for the penalised least squares problem. Some of these limitations are overcome by the recently introduced Opticom method.
Then I will give an update on a joint project with Martin Gutknecht and Gene Golub on the backfitting algorithm for ANOVA decomposition based models.
Marlis Hochbruck: A multilevel Jacobi-Davidson method for polynomial eigenvalue pde problems
The simulation of drift instabilities in the plasma edge leads to cubic polynomial eigenvalue pde problems with parameter dependent coefficients. The aim is to determine the wave number which leads to the maximum growth rate of the amplitude of the wave. This requires the solution of a large number of eigenvalue pde problems. Since we are only interested in a smooth eigenfunction corresponding to the eigenvalue with largest imaginary part, the Jacobi-Davidson method can be applied. Unfortunately, a naive implementation of this method is much too expensive for the large number of problems that have to be solved.
In this talk we will present a multilevel approach for the construction of an appropriate initial search space. We will also discuss the efficient solution of the correction equation and we will present how optimal scaling helps to accelerate the convergence.
This is joint work with Dominik Löchel, Düsseldorf
Jürg Hutter: Solving non-linear Schrödinger equations in physics and chemistry
Non-linear Schrödinger equations appear in independent particle models in quantum chemistry and solid-state physics. The Kohn-Sham method, the dominant electronic structure method currently used for ground state problems, is of this type. When its solutions are expanded in a finite set of basis functions, the Kohn-Sham equations can be viewed as an optimization problem. Since 50 years, first mostly isolated, recently also in collaborations with the numerical mathematics community, quantum chemists have studied algorithms to solve this problem. I will give an overview over the history of algorithms used in the field and show some of the state-of-the-art methods in use in modern quantum chemistry programs. Finally, I will present open problems and current directions of research.
Eric de Sturler: Methods for the efficient solution of sequences of linear systems
In many computational science and engineering problems we have to solve a sequence of large, sparse, linear systems that change slowly from one system to the next or change in an algebraically structured way. This is often the case in nonlinear systems, optimization, evolutionary problems, eigenvalue problems, Markov Chain Monte Carlo methods, methods involving adaptive meshes, and uncertainty quantification. We have developed a number of techniques to significantly reduce the cost of solving such a sequence of linear systems, such as recycling subspaces generated for previous linear systems and updating preconditioners. We will present a brief overview of these techniques with applications and discuss new convergence theory. Applications include crack propagation, computed tomography, optimal design of structures, model reduction, and problems in computational physics and materials science.
The earliest work on these methods was carried out at the Interdisciplinary Project Center for Supercomputing.
Nick Trefethen: Carathéodory-Fejér approximation 30 years later
Carathéodory-Fejér approximation is a technique for constructing near-best approximations of functions via the SVD of a Hankel matrix of Taylor or Chebyshev coefficients; it is related to the Nevanlinna-Pick problem, to H-infinity control, and to the work of Adamjan, Arov, and Krein. When Martin Gutknecht and I invented CF approximation thirty years ago, we regarded it as a tool of theoretical interest. Time passes, computers speed up, and suddenly these methods are eminently practical! Indeed, for classical approximation on an interval, CF approximation is usually much more efficient than the Remez algorithm. This talk represents joint work with Ricardo Pachon.
Matthias Troyer: The many-body Schrödinger equation: a deceptively simple linear eigenvalue problem
The Schrödinger equation is a deceptively simple linear eigenvalue problem describing the quantum mechanical many body problem. Solutions of this simple 1-line equation include well-known material properties like insulators, metals, semiconductors, superconductors, but also biomolecules, and even cells and full organisms. The problem, however, is that the Schrödinger equation is a 3N-dimensional PDE for N particles, and the complexity thus grows exponentially with N. In this talk I will review two topics:
a) recent developments in quantum information theory show that although the dimension of the Hilbert space is exponential, the ground state and thermal properties of physically relevant Hamiltonians can be parametrized by a an only polynomially large number of parameters, however finding these states can still be QMA-complete.
b) in many models stochastic (Monte Carlo) solvers are very efficient, and I will review recent calculations using millions of particles. However, for the interesting fermionic (electronic) problems, the complexity remains exponential in many interesting cases.