It is God's privilege to conceal things,
but the kings' pride is to research them.
A= Analysis, C = Combinatorics, F = Foundations,
L = Linear Algebra, N = Numerical Analysis, O = Optimization,
P = Physics/Chemistry, S = Statistics
In many cases, there are associated publications, see Recent Papers and Preprints
A. Neumaier, VXQR: Derivative-free unconstrained optimization based on QR factorizations, Slides, 2010.
pdf file (105K)
The method is based on computable bounds for the inverse of linear elliptic operators. Like in the dual weighted residual (DWR) method, our error bounds for response functionals have the quadratic approximation property (so that they are asymptotically optimal), but in contrast to DWR, our bounds are rigorous and also capture the higher order contributions to the error.
Using global optimization techniques, bounds can be found that not only cover the errors in solving the differential equations but also the errors caused by the uncertainty in the parameters. This provides reliable tools for the assessment of uncertainty in the solution of elliptic partial differential equations. Our bounds are independent of the way the approximations are obtained, hence can be used to independently verify the quality of an approximation computed by an arbitrary solver. The bounds not only account for discretization errors but also for other numerical errors introduced through numerical integration and boundary aproximations.
We also discuss how to represent model uncertainty in terms of so-called clouds, which describe the rough shapes of typical samples of various size, without fixing the details of the distribution. Clouds use only information from 1- and 2-dimensional marginal distributions, readily available in practice.
A. Neumaier, Constrained global optimization, Slides, 2005.
pdf file (225K)
A. Neumaier, Worst case analysis of mechanical structures by interval methods, Slides, 2005.
pdf file (263K)
A. Neumaier, Uncertainty modeling for robust verifiable design, Slides, 2004.
pdf file (318K)
A. Neumaier and J.-P. Merlet, Constraint satisfaction and global optimization in robotics, Slides, 2002.
pdf file (449K)
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)