*It is God's privilege to conceal things,
but the kings' pride is to research them.
(Proverbs 25:2)*

A= Analysis, C = Combinatorics, F = Foundations,

L = Linear Algebra, N = Numerical Analysis, O = Optimization,

P = Physics/Chemistry, S = Statistics

In some cases, there are associated publications, see Recent Papers and Preprints

LP.

F.

N.

N.

F.

P.

In this lecture, a timeline is traced from Huygens' wave optics to the modern concept of light according to quantum electrodynamics. The lecture highlights the closeness of classical concepts and quantum concepts to a surprising extent. For example, it is shown that the modern quantum concept of a qubit was already known in 1852 in fully classical terms.

P.

In this lecture the results of the historical review given in my lecture ''Classical models for quantum light'' are utilized to reassess the meaning of observables and stochastic processes for the classical and quantum description of light.

In particular we discuss the description of partially coherent, fluctuating light through classical stochastic Maxwell equations (with uncertainty in the initial conditions only), and look at a generalization that works for all quantum aspects of arbitrary quantum systems.

N.

O.

O.

F.

Slides for a lecture at the AUTOMATHEŘ 2009 workshop, giving an overview of our work in the FMathL project for creating a modeling and documentation language for mathematics, suited to the habits of mathematicians

NO.

Using tools from functional analysis and global optimization, methods are presented for obtaining, given an approximate solution of a partial differential equation, realistic error bounds for some response functional of the solution.

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.

P.

This lecture (the second of three) discusses work towards a new, classical view of quantum mechanics. It is based on an analysis of polarized light, of the meaning of quantum ensembles in a field theory, of classical simulations of quantum computing algorithms, and resulting optical models for the simulation of quantum mechanics.

In particular, it is shown that classical second-order stochastic optics is precisely the quantum mechanics of a single photon, with all its phenomenological bells and whistles.

P.

This lecture discusses foundational problems on the nature of light revealed by 1. attempts to define a probability concept for photons, 2. quantum models for photons on demands (and their realization through laser-induced emission by a calcium ion in a cavity), 3. models explaining the photo effect, and 4. Bell-type experiments for single photon nonlocality.

O.

pdf file (173K)

O.
**
A. Neumaier,
Constrained global optimization**,
Slides, 2005.

pdf file (225K)

O.
**
A. Neumaier,
Worst case analysis of mechanical structures by interval methods**,
Slides, 2005.

pdf file (263K)

OS.
**
A. Neumaier,
Uncertainty modeling for robust verifiable design**,
Slides, 2004.

pdf file (318K)

O.
**
A. Neumaier and J.-P. Merlet,
Constraint satisfaction and global optimization in robotics**,
Slides, 2002.

pdf file (449K)

Global Optimization

Protein Folding

Interval Methods

Regularization

Mathematical Software

Computational Mathematics Links

Mathematics Links

Statistics Links

my home page (http://www.mat.univie.ac.at/~neum)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)