Applications for Physical Sciences

From the beginning of the planning process for the CMA, it has been our strong belief that a Centre devoted to applications of mathematics would benefit substantially by including some activity on advanced scientific computing. Astrophysics and Computational Quantum Mechanics are strong research groups at The University of Oslo, both depending heavily on computations, and representing research areas where new mathematical tools are necessary for further progress. The participation of these groups has therefore been vital for the research profile of the CMA, and we will continue to UiO / CMA Winther 146077/V30 Page 9 strengthen the interactions between mathematics and these areas of physical science in the next five year period.

Astrophysics

The activity in modelling the solar outer atmosphere has followed the research plan closely. The goal of completing a 3D Radiation-Magneto-HydroDynamic code with non-restrictive boundary conditions has been achieved. A number of significant problems in solar/stellar astrophysics are now within reach using these tools. We plan to use the code to study the excitation of waves of different types (slow and fast mode magneto-acoustic waves, Alfvén waves) in the convection zone. How much power that is generated in these wave modes is not known, especially not at high frequencies. The methods developed make it possible to run simulations at sufficiently high resolution to answer several of the outstanding problems. The possible mode-coupling of these waves when they propagate through the upper atmosphere will also be studied. In this matter, it is crucial to have a code with minimal reflections from the upper boundary and our new code is world-leading in this respect. Joint work with the PDE group is planned to further improve the boundary conditions. We will also look at the fully coupled system extending from the convection zone to the corona, building on the pioneering work by B. Gudiksen and V. Hansteen. A new postdoc researcher will be hired on a RCN grant to further strengthen this activity. Specific problems that will be addressed include the structure, dynamics and energy balance of the chromosphere and corona. We will here work on new methods to describe the radiation coupling in the chromosphere, again in collaboration with the PDE group.

Within cosmology, we have, in accordance with the research plan, developed new methods for estimating real-space higher-order correlation functions, that have enabled the application of these to current Wilkinson Microwave Anisotropy Probe (WMAP) and future Planck experiments. We have also devised new, mostly real-space, methods for testing the signals for deviation from Gaussianity. These will be developed further, especially for application to polarization data, and applied to the three-year WMAP data released March 2006 and to the Planck data when we get access to this during 2009 – 2010. In addition to the research mentioned in the research plan, we have started a major effort on improving estimation techniques for the Cosmic Microwave Background (CMB) power spectrum and the cosmological parameters from the CMB maps. These techniques are all based on modern methods of computational Bayesian statistics, e.g., methods building upon the Metropolis-Hastings algorithm, like Gibbs sampling. The preliminary results are extremely promising, and may point towards the methods that will eventually be used to analyse the Planck data. Very recently, we have also started implementing similar methods to the problem of separating the different diffuse components in the CMB maps, i.e., the real CMB signal and the different contributions from gas and dust in the Galaxy. Our aim is to utilise these Bayesian methods as much as possible in the process of getting from the raw data to the final results. This will enable much better control on errors and the probability distributions (i.e., significance limits) of the final results. However, the methods need very substantial further refinements, and collaboration with experts in the research front of modern Monte Carlo methods is necessary, something that is sought in collaboration with the stochastic analysis group.

Computational quantum mechanics

Based on our newly developed diffusion and variational Monte Carlo code, we plan to build a Green's function Monte Carlo program and allow for the implementation of fermionic wave functions for our diffusion Monte Carlo codes. We also plan studies of quasi Monte Carlo methods, used in computational finance to sample points in a deterministic way. These are techniques which reduce the variance and are useful in the Monte Carlo optimization part. Within the framework of stochastic partial differential equations we will study the numerical stability and convergence properties of stochastic partial differential equations. We will generalize these numerical results for other dimensions, domains and for SPDE's with nonlinear and integral terms, and to other types of PDEs, for example the heat equation.

For the nuclear many-body problem our next plans will focus especially on extending the coupled cluster methods to include three-body forces, expected to be crucial for our understanding of shell closures. Parts of this have already been achieved by Hagen and Dean and are expected to be published in Physical Review Letters. Moreover, since a major breakthrough was made recently by Hagen and Hjorth-Jensen on studies of the coupling to weakly bound systems and resonances, we are now able to compute effective interactions for nuclear systems which include weakly-bound states and resonances starting from realistic nucleon-nucleon interaction models. These interactions will in turn be included in our coupled-cluster and shell-model studies. We plan also to extend the coupled-cluster methods to studies of infinite matter such as neutron star matter.