Computational Methods for Direct Field Problems (CM) - Ulrich Langer
Group Leader: Ulrich Langer
The "Computational Mathematics Group" (CMG) will mainly focus on the development, analysis and implementation of novel computational methods for complicated multiple field problems (e.g., but not exclusively, modelled by systems of partial differential equations) arising in classical and novel application fields like computational fluid mechanics, structural mechanics, electro-magnetics, computational physics and chemistry, computational finance and, in the future, computational life sciences. Methods currently studied include domain decomposition and multigrid methods with special emphasis on adaptivity. Multi-level approximation techniques are among the most powerful tools for constructing fast and efficient (mostly parallel) methods for solving discretised field problems of high complexity. The development of the numerical methods is usually driven by hard applications. The numerical handling of hard applications requires some kind of Scientific Problem Solver Environment (PSE) that contains a set of interacting scientific computing tools such as:
- Geometry and problem editor
- Mesh generator and mesh handler
- Parallelisation tools
- Solver tools (adaptive discretisation and solution techniques)
- Optimisation tools
Topics where this group plans to do research includes:
Numerical Methods: Construction -Analysis -Parallelisation
At the Radon Institute, the CMG will focus its main methodological research activities on the following advanced numerical methods with different fields of applications:
Robust Adaptive Multilevel FEM Methods:
Multigrid and multilevel methods are at least potentially the most powerful methods for solving large scale systems of finite element equations. The weak point of the standard Geometrical Multigrid (GMG) methods is the lack of the robustness that makes the direct methods so attractive to commercial users. To make multigrid convergence rate robust to bad parameters typical for some class of applications (e.g. elasticity propels for almost incompressible materials, thin plate or shell problems, anisotropies etc.), one needs deep theoretical insight into the interplay of the multigrid components. The construction and the analysis of robust GMG methods for different clauses of problems is our first main research direction.
Algebraic Multigrid Methods:
The CMG has investigated AMG methods mainly for SPD problems where we have developed a special technique for constructing the AMG components from an auxiliary problem. This approach was especially successful for Maxwell equations discretised by edge elements. At the Radon Institute, we will look for successful applications of these techniques including symmetric and indefinite problems as well as non-symmetric system of algebraic equations which are typically arising from the discretisation of Navier-Stokes equations.
Domain Decomposition (DD) Methods:
DD techniques are not only the basic parallelisation tool but also the basic method for handling different discretisation techniques in one scheme and multifield problems. The CMG has a lot of experience in handling the BEM-FEM coupling via DD techniques. We will develop DD techniques especially for surface coupled multifield problems as well as for coupling of different discretisation techniques.
Solvers for Discretised Optimisation Problems:
The construction of fast so-called All-At-Once solvers for large-scale optimisation problems arising typically from the finite element discretisation of optimisation problems with PDE constrains is a hot topic in the research with enormous potential use for a lot of practical applications. A successful work on these problems requires the combination of optimisation methods, regularisation techniques and preconditioning techniques. All the methods mentioned above (i.e. GMG, AMG and DD methods) can be used for constructing preconditioners for the KKT system or for parts (blocks) of the KKT system. In close co-operation with the Inverse Problems Group we will continue research in this direction, that we have already started in the SFB "Numerical and Symbolic Scientific Computing".
Our parallelisation approach is based on a special calculus with distributed data that is an formalisation of the data distribution and the corresponding operation that we used earlier in the non-overlapping domain decomposition method, This parallelisation strategy was used for developing a parallel geometrical multigrid 3D Maxwell solvers.
Inverse Problems (IP) - Otmar Scherzer
Group Leader: Otmar Scherzer
Inverse problems are concerned with determining (usually numerically) causes for desired or observed effects, which is a type of problems frequently arising in science and engineering. Application areas where the Linz group has been active include inverse scattering, parameter identification in partial differential equations, imaging science, deconvolution problems in physics, inverse problems in refractive optics, inverse heat conduction problems with applications in steel processing, and non-destructive testing.
The property that makes inverse problems mathematically challenging is their "ill-posedness", which renders traditional numerical methods inherently unstable. This means that a solution to an inverse problem might neither exist nor be unique, and even if some generalised concept of solution is introduced, then this solution depends in a discontinuous way on the data. Since in applications, data always contain noise, which is highly amplified due to the ill-posedness of the inverse problem, the (in)stability issue is of high practical relevance. Special "regularisation" methods have to be developed to stabilise the solution process of inverse problems. Until about a decade ago, mathematical research concentrated on linear ill-posed problems. For non-linear problems, a variational method carried over from the linear theory, namely Tikhonov regularisation, had long been used and mathematically analysed. The main emphasis now, however, seems to be on iterative regularisation methods. We aim at developing iterative regularisation methods in a "theory-based" way, i.e., at analysing their stability, convergence and convergence rates and, on this basis, develop a-posteriori stopping rules for optimal convergence rates: a peculiar feature of iterative methods for inverse problems is that, while the error initially decreases with progressing iterations, in the presence of noise, after some time, the error starts to increase again, finally without bound. This is due to noise propagation; for practical success of an iterative regularisation method, it is therefore of high importance to stop the iteration at the right time, which is possible only based on a mathematical analysis as indicated. Methods to be used come from non-linear functional analysis. When solving a complex inverse problems, e.g., a parameter identification problem for a transient and spatially three-dimensional partial differential equation, the efficient coupling of the "inverse iteration" with the "forward solver" (in this case, the method used for solving the pde with a given parameter) is crucial. The final aim is to obtain an accurate solution in as little overall computing time as possible, and in a way which is stable with respect to data noise. To reach this aim, one has to analyse the direct and the inverse problems as an entity; there are various ways to do this, which we are both currently investigating: either, one can control the necessary accuracy with which the direct solver has to work via its a posteriori error estimates from the inverse iteration, or one can use an all-at-once approach, where the inverse problem is formulated as an optimisation problem with the forward equation as a constraint. This issue is the topic of an ongoing co-operation, which we will further intensify in the Radon Institute, between the groups of Prof. Langer and Prof. Engl.
Symbolic Computation (SC) - Josef Schicho
Group Leader: Josef Schicho
Symbolic computation is concerned with algorithmic manipulations of symbolic objects. These could be objects in formal language, such as formulas or programs, or algebraic objects, such as polynomials or residue classes, or geometric objects, such as lines or circles. Research in symbolic computation combines advanced mathematics with advanced computing techniques.
Linz has a strong tradition in the field of symbolic computation going back at least to the foundation of RISC by Bruno Buchberger in 1987. The symbolic computation group at RICAM works mainly on the following topics:
Symbolic-numeric methods for polynomial system solving:
Fo many applications of polynomial system solving - for instance in computer aided geometric design - both the given data and the requested result are modelled by floating point numbers; small numerical errors are unavoidable. On the other hand, many algorithms for polynomials, such as implicitization or parametrization, are severely ill-posed: a slight change in the coefficient may cause a qualitative change. We devise algorithms for approximate polynomials and study their numerical behavior.
Exact methods for polynomial system solving:
There are, on the other hand, also instances of polynomial system solving where the existence of an exact solution is guaranteed by number theory, and the size of the solutions can be controlled. Our goal is to compute the solutions exactly in these cases, and to extend the domain of problems where exact methods can be used.
Symbolic Functional Analysis:
Here, symbolic methods are applied to various problems in functional analysis. The main objects of study are operators between function spaces, as arising in differential equations, particularly boundary problems. Representing the relevant operator identities by noncommutative polynomials, one can apply Groebner bases for manipulating certain operators. In the case of two-point boundary problems, this leads to symbolic algorithms for solving and factoring such problems.
Here, the goal is to develop tools and techniques for constructing and geometric understanding of the singular solution variety of polynomial equations. We use resolution, parametrization by multivariate power series or Puiseux series, differential forms, arc spaces.
Piecewise algebraic representations of curves and surfaces:
In the simulation and optimization of free form shapes, one frequently uses polynomial or rational spline functions, both implicitly, as the zero contour of spline functions, and explicitly, as image of parametrizations. We develop algorithms for the design and analysis of such shapes.
Analysis of Partial Differential Equations (PDE)
Groupleader position currently vacant.
This research group will work mainly on modelling with and analysis of partial differential equations. It will provide a direct interface to the groups of the Radon Institute in financial mathematics (Schachermayer) and of Engl/Langer in inverse problems/numerical analysis and simulation. The interaction will be very tight since modern pde modelling/analysis and numerics/simulation cannot survive without one another in the near future. Although we fully appreciate the fact that research guidelines have to be extremely flexible (also pushed by an unprecedented hardware development which nowadays makes simulations possible which nobody dreamt about only ten years ago), some main new topics, which will be developed within the Radon Institute, have emerged.
1) The 'next' generation of pde-analysts will to a large extent be concerned with non-linear system of pdes. It will be a main challenge to overcome already well-developed tools for scalar problems (like maximum principles) pushing the so far underdeveloped theory of pde systems.
2) Continuing research on (fully) non-linear pdes: Monge-Ampere equation, Monge-Kantorovich mass transportation problems, Wasserstein metric, entropies for diffusive systems, convex Sobolev inequalities, Hamilton-Jacobi equations, free boundary problems (typical technique: viscosity solutions), connections to control theory, geometric motions (mean curvature flow, pattern formation), Hamiltonian-Lagrangian (quantum) mechanics, homogenisation of non-linear pdes. There are close links to level set methods as mentioned in the description of the Inverse Problems Group.
3) Stochastic pdes with space-time stochasticity (random Schrödinger equations, Brownian motions on manifolds, random diffusion systems, stochastic non-linear pdes). Typical applications are in quantum mechanics (impurity scattering), geophysics, finance mathematics, mathematical biology.
Already ongoing research of the group of Peter Markowich on models for the dynamics of large particle ensembles will continue. This covers the range from basic quantum mechanical models (e.g., (non-linear) Schrödinger and Dirac equations) to classical macroscopic fluid models, including (semi-)classical and quantum kinetic models (e.g., Boltzmann, Fokker-Planck, Wigner equations). A particular emphasis will be on scaling limits and the analysis of large time behaviour. Specific topics are:
4) Kinetic models in population dynamics and mathematical biology: the dynamic interaction of genotype/phenotype and spatial (geography) effects will be studied by means of kinetic modelling (the "genotype/phenotype" variables play the role of the momentum variable, dual to the spatial variable). Chemotaxis will be studied on the kinetic level, with particular emphasis ondiffusion limits and microscopic modelling of the motion of bacteria. It is conceivable that finite-time blow-up can be avoided on the kinetic level by careful modelling of reorientation processes.
5) Macroscopic limits of kinetic models: Applications include charge transport in semiconductors, ionisation of gases, and kinetic models for wave-particle scattering. An understanding of fluid limits of the latter might lead to a new approach for understanding macroscopic models of gas dynamics.
6) Modelling and Simulation of Bose-Einstein condensation: Bose-Einstein condensation is one of the extremely hot topics in low-temperature atomic physics (cf. the Nobel Prize in physics 2001). The typical model after condensation is the Gross-Pitaevskii equation (cubically non-linear Schrödinger equation with harmonic confinement), while the bosonic Boltzmann equation is used to describe the condensation process. We expect to continue numerical simulations and the analysis of the condensation process (transition regime of the bosonic Boltzmann equation to the Gross-Pitaevskii equation).
7) Derivation, analysis and numerics of quantum Boltzmann equations by means of Wigner transforms: This work is supposed to shed new light on the origin of irreversibility in semi-classical and other scaling limits. New insights into open quantum systems are expected by employing quantum entropy and quantum entropy-dissipation techniques similarly to the classical-mechanics case.
8) Inverse problems in fluid-type pdes: applications in semiconductors, nano-technology, geophysics. In this area there will be significant interaction with semiconductor device engineers outside the Radon Institute and with the in-house groups of Engl and Langer. Main topics will be doping profile identification and device performance optimisation.
Optimization and Optimal Control (OOC) - Karl Kunisch
Group Leader: Karl Kunisch
The research of this group focuses on optimal control of infinite dimensional systems typically with constraints given by partial differential equations. The mathematical analysis of particular problems as well as their numerical treatment are investigated. Open loop optimal control problems typically involve large systems of coupled partial differential equations. Recently significant progress has been made in their solution based on both, improved concepts from optimisation theory (SQP-methods and semi-smooth Newton methods in function space) and numerical analysis (hierarchical methods, adaptivity). These new techniques now allow the control of problems in fluid dynamics which were unsolvable only a few years ago. Future work will encompass the control of coupled systems (control by magnetic fields, fluid-structure interaction), control of quantum mechanical systems, as well as closed-loop control. The latter requires to consider the Hamilton-Jacobi-Bellman equation which is still untractable for systems of practical size. Therefore model reduction techniques, including proper orthogonal systems, balanced truncation and the center manifold theorem (nonlinear Galerkin methods) will be investigated. These methods are important also in their own right, for instance for real-time optimal control.
Mathematical Imaging (MI) - Otmar Scherzer
Group Leader: Otmar Scherzer
The objective of imaging is to make recorded data accessible for efficient visual and automatic inspection. We are interested in the design process of imaging devices for thermoacoustical imaging, and for computational image processing of well established imaging modalities, where we focus on the design and efficient implementation of differential equations and variational methods for data processing, like noise reduction, pattern recognition, enhancing, and segmentation.
Mathematical Methods in Molecular and Systems Biology (BIO) - Philipp Kügler and Christian Schmeiser
The group on "Mathematical Methods in Molecular and Systems Biology"
(M3SB) focusses on interdisciplinary projects, motivated by questions
in Molecular and Cell Biology, sometimes including biomedical aspects.
The group is located in Vienna (3rd district, Apostelgasse 23), since
most of the (actual and potential) national cooperation partners in the
Life Sciences are located here. This includes the "Institute for Molecular
Biotechnology" (IMBA, Austrian Academy of Sciences), the "Center for
Molecular Medicine" (CeMM, Austrian Academy of Sciences), the "Max F.
Perutz Laboratories" (MFPL, University of Vienna and Medical University
of Vienna), and the "Institute for Science and Technology, Austria"
The mathematical expertise of M3SB ranges from parameter identification to partial differential equations and image analysis. Biological expertise has been acquired in
- biochemical reaction networks,
- cell motility, in particular the chemo-mechanics of the cytoskeleton.
Various related subjects, such as asymmetric cell division, ion channels,
chemotaxis, and cell membranes have also been investigated, partially as
reaction to requests from the Life Science community. The main projects,
M3SB is involved in at the moment:
Actin driven cell motility
The subject of an ongoing cooperation with IMBA is the imaging and mathematical modeling of the lamellipodium, a flat cell protrusion, driven by actin polymerization and acting as the motility machinery of many cell types. The architecture of the cytoskeleton, as observed by electron tomography, is explained by mathematical models of the cytoskeletal dynamics. A specialization to leukocytes and an extension to motility in three-dimensional environments is considered in a cooperation with IST.
Chemical reaction network theory
An internal cooperation with the RICAM group "Symbolic Computation" aims at an open software platform for symbolic analysis of biochemical networks. Another activity deals with inverse problems in stochastic models. Interdisciplinary cooperations deal with various subjects, such as metabolic networks.
Transfer Group (TG) - Ronny Ramlau
Group Leader: Ronny Ramlau
Research cooperations with industry have been a long-standing strength of mathematics in Linz. The RICAM transfer group works on mathematical problems motivated by industrial applications arising in the partner companies. Together with the Industrial Mathematics Competence Center at MathConsult, research activities are currently carried out in the following fields:
- Fast numerical solvers for differential equations in industrial applications
- Surrogate models in industrial applications
- Inverse problems and optimization
Motivation for these problems arises, e.g., in multiphysics problems, in automotive simulation or in mathematical finance. Frequently, similar mathematical techniques can successfully be applied to quite different fields of industrial background.
Project: Applied Discrete Mathematics and Cryptography (DC) - Arne Winterhof
Project Leader: Arne Winterhof
This research project focuses on application areas as cryptography, coding theory and the analysis of pseudorandom number generators where methods from discrete mathematics, as for example exponential sum techniques, are promising. Special focus is put on finite fields and their applications, in particular sequences in cryptography and numerical integration.
NFG Project: Multiscale modeling and simulation of crowded transport in the life and social sciences - Marie-Therese Wolfram
Project Leader: Marie-Therese Wolfram
The New Frontiers Group focuses on the mathematical modeling and simulation of crowded transport on multiple scales. The research objectives include general modeling aspects of crowded motion, i.e. translation of the microscopic interactions on the macroscopic scale, as well as their application in biology and social sciences. Examples include pedestrian motion, cell motility, animal herding or the transport of charged particles in biological and synthetic channels. Based on the derived mathematical models and their analytic behavior, we focus on the development of efficient numerical methods on the macroscopic scale and their consistent coupling with microscopic approaches.
Start Project: Sparse Approximation and Optimization in High Dimensions - Massimo Fornasier
Project Leader: Massimo Fornasier
Aims and research
The dimension scale of problems arising in our modern information society became very large.
A new area of science and engineering is now urgently needed in order to extract and interpret
significant information from the universe of data collected from a variety of modern sources
(Internet, physics experiments, medical diagnostics, etc.).
Numerical simulations at the required scale will be one of the great challenges of the 21st century. In short, we need to become capable of organizing and understanding complexity.
The most notable recent advances in data analysis and numerical simulation are based on the
observation that in several situations, even for very complex phenomena, only a few governing
components are required to describe the whole dynamics; a dimensionality reduction can be achieved
by demanding that the solution be "sparse" or "compressible".
Since the relevant degrees of freedom are not prescribed, and may depend on the particular solution, we need efficient optimization methods for solving the hard combinatorial problem of identifying them.
In this project we will first address the problem of designing efficient algorithms which allow us to
achieve sparse optimization in high-dimensions.
Secondly, the tools which we will develop for achieving adaptive dimensionality reductions will subsequently be used as building blocks for solving large-scale partial differential equations or variational problems arising in various contexts.
Finally, we will apply the whole machinery to interesting applications in image processing, numerical simulation, and we will explore new applications in innovative fields such as automatic learning of dynamical systems.
The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)
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