Dr. Konrad Simon
Research scientist Numerical Methods in Geosciences
CliSAP/CEN, Universität Hamburg
keywords: multiscale numerics, advection-dominated problems, finite elements, Galerkin methods, inverse problems, basis reconstruction, numerics of atmosphere and ocean, Lagrangian fluid dynamics, semi-Lagrangian methods
problem: Modeling and solving complex physical systems described by partial differential equations poses many challenges to mathematicians and practitioners. Typically, different parts of the system are modeled by processes that interact on various spacial and temporal scales. Simulations of such systems, e.g., climate simulations, is extremely challenging.
Since every numerical simulation possesses a truncation scale important subgrid processes can not be resolved and must be taken care of by different means, e.g., by so-called parametrizations. The coupling between parametrized processes and the computational model (e.g., the dynamical core) is, however, often just heuristic. For reliable simulations, in particular over long time, coupling strategies to the (main) computational model/scale are essential.
goal: Our current goal is to improve the mathematical consistency of coupling different scales for transient systems with advection-dominated behavior. Current multiscale methods have problems with advection dominated systems since these break essential assumptions on locality. The goal is to come up with a new framework of methods that retains this locality and yet does not loose any power of "standard" multiscale methods (which themselves are still subject to current research). It is out believe that this can be achieved by "doing something physical", i.e., a numerical method connecting multiple scales must take into account (possibly different) physical behavior on all of its scales. This involves in particular the question of what boundary conditions one can impose on subgrid models.
outcome: We used a simple 1-dimensional model for advection-dominated but yet diffusive tracer transport with multiple scales to demonstrate that correct boundary conditions for subgrid problems are essential and can be adjusted such that upscaling is meaningful in the sense that the way information travels is respected. The difficulty here is that subgrid problems need to be adjusted as well as their coupling to (coarse) computational scales. Our method is not practical but helps to understand, first, generalizabilty of our idea to higher dimensions and, secondly, it helps to come up with solutions for problems encountered using, for example, full Lagrangian techniques. Thirdly, our method suggests a way to separate subgrid problems strongly which makes our idea suitable for strong parallelization. The latter is clearly desired due to modern high-performance computer architectures.
Currently we investigate generalizations of our idea to PDEs in a semi-Lagrangian setting in two and three dimensions. We also investigate their applicability to systems of multiscale PDEs. First results for advectin-diffusion-reaction equations are very promising.
other research interests: domain decomposition methods for multi-physics systems, isogeometric methods, computer vision, medical imaging, pattern recognition, exploration and understanding of high-dimensional data
long-term research goals:
- A framework of Galerkin methods (finite elements, discontinuous Galerkin, etc.) for complex advection-dominated physical systems
- An improved understanding of (numerical) scale interactions
- An improved understanding of how to design consistent and stable numerical methods for multiscale problems (a bit differential geometry and physics might help here)
- A rigorous analysis of our method and possible a-posteriori error estimators
- Implementing the method on near-spherical geometries
- Pushing these methods towards applicability
keywords: shape analysis and matching, inverse problems, optimization, (nonlinear) elasticity theory, (nonlinear) finite elements, computer vision and graphics, nonlinear dispersive PDEs, functional analysis, monotone operators, unbounded operators
PhD work: During my PhD studies at the Weizmann Institute of Science, Israel, I was part of the computer vision and graphics group. My PhD thesis was focused on the development of physics based methods (PDE models) for the shape alignment problem which has applications in medical imaging, computer graphics and vision.
outcome: A set of three methods for (pairwise) shape matching that account for unknown (observed) large deformations of shapes, e.g., anatomical shapes, in 2D and 3D. An improved understanding of the origin of deformations can be provided by finding a possible physical cause (external forces) in a suitable optimization framework.
work on PDEs: We investigated the applicability of novel so-called time averaging methods to systems of dispersive PDEs. These methods are simpler than those coming from harmonic analysis and by-pass the use of complicated dispersive Sobolev spaces. Furthermore, we believe that these methods reveal the regularizing character of dispersion (e.g., fast rotations) much clearer than powerful but complicated methods form harmonic analysis.
outcome: We can show well-posedness of a dispersive system that models the interaction of baroclinic and barotropic waves in the mid-latitudes with relatively simple means. Furthermore, we show that it is not necessary to invert a relatively complicated ODE analytically which was done in the original paper. Instead, we employ a mode splitting into high and low modes and achieve well-posedness in spaces of relatively rough functions.
Diploma thesis: My diploma thesis (German equiv. to master thesis) is about well-posedness of a class of integro-differential equations that arise in aeronautical engineering of aircraft wings. These equations involve singular integrals and often fairly complicated nonlinearities.
outcome: My thesis summarizes results in the literature in a comprehensive and detailed way in order to make the results and methods accessible and clearer. Furthermore, we point out the essential difficulties and necessities to obtain well-posedness of the relevant models and suggest generalizations.
- K.S., J. Behrens, Multiscale finite elements for transient advection-diffusion equations through advection-induced coordinates, submitted (2018), Preprint
- K.S., R. Basri, Elasticity-based Matching by Minimizing the Symmetric Difference of Shapes, IET Computer Vision (2017), PDF
- K.S., S. Sheorey, D. Jacobs, R. Basri, A Hyperelastic Two-Scale Optimization Model for Shape Matching, SIAM Journal on Scientific Computing SISC 2016, in press PDF
- K.S., S. Sheorey, D. Jacobs, R. Basri, A Linear Elastic Force Optimization Model for Shape Matching, Journal of Mathematical Imaging and Vision (2015) PDF
- Y. Guo, K.S. and E.S. Titi, Global Well-posedness of a System of Nonlinearly Coupled KdV equations of Majda and Biello, Communications in Mathematical Sciences (2014) PDF
- K.S., Linear and Nonlinear Elasticity Models for Shape Matching, PhD thesis, The Weizmann Institute of Science (Israel), 2015
- K.S., Methoden der Theorie Monotoner Operatoren und ihre Anwendung auf Integral- und Integro-Differenzialgleichungen, Diploma thesis (in German), University of Leipzig (Germany), 2009
Talks and Conferences
Invited Talk on "Flow-induced Coordinates and Reconstruction Methods for Transient Advection-Dominated Problems with Multiple Scales", KTH Stockholm, Stockholm, Sweden, May 2018
Talk on "Flow-induced Coordinates and Reconstruction Methods for Transient Advection-Dominated Problems with Multiple Scales", European Geophysical Union General Assembly 2018, Vienna, Austria, April 2018
Talk on "Flow-induced Coordinates and Reconstruction Methods for Transient Advection-Dominated Multiscale Problems", PalMod Open Science Conference 2018, Vienna, Austria, April 2018
Talk on "Flow-induced Coordinates and Reconstruction Methods for Transient Advection-Dominated Multiscale Problems", Scales and Scaling Cascades in Geophyscal Systems 2018, Hamburg, Germany, April 2018
- Talk on "Flow-induced Coordinates for Transient Advection-Diffusion Equations with Multiple Scales", conference on "Mathematics of the Weather", Erqy, France, October 2017
- Talk on "Flow-induced Coordinates for Transient Advection-Diffusion Equations with Multiple Scales", workshop on "Multi-scale modelling of ice characteristics and behaviour", Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, September 2017
Talk on “A Langrangian multiscale FEM for transient passive advection-diffusion equations with strong transport”, Conference “PDEs on the Sphere 2017”, Paris, France, April 2017
Workshop on “Scaling Cascades in Complex Systems 2017”, Berlin, Germany, March 2017
Talk at the PalMod annual meeting 2017 on “A Multiscale Finite Element Method for Subgrid Scale Representation”, Lübeck, Germany, February 2017
Workshop on “Different Mathematical Perspectives on Description of Unresolved Scales in Multiscale Systems”, Oberwolfach, Germany, November 2016
Dynamical Core Intercomparison Project 2016 (DCMIP), NCAR, Boulder (CO), USA, June 2016
Talk on “Linear and Nonlinear Elasticity Methods for Shape Matching” Technion, Israel, June 2015
Conference on “Numerical Methods and Applied Geometry”, The Weizmann Institute of Science, Israel, May 2015
Workshop on “Mathematical Imaging and Statistical Machine Learning”, The Weizmann Institute of Science, Israel, March 2012
Spring School on “Mathematical Fluid Dynamics“, Technische Universität Darmstadt, Germany, February/March 2011
Conference on “Recent Advances in Nonlinear Evolutionary Equations and Analysis of Multi-Scale Phenomena“, The Weizmann Institute of Science, Israel, July 2010
International Summer School on “Mathematical Fluid Dynamics“, Levico Terme, Italy, June/July 2010
I wrote a Julia wrapper for generation and local refinement of 2D triangular meshes (get official Julia package).
I recently started implementing ideas in Julia - a cross-platform language designed at MIT for scientific computing with great interfacing to C/C++ and Python. If you are often prototyping numerical algorithms I really recommend to have a look here. Note that this language is not only suitable for prototyping and can be used for large scale scientific computations.
For "more serious" and scalable implementations the open source C++-library deal.ii offers a great framework for solving PDEs with finite elements and discontinuous Galerkin methods. In terms of C++ and numerical methods you have to know (at least a bit) what you are doing but the overhead of learning the library is worth it and its documentation is very good.
In my free time I like to enjoy wakeboarding.
Numerical Methods in Geosciences
Integrated Climate System Analysis and Prediction (CliSAP)
University of Hamburg
20144 Hamburg, Germany
Fax : +49-40-42838-7712