Forschungsschwerpunkte - Research

Forschungsbericht - Research Report

Marie Curie International Training Network
Multi-ITN STRIKE - Novel Methods in Computational Finance

The computational complexity of mathematical models employed in financial mathematics has witnessed a tremendous growth that requires the development of advanced numerical techniques appropriate for the most present-day applications in financial industry. Besides a series of internationally recognized researchers from academics, leading quantitative analysts from the financial industry also participate in this network. The challenge lies in the necessity of combining transferable techniques and skills such as mathematical analysis, sophisticated numerical methods and stochastic simulation methods with deep qualitative and quantitative understanding of mathematical models arising from financial markets. The main training objective is to prepare, at the highest possible level, young researchers with a broad scope of scientific knowledge, in particular computational finance, and to teach transferable skills, like social awareness which is very important in view of the recent financial crises. In this research training network our aim is to deeper understand complex nonlinear financial models and to develop effective and robust numerical schemes for solving problems arising from the mathematical theory of pricing financial derivatives and related financial products. This aim will be accomplished by means of financial modeling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. Within the ITN Strike Network, the Chair of Scientific Computing in Würzburg is responsible of the investigation of new robust and efficient optimal control techniques for financial market models. Our contribution to the formulation of optimization and control strategies for stochastic models and models with random data is summarized in the following papers:

M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes
Journal of Computational and Applied Mathematics, 237 (2013), 487-507.

M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes, Mathematical Modelling and Analysis, 15 (2010), 393-407.

A. Borzì and G. von Winckel, A POD framework to determine robust controls in PDE optimization, Computing and Visualization in Science, 14 (2011), 91-103.

A. Borzì, V. Schulz, C. Schillings, and G. von Winckel, On the treatment of distributed uncertainties in PDE constrained optimization, GAMM Mitteilungen, 33 (2010), 230-246.

A. Borzì, Multigrid and sparse-grid schemes for elliptic control problems with random coefficients, Computing and Visualization in Science, 13 (2010), 153-160.

A. Borzì and G. von Winckel, Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients, SIAM J. Sci. Comp., 31 (2009), 2172-2192


Steuerung von Quantensystemen - Control of Quantum Systems

Control of quantum phenomena is becoming central in a variety of systems with present and perspective applications ranging from quantum optics and quantum chemistry to semiconductor nanostructures. In most cases, quantum control is aiming at quantum devices and molecular systems where there is the need to manipulate quantum states with highest possible precision. The successful application of optimal control theory in this field together with the enormous effort towards nanosciences explains the large growing interest in quantum control problems. Our focus is the development and investigation of fast and robust solvers of finite- and infinite-dimensional quantum control problems within an interdisciplinary scientific environment including biology, chemistry, mathematics, and physics.
In the realm of quantum control problems, one can roughly identify classes of problems ranging from the realization of laser pulses to break a bond in a molecule or to drive a certain chemical reaction, to the control of photochemical processes and the control of quantum dots. More generally, control may be required to drive state transitions, maximize observable expectation, and obtain best performance of quantum operators. Our contribution to the solution of these problems is summarized in the following papers:

G. von Winckel and A. Borzì, QUCON: A fast Krylov-Newton code for dipole quantum control problems, Computer Physics Communications, 181 (2010), 2158-2163.

G. von Winckel, A. Borzì, and S. Volkwein, A globalized Newton method for the accurate solution of a dipole quantum control problem, SIAM J. Sci. Comp., 31 (2009), 4176-4203

G. von Winckel and A. Borzì, Computational techniques for a quantum control problem with H1-cost, Inverse Problems, 24 (2008) 034007.

A. Borzì and U. Hohenester, Multigrid optimization schemes for solving Bose--Einstein condensates control problems, SIAM J. Sci. Comp., 30 (2008), 441-462

P. Ditz and A. Borzì, A cascadic monotonic time-discretized algorithm for finite-level quantum control computation, Computer Physics Communications, 178 (2008), 393-399.

A. Borzì, J. Salomon, and S. Volkwein, Formulation and numerical solution of finite-level quantum optimal control problems, Journal of Computational and Applied Mathematics,216 (2008), 170-197

U. Hohenester, P.K. Rekdal, A. Borzì, J. Schmiedmayer, Optimal quantum control of Bose-Einstein condensates in magnetic microtraps, Phys. Rev. A 75, 023602 (2007)

A. Borzì and E. Decker, Analysis of a leap-frog pseudospectral scheme for the Schroedinger equation, Journal of Computational and Applied Mathematics, 193(1) (2006), 65-88.

A. Borzì, G. Stadler, and U. Hohenester, Optimal quantum control in nanostructures: Theory and application to a generic three-level system , Phys. Rev. A 66, 053811 (2002).

Mehrgitterverfahren - Multigrid Methods

Multigrid methods represent one of the most important breakthrough in the development of fast solvers of large-scale algebraic problems. In particular, multigrid principles allow to define iterative schemes with optimal computational complexity for solving discretized partial differential equations (PDEs). More recently, multigrid schemes have been applied successfully to solve PDE optimization problems and inverse problems. Our purpose is the development and analysis of multigrid methods for simulation and optimization with PDE models and the formulation of multigrid optimization schemes for unconstrained optimization. Our contribution to the development of multigrid schemes is summarized in the following papers:

S. Gonzalez Andrade and A. Borzì, Multigrid solution of a Lavrentiev-regularized state-constrained parabolic control problem,
Numerical Mathematics: Theory, Methods and Applications, 5 (2012), 1-18.

M. M. Butt and A. Borzì, A full multigrid solution of control-constrained Cauchy-Riemann optimal control problems ,
Journal of Numerical Mathematics, 19 (2011), 189-214.

S. Gonzalez Andrade and A. Borzì, Multigrid second-order accurate solution of parabolic control-constrained problems,
Computational Optimization and Applications, 2010.

A. Borzì, Multigrid and sparse-grid schemes for elliptic control problems with random coefficients,
Computing and Visualization in Science, 13 (2010), 153-160.

A. Borzì and G. von Winckel, Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients,
SIAM J. Sci. Comp., 31 (2009), 2172-2192

A. Borzì and V. Schulz, Multigrid methods for PDE optimization,
SIAM Review, 51 (2009), 361-395.

O. Lass, M. Vallejos, A. Borzì, and C.C. Douglas, Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems,
Computing, 84 (2009), 27-48

M. Vallejos and A. Borzì, Multigrid optimization methods for linear and bilinear elliptic optimal control problems,
Computing, 82 (2008), 31-52

A. Borzì, Smoothers for control- and state-constrained optimal control problems,
Computing and Visualization in Science, 11 (2008), 59-66.

A. Borzì, Space-time multigrid methods for solving unsteady optimal control problems,
(Chapter 5) in L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes and B. van Bloemen Waanders (Eds.),
Real-Time PDE-Constrained Optimization, Computational Science and Engineering, Vol. 3, SIAM, Philadelphia, 2007.

A. Borzì, High-order discretization and multigrid solution of elliptic nonlinear constrained optimal control problems,
Journal of Computational and Applied Mathematics, 200 (2007), 67-85.

A. Borzì and K. Kunisch, A globalization strategy for the multigrid solution of elliptic optimal control problems,
Optimization Methods and Software, 21(3) (2006), 445-459

A. Borzì and R. Griesse, Experiences with a space-time multigrid method for the optimal control of a chemical turbulence model,
Int. J. Numer. Meth. Fluids., 47 (2005), 879-885.

A. Borzì, On the convergence of the MG/OPT method,
PAMM, 5(1) (2005), 735-736

A. Borzì and K. Kunisch, A multigrid scheme for elliptic constrained optimal control problems,
Computational Optimization and Applications, 31 (2005), 309-333.

A. Borzì and G. Borzì, Algebraic multigrid methods for solving generalized eigenvalue problems,
International Journal for Numerical Methods in Engineering, 65(8) (2006), 1186-1196.

A. Borzì, Solution of lambda-omega systems: Theta-schemes and multigrid methods,
Numerische Mathematik, 98(4) (2004), 581-606.

A. Borzì and G. Borzì, An efficient algebraic multigrid method for solving optimality systems,
Computing and Visualization in Science, 7(3/4) (2004), 183-188.

A. Borzì, Multigrid methods for parabolic distributed optimal control problems,
J. Comp. Appl. Math, 157 (2003), 365-382.

A. Borzì, K. Kunisch, and Do Y. Kwak, Accuracy and Convergence Properties of the Finite Difference Multigrid Solution of an Optimal Control Optimality System,
SIAM J. Control Opt., 41(5) (2003), 1477-1497.

A. Borzì and G. Borzì, An algebraic multigrid method for a class of elliptic differential systems,
SIAM J. Sci. Comp., 25(1) (2003), 302-323.

A. Borzì, K.W. Morton, E. Süli and M. Vanmaele, Multilevel Solution of Cell Vertex Cauchy-Riemann Equations, SIAM J. Sci. Comp.,18(2) (1997),441-459.

A. Borzì and A. Koubek, A multi-grid method for the resolution of thermodynamic Bethe ansatz equations,
Comp. Phys. Commun., 75, 118-126,(1993).

Robuste Energie-Optimierung bei Gärprozessen in der Produktion von Biogas und Wein: ROENOBIO - Robust energy optimization of fermentation processes for the production of biogas and wine: ROENOBIO

Gärung spielt als grundlegender biochemischer Prozess in vielfältigen technologischen Prozessen eine herausragende Rolle. In diesem Projekt sollen Gärprozesse bei der Produktion von Biogas und Wein räumlich und zeitlich detailliert modelliert und simuliert werden. Durch die kombinierte Untersuchung der Biogas- und der Weinproduktion ist es möglich, Synergieeffekte zu nutzen, da die leichter kontrollierbare Weingärung als Prototyp für die komplexere Biogasgärung betrachtet werden kann. Die Modelle werden basierend auf einem gemeinsamen generischen Grundmodell entwickelt mit dem Ziel, die Prozesse v.a. hinsichtlich des thermischen Energieaufwands zu optimieren, wobei produktspezifische Aspekte wie Gasausbeute (Biogas) oder Aromaprofil (Wein) berücksichtigt werden. Damit gehört dieses Projekt zum Bedarfsfeld Klima/Energie der Hightech-Strategie der Bundesregierung. Es werden Modelle zur Beschreibung der Dynamik des Gärprozesses als biologisches Phänomen in Form von Integro-Differentialgleichungen mit räumlich strukturierten Strömungsund Diffusionsmodellen zusammengeführt. Als Resultat wird ein 3D Multiphysics-System zusammen mit biologischen Modellaspekten simuliert. Auf dem Weg zu einem validen Modell gilt es, Modellparameter aus Versuchen und Messdaten der Industriepartner zu schätzen und die Daten/Versuche hinsichtlich ihrer statistischen Qualität einzuordnen, um verbesserte Versuchspläne vorzuschlagen. Aufbauend auf den erzielten numerischen Simulationsmodellen in verschiedenen Detaillierungstiefen werden Methoden zur optimalen Steuerung der Prozesse entwickelt, um hiermit modellprädiktive Regelung durchzuführen. Darüber hinaus sollen auch die Form und Platzierung von Sensoren und Aktuatoren optimiert werden. Die drei beteiligten Hersteller von Biogasanlagen werden die erzielten Ergebnisse sowohl zur optimierten Steuerung existierender Anlagen als auch zur Optimierung neu konzipierter Anlagen einsetzen. Die beiden Weinbauversuchsanstalten werden die Ergebnisse in Form von Beratungsdienstleistung für die Weinindustrie verwerten. Die Firma fp sensorsystems wird die Ergebnisse in ihrem Regelungskonzept umsetzen. Durch die FuE-Leistung dieses Projektes erwarten wir sowohl eine erhebliche Stärkung der internationalen Wettbewerbsfähigkeit der deutschen Wirtschaft als auch eine Weiterentwicklung der MSO-Technologie im Bio-Multiphysics-Kontext.


Optimierung mit partiellen Differentialgleichungen - PDE Optimization

The purpose of PDE optimization is to define ways of how optimally change, influence, or estimate features of systems modeled by partial differential equations. An important class of problems in PDE optimization results from optimal control applications. Other important classes of optimization problems are shape design, topology, and parameter optimization. Optimization is also an essential tool for solving many inverse problems. Our research work aims at the formulation, approximation, and efficient solution of PDE optimization problems also in the presence of data and model uncertainty. Our contribution to the formulation and solution of PDE optimization problems is summarized in the following papers:

A. Borzì and G. von Winckel, A POD framework to determine robust controls in PDE optimization,
Computing and Visualization in Science, 14 (2011), 91-103.

M. M. Butt and A. Borzì, Formulation and multigrid solution of Cauchy-Riemann optimal control problems,
Computing and Visualization in Science, 14 (2011), 79-90.

M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes,
Mathematical Modelling and Analysis, 15 (2010), 393-407.

A. Borzì, V. Schulz, C. Schillings, and G. von Winckel, On the treatment of distributed uncertainties in PDE constrained optimization,
GAMM Mitteilungen, 33 (2010), 230-246.

M. Annunziato and A. Borzì, Fast solvers of Fredholm optimal control problems,
Numerical Mathematics: Theory, Methods and Applications, 3 (2010), 431-448.

A. Borzì and R. Griesse, Distributed optimal control of lambda-omega systems,
Journal of Numerical Mathematics, 14 (2006), 17-40.

A. Borzì, K. Ito, and K. Kunisch, Optimal Control Formulation for Determining Optical Flow,
SIAM J. Sci. Comput., 24(3) (2002), 818-847.

A. Borzì, K. Ito, and K. Kunisch, An Optimal Control Approach to Optical Flow Computation,
Int. J. Numer. Meth. Fluids., 40 (2002), 231-240.

A. Borzì and K. Kunisch, The Numerical Solution of the Steady State Solid Fuel Ignition Model and Its Optimal Control,
SIAM J. Sci. Comp., 22(1) (2000), 263-284.

Bildverarbeitung - Image analysis

The development of tomographic and other non-invasive imaging devices represent an exceptional contribution to medicine, material sciences, and arts. This development has led to a fast growth in the interdisciplinary field of imaging science where mathematics plays a central role providing a rigorous basis for imaging analysis. We investigate mathematical methods and fast algorithms for image processing and computer vision. The main focus is on techniques using partial differential equations and optimization. Our contribution to the formulation of image analysis methodologies is summarized in the following papers:

Davide Calebiro, Finn Rieken, Julia Wagner, Titiwat Sungkaworn, Ulrike Zabel, Alfio Borzì, Emanuele Cocucci, Alexander Zuern, Martin J. Lohse, Single-molecule analysis of fluorescently labeled GPCRs reveals receptor-specific complexes with distinct dynamics and organization,
Proceedings of the National Academy of Sciences (PNAS), to appear

A. Borzi, K. Ito, and K. Kunisch, Optimal Control Formulation for Determining Optical Flow,
SIAM J. Sci. Comput., 24(3) (2002), 818-847.

A. Borzì, K. Ito, and K. Kunisch, An Optimal Control Approach to Optical Flow Computation,
Int. J. Numer. Meth. Fluids., 40 (2002), 231-240.

A. Borzì, M. di Bisceglie, C. Galdi, G. Giangregorio, Robust registration of satellite images with local distorsions,
Proceedings IEEE International Geoscience & Remote Sensing Symposium July 12-17, 2009, Cape Town, South Africa

A. Borzì, H. Grossauer, and O. Scherzer, Analysis of Iterative Methods for Solving a Ginzburg-Landau Equation,
Int. Journal of Computer Vision, 64 (2005), 203-219.

B. Harrach, On uniqueness in diffuse optical tomography,
Inverse Problems 25 (5), 055010 (14pp), 2009.

B. Harrach, Simultaneous imaging of absorption and scattering in dc diffuse optical tomography,
in: O. Dössel and W. C. Schlegel (Eds.): WC2009, IFMBE Proceedings 25/II, 776-779, 2009.

B. Harrach and J. K. Seo,
Detecting inclusions in electrical impedance tomography without reference measurements,
SIAM J. Appl. Math. 69 (6), 1662-1681, 2009.

B. Harrach, J. K. Seo and E. J. Woo
Factorization Method and Its Physical Justification in Frequency-Difference Electrical Impedance Tomography IEEE Trans. Med. Imaging 29 (11), 1918-1926, 2010.

Projects: Entwicklung einer funktionellen 3D MR-basierten Diagnostik der intestinalen Obstruktion - Parallel Multigrid Imaging and Compressed Sensing for Dynamic 3D Magnetic Resonance Imaging, IZKF (Interdisziplinäres Zentrum für Klinische Forschung der Universität Würzburg)

Inverse scattering and impedance imaging

The common aim in inverse scattering and impedance imaging is to determine unknown material properties inside an object or the distribution of certain unknown obstacles inside an otherwise known background medium from observations of acoustic or electromagnetic waves or electric currents and voltages. In practice such questions manifest themselves in numerous applications, such as, e.g., nondestructive testing, medical imaging, geophysical prospecting or radar imaging. Mathematically they can be formulated as inverse problems for partial differential equations, which are typically ill-posed, as the cause-to-effect map tends to annihilate information and thus is not continuously invertible. Besides standard regularization methods a completely different class of reconstruction methods, which recover specific qualitative features of the unknown quantity, such as, e.g., the support of localized anomalies, has been developed during the past years. These methods are extremely fast, but they require many measurement data. Our research in this direction currently focuses on the analysis and implementation of qualitative reconstruction methods for reduced data sets as well as on variants of qualitative reconstruction methods that can efficiently handle multi-frequency information. Some recent results have been summarized in the following papers:

R. Griesmaier, M. Hanke und J. Sylvester, Far field splitting for the Helmholtz equation, SIAM J. Numer. Anal., 52 (2014), 343–362. (DOI)

R. Griesmaier, M. Hanke und T. Raasch, Inverse source problems for the Helmholtz equation and the windowed Fourier transform II, SIAM J. Sci. Comput., 35 (2013), A2188-2206. (DOI)

R. Griesmaier, M. Hanke und T. Raasch, Inverse source problems for the Helmholtz equation and the windowed Fourier transform, SIAM J. Sci. Comput., 34 (2012), A1544-A1562. (DOI)

R. Griesmaier, Multi-frequency orthogonality sampling for inverse obstacle scattering problems, Inverse Problems, 27 (2011), 085005 (23pp). (DOI)

Formoptimierung - Shape optimization

Shape optimization is a special class of PDE constrained optimization, where the domain is the optimisation variable. Besides classical applications in CFD, as control of fluids and aerodynamic design, shape optimisation appears in many other application fields as computational acoustics, tomography, inverse design, uncertainties quantification and elasticity. Shape optimization problems are among the most challenging applied mathematics problems that are based on rigorous theoretical ground and involve sophisticated algorithms. Thus, this research area combines analysis and geometry with numerics and scientific computing to form structure exploiting algorithms which enable the consideration of large scale problems. Our contribution to the shape optimization is summarized in the following papers:

S. Schmidt and V. Schulz. Impulse response approximations of discrete shape Hessians with application in CFD. SIAM Journal on Control and Optimization, 48(4):2562–2580, 2009.DOI | .pdf ]

S. Schmidt and V. Schulz. Pareto-curve continuation in multi-objective optimization. Pacific Journal of Optimization, 4(2):243–257, 2008.

P.F. Antonietti, A. Borzì, and M. Verani, Multigrid Shape Optimization Governed by Elliptic PDEs,
SIAM J. Control Optim., 51 (2013), 1417-1440

A. Borzi, V. Schulz, C. Schillings, and G. von Winckel, On the treatment of distributed uncertainties in PDE constrained optimization, GAMM Mitteilungen, 33 (2010), 230-246.

Advanced nebular spectral synthesis for Supernovae

Supernovae are bright optical "transient stars", appearing and disappearing again within some 100 days. In fact, these objects are stars which explode after having reached the final stages of their lives. How Supernova explosions exactly take place is still unclear. While some Supernovae are certainly due to the collapse of a massive stellar core (Supernovae of Type Ib/Ic/II), others are most probably thermonuclear explosions of white dwarfs stars (Type Ia). With our project, we aim at an improved understanding of Supernovae by calculating accurate Supernova spectra from numerical Supernova models (synthetic spectra) and checking whether these match observed spectra. At the Astronomy Chair of the University of Würzburg, the group of Prof. Röpke focuses on numerical modelling (hydrodynamical simulations) of Type Ia Supernovae (SNe Ia) - or rather of possible explosion scenarios, such as the explosion of one white dwarf star accreting mass from an extended star, or explosions of two merging white dwarf stars [for a review, see Hillebrandt and Niemeyer (2000)]. Whether each such simulation - and in the end the respective scenario - is realistic or not must be tested by comparison to observations. This is done simulating radiative transfer and thus yielding a synthetic spectrum, i.e. the spectrum which one expects from a model supernova. The spectral shape of a supernova critically depends on the chemical elements synthesised in the explosion, such as Oxygen, Silicon, Iron and Nickel (Figure 1). Thus, comparing synthetic and observed spectra, one tests if the chemical stratification and the density structure produced by an explosion model is realistic. One of the group's radiative transfer codes, under on-going development, will calculate "nebular" spectra of SNe Ia - i.e. spectra some 100-300d after explosion. These spectra are sensitive to the ejecta core, where e.g. Nickel shows a very different distribution in white-dwarf mergers as compared to explosions of single white dwarfs with an extended companion star (Röpke et al. 2012). Using basic methods of the NERO code (Maurer et al. 2011), the nebular spectral synthesis code will calculate the stationary excitation/ionisation state of the plasma (atoms, ions, electrons) as well as the radiation field. To this end, the atomic rate equations have to be solved far from thermal equilibrium - in non-LTE (LTE: local thermodynamic equilibrium). This means that the number densities of atoms in each excitation state have to be found such that excitations and deexcitation happen at equilibrated rates (cf. Hachinger et al. 2012). Furthermore, in an outer iteration cycle, the free-electron temperature is determined such that the electron gas is in heating-cooling balance. Both the solution of the rate equations, and the solution to the electron's energy balance amount to finding the root of functions which only known numerically and expensive to evaluate. With our ongoing work, we aim at understanding the numerical behavior of these nonlinear problems and at finding optimal numerical solvers. First attempts with Newton-Raphson iterations using approximative Jacobians have already brought a much better-controlled behavior of the rate solver, and a faster convergence of the outer iteration cycle. First synthetic spectra we are calculating from relatively crude, spherically-symmetric ejecta models show an encouraging qualitative match to typical observations (Figure 1). "Our and others' work on optimising the model caluclations is summarized in the following papers:":

Maurer J. I. et al., "NERO - a post-maximum supernova radiation transport code", 2011, MNRAS, 418, 1517

Hachinger S. et al.: How much H and He is "hidden" in SNe Ib/c? I. - low-mass objects, 2012, MNRAS, 422, 70

Hillebrandt W. and Niemeyer J. C., "Type IA Supernova Explosion Models", 2000, ARA&A, 38, 191

Iwamoto K. et al., "Nucleosynthesis in Chandrasekhar Mass Models for Type IA Supernovae and Constraints on Progenitor Systems and Burning-Front Propagation", 1999, ApJS, 125, 439

Röpke F. K. et al., "Constraining Type Ia Supernova Models: SN 2011fe as a Test Case", 2012, ApJL, 750, L19

Stanishev V. et al., "SN 2003du: 480 days in the life of a normal type Ia supernova", 2007, A&A, 469, 645