Examples of current research projects
- Structure-preserving variational methods for nonlinear partial differential equations with an underlying gradient flow structure
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Various well-known non-linear evolution equations allow for a gradient flow formulation with respect to the Wasserstein metric. Examples include the porous medium equation, the Cahn-Hilliard equation, and the thin-film equation. Efficient numerical methods for their accurate approximation are notoriously difficult to derive. We develop and analyse novel schemes which respect the equations' underlying gradient flow structure. These schemes are able to preserve important properties of the continuous solution, as strict monotonicity of the energy functional, preservation of mass, and positivity, at the discrete level in order to obtain accurate and stable algorithms.
Numerical solution of the nonlinear fourth order Derrida-Lebowitz-Speer-Spohn equation using a structure-preserving Lagrangian scheme. The method [Düring, Matthes and Milišić, 2010] exploits the underlying gradient flow structure and is based on a fully discrete variant of the minimising movement scheme.
Numerical solution of the porous medium equation using a structure-preserving Lagrangian scheme whose key ingredient is the gradient flow structure of the dynamics. The scheme [Carrillo, Düring, Matthes and McCormick, 2018] approximates the associated Lagrangian maps using a finite dimensional subspace of linear maps in space, while the time discretisation is based on the minimising movement scheme.
For details on this project and related research areas, contact Bertram Düring.
- Numerical methods for fully nonlinear equations
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Investigations focus on deriving novel numerical methods for fully nonlinear equations such as Monge--Ampère and Bellman equations. One research led by Max Jensen studies the convergence of semi-Lagrangian methods [] for the Monge--Ampère equation $\det D^2 u(x)=f(x,u(x),\nabla u(x))$, while Omar Lakkis and Tristan Pryer looked at Hessian recovery methods and a posteriori residual estimation adaptive methods [,,]. Max Jensen et al. [,] prove the uniform and $L^2(H^1)$ convergence of finite element approximations to viscosity solutions of Bellman equations.
For details on this project and related research areas, contact Max Jensen and Omar Lakkis
- Optimal control
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Option pricing problems in financial mathematics frequently lead to anisotropic parabolic partial (integro-)differential equations. Bertram Düring works on high-order methods for their efficient solution as well as on PDE-constrained optimal control problems for calibrating model option prices to observed market prices.
Max Jensen works on numerical methods for stochastic dynamic programming equations. A focus is on the convergence behaviour of numerical approximations to viscosity solutions of Hamilton—Jacobi—Bellman equations. His work is also concerned with the interplay of optimal transport and optimal control and discretisation of convexity constraints through Bellman equations.
For details on this project and related research areas, contact Max Jensen
- Free boundary problems
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Research concentrates on the analysis and numerical treatment of free boundary problems motivated by shape and topology optimization, parameter estimation and control of differential variational inequalities. Besides the introduction of level-set based shape optimization methods, this led to the development of semismooth Newton techniques with many application areas ranging from contact problems in elasticity through computational finance to mathematical image processing.
For details on this project and related research areas, contact Vanessa Styles
- Finite element methods for complex flow problems
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Research comprises, for example, a priori and a posteriori error estimates for stabilised finite element methods using continuous or discontinuous approximation spaces, the design of monotone finite element methods and the analysis of the dissipative structure of continuous or discontinuous Galerkin methods.
- Nonlinear parabolic stochastic PDEs
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Particular interest lies in geometrically based motions arising from free boundary problems; numerical methods for stochastic PDEs in phase transition [].
For details on this project and related research areas, contact Omar Lakkis
Recent publications
2023
Jensen, Max and Jaroszkowski, Bartosz (2023) Applied Mathematical Finance, 29 (3). pp. 213-226. ISSN 1350-486X
2022
Jensen, Max, Kiss, Istvan, Rempała, Grzegorz A, Di Lauro, Francesco, KhudaBukhsh, Wasiur R and Kenah, Eben (2022) Journal of the Royal Society Interface, 19 (191). pp. 1-16. ISSN 1742-5662
Jaroszkowski, Bartosz and Jensen, Max (2022) Applied Mathematics and Optimization, 85. a8 1-32. ISSN 1432-0606
2019
Carrilo, José A, Düring, Bertram, Kreusser, Lisa Maria and Schöenlieb, Carola-Bibiane (2019) SIAM Journal on Applied Dynamical Systems, 18 (4). pp. 1798-1845. ISSN 1536-0040
Düring, Bertram and Pitkin, Alexander (2019) Journal of Computational and Applied Mathematics, 355. pp. 201-217. ISSN 0377‐0427
Jensen, Max, Majee, Ananta K, Prohl, Andreas and Schellnegger, Christian (2019) Journal of Scientific Computing, 80 (1). pp. 351-375. ISSN 0885-7474
Düring, Bertram, Torregrossa, Marco and Wolfram, Marie-Therese (2019) Journal of Nonlinear Science, 29 (3). pp. 1095-1128. ISSN 0938-8974
Düring, Bertram, Gottschlich, Carsten, Huckemann, Stephan, Kreusser, Lisa Maria and Schönlieb, Carola-Bibiane (2019) Journal of Mathematical Biology, 78 (7). pp. 2171-2206. ISSN 0303-6812
Deckelnick, Klaus, Elliott, Charles M, Miura, Tatsu-Hiko and Styles, Vanessa (2019) Mathematics of Computation. ISSN 0025-5718
Düring, Bertram and Pitkin, Alexander (2019) In: Faragó, István, Izsák, Ferenc and Simon, Péter L (eds.) Progress in Industrial Mathematics at ECMI 2018. The European Consortium for Mathematics in Industry . Springer Verlag. ISBN 9783030275495
2018
Düring, Bertram and Pitkin, Alexander (2018) The 5th AMMCS International Conference, Waterloo, Ontario, Canada, August 18-23, 2019. Published in: Makarov, Roman, (ed.) Recent Advances in Mathematical and Statistical Methods for Scientific and Engineering Applications. 259 489-498. Springer Verlag ISSN 2194-1009
D¨uring, Bertram, Pareschi, Lorenzo and Toscani, Giuseppe (2018) European Physical Journal B: Condensed Matter and Complex Systems, 91 (10). 265 1-12. ISSN 1434-6028
Garcke, Harald, Lam, Kei Fong and Styles, Vanessa (2018) SIAM Journal on Imaging Sciences, 11 (3). pp. 2064-2089. ISSN 1936-4954
Jensen, Max (2018) In: Dante, Kalise, Kunisch, Karl and Zhiping, Rao (eds.) Hamilton-Jacobi-Bellman Equations: Numerical Methods and Applications in Optimal Control. Radon Series on Computational and Applied Mathematics . De Gruyter, Berlin,, pp. 129-142. ISBN 9783110543599
Jensen, Max and Smears, Iain (2018) In: Kalise, Dante, Kunisch, Karl and Rao, Zhiping (eds.) Hamilton-Jacobi-Bellman Equations: Numerical Methods and Applications in Optimal Control. Radon Series on Computational and Applied Mathematics . De Gruyter, Berlin, pp. 143-154. ISBN 9783110543599
Deckelnick, Klaus and Styles, Vanessa (2018) Numerische Mathematik, 139 (3). pp. 709-741. ISSN 0029-599X
Carrillo, José A, Düring, Bertram, Matthes, Daniel and McCormick, David S (2018) Journal of Scientific Computing, 75 (3). pp. 1463-1499. ISSN 0885-7474
Burger, Martin, Düring, Bertram, Kreusser, Lisa Maria, Markowich, Peter A and Schönlieb, Carola-Bibiane (2018) Mathematical Models and Methods in Applied Sciences, 28 (3). pp. 409-451. ISSN 0218-2025
2017
During, Bertram, Georgiou, Nicos and Scalas, Enrico (2017) In: Akura, Yuji and Kirman, Alan (eds.) Economic Foundations of Social Complexity Science. Springer Singapore, Singapore, pp. 95-117. ISBN 9789811057045
Dűring, Bertram and Heuer, Christof (2017) In: Erhhardt, Matthias, Gunther, Michael and ter Maten, E. Jan W (eds.) Novel Methods of Computational Finance. The European Consortium of Mathematics in Industry, 25 . Springer International, pp. 313-319. ISBN 9783319612829
Düring, Bertram, Miles, James and Hendricks, Christian (2017) In: Erhhardt, Matthias, Gunther, Michael and ter Maten, E. Jan W. (eds.) Novel Methods of Computational Finance. The European Consortium of Mathematics in Industry, 25 . Springer International, pp. 295-312. ISBN 9783319612829
Barrett, John W, Deckelnick, Klaus and Styles, Vanessa (2017) SIAM Journal on Numerical Analysis (SINUM), 55 (2). pp. 1080-1100. ISSN 0036-1429
Feng, Xiaobing and Jensen, Max (2017) SIAM Journal on Numerical Analysis, 55 (2). pp. 691-712. ISSN 0036-1429
Düring, Bertram, Jüngel, Ansgar and Trussardi, Lara (2017) Kinetic and Related Models, 10 (1). pp. 239-261. ISSN 1937-5093
Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2017) Archive for Rational Mechanics and Analysis, 223 (2). pp. 677-691. ISSN 0003-9527
Richardson, G, Please, C and Styles, V (2017) European Journal of Applied Mathematics, 28 (6). pp. 973-1014. ISSN 0956-7925
Yang, Feng Wei, Venkataraman, Chandrasekhar, Styles, Vanessa and Madzvamuse, Anotida (2017) Communications in Computational Physics, 21 (1). pp. 65-92. ISSN 1815-2406
2016
Düring, Bertram and Miles, James (2016) Journal of Computational and Applied Mathematics, 316. pp. 109-121. ISSN 0377-0427
McCormick, David S, Olson, Eric J, Robinson, James C, Rodrigo, Jose L, Vidal-López, Alejandro and Zhou, Yi (2016) SIAM Journal on Mathematical Analysis (SIMA), 48 (3). pp. 2119-2132. ISSN 0036-1410
Düring, Bertram, Schönlieb, Carola-Bibiane and Wolfram, Wolfram, eds. (2016) ESAIM: Proceedings and Surveys, 54 . EDP Sciences, SMAI.
Chemin, Jean-Yves, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2016) Advances in Mathematics, 286. pp. 1-31. ISSN 0001-8708
2015
McCormick, David S, Robinson, James C and Rodrigo, Jose L (2015) Archive for Rational Mechanics and Analysis, 214 (2). pp. 503-523. ISSN 0003-9527
Düring, Bertram and Heuer, Christof (2015) SIAM Journal on Numerical Analysis, 53 (5). pp. 2113-2134. ISSN 0036-1429
Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2015) Journal of Functional Analysis, 267 (4). pp. 1035-1056. ISSN 0022-1236
Düring, Bertram and Heuer, Christof (2015) In: Progress in industrial mathematics at ECMI 2014. Mathematics in industry . Springer. ISBN 9783642427596
2014
Düring, Bertram, Fournié, Michel and Heuer, Christof (2014) Journal of Computational and Applied Mathematics, 271. pp. 247-266. ISSN 0377-0427
Karakashian, Ohannes and Makridakis, Charalambos (2014) Mathematics of Computation, 84. pp. 1145-1167. ISSN 0025-5718
Calatroni, Luca, Düring, Bertram and Schönlieb, Carola-Bibiane (2014) Discrete and Continuous Dynamical Systems - Series A, 34 (3). pp. 931-957. ISSN 1078-0947
Giesselmann, Jan, Makridakis, Charalambos and Pryer, Tristan (2014) Mathematics of Computation, 83 (289). pp. 2071-2099. ISSN 0025-5718
2013
McCormick, David S, Robinson, James C and Rodrigo, Jose L (2013) Milan Journal of Mathematics, 81 (2). pp. 265-289. ISSN 1424-9286
Brett, C E A, Lam, K F, Law, K J H, McCormick, D S, Scott, M R and Stuart, A M (2013) Physica D: Nonlinear Phenomena, 245 (1). pp. 34-45. ISSN 0167-2789
