Examples of current research projects
 Structurepreserving variational methods for nonlinear partial differential equations with an underlying gradient flow structure

Various wellknown nonlinear evolution equations allow for a gradient flow formulation with respect to the Wasserstein metric. Examples include the porous medium equation, the CahnHilliard equation, and the thinfilm 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 DerridaLebowitzSpeerSpohn equation using a structurepreserving 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 structurepreserving 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

Investigations focus on deriving novel numerical methods for fully nonlinear equations such as MongeAmpère and Bellman equations. One research led by Max Jensen studies the convergence of semiLagrangian methods [arXiv:1602.04758] for the MongeAmpère 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 [arXiv:1003.0292,arXiv:1103.2970,arXiv:1311.3930]. Max Jensen et al. [arXiv:1201.3581,arXiv:1507.00140] 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

Option pricing problems in financial mathematics frequently lead to anisotropic parabolic partial (integro)differential equations. Bertram Düring works on highorder methods for their efficient solution as well as on PDEconstrained 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

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 levelset 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

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

Particular interest lies in geometrically based motions arising from free boundary problems; numerical methods for stochastic PDEs in phase transition [arXiv:1111.6312].
For details on this project and related research areas, contact Omar Lakkis
Recent publications
2019
Carrilo, José A, Düring, Bertram, Kreusser, Lisa Maria and Schöenlieb, CarolaBibiane (2019) Stability analysis of line patterns of an anisotropic interaction model. SIAM Journal on Applied Dynamical Systems, 18 (4). pp. 17981845. ISSN 15360040
Düring, Bertram and Pitkin, Alexander (2019) Highorder compact finite difference scheme for option pricing in stochastic volatility jump models. Journal of Computational and Applied Mathematics, 355. pp. 201217. ISSN 0377‐0427
Jensen, Max, Majee, Ananta K, Prohl, Andreas and Schellnegger, Christian (2019) Dynamic programming for finite ensembles of nanomagnetic particles. Journal of Scientific Computing, 80 (1). pp. 351375. ISSN 08857474
Düring, Bertram, Torregrossa, Marco and Wolfram, MarieTherese (2019) Boltzmann and FokkerPlanck equations modelling the Elo rating system with learning effects. Journal of Nonlinear Science, 29 (3). pp. 10951128. ISSN 09388974
Düring, Bertram, Gottschlich, Carsten, Huckemann, Stephan, Kreusser, Lisa Maria and Schönlieb, CarolaBibiane (2019) An anisotropic interaction model for simulating fingerprints. Journal of Mathematical Biology, 78 (7). pp. 21712206. ISSN 03036812
Deckelnick, Klaus, Elliott, Charles M, Miura, TatsuHiko and Styles, Vanessa (2019) HamiltonJacobi equations on an evolving surface. Mathematics of Computation. ISSN 00255718
Düring, Bertram and Pitkin, Alexander (2019) Highorder compact finite difference scheme for option pricing in stochastic volatility with contemporaneous jump models. 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) Efficient hedging in Bates model using highorder compact finite differences. The 5th AMMCS International Conference, Waterloo, Ontario, Canada, August 1823, 2019. Published in: Makarov, Roman, (ed.) Recent Advances in Mathematical and Statistical Methods for Scientific and Engineering Applications. 259 489498. Springer Verlag ISSN 21941009
D¨uring, Bertram, Pareschi, Lorenzo and Toscani, Giuseppe (2018) Kinetic models for optimal control of wealth inequalities. European Physical Journal B: Condensed Matter and Complex Systems, 91 (10). 265 112. ISSN 14346028
Garcke, Harald, Lam, Kei Fong and Styles, Vanessa (2018) CahnHilliard inpainting with the double obstacle potential. SIAM Journal on Imaging Sciences, 11 (3). pp. 20642089. ISSN 19364954
Jensen, Max (2018) Numerical solution of the simple Monge–Ampère equation with nonconvex dirichlet data on nonconvex domains. In: Dante, Kalise, Kunisch, Karl and Zhiping, Rao (eds.) HamiltonJacobiBellman Equations: Numerical Methods and Applications in Optimal Control. Radon Series on Computational and Applied Mathematics . De Gruyter, Berlin,, pp. 129142. ISBN 9783110543599
Jensen, Max and Smears, Iain (2018) On the notion of boundary conditions in comparison principles for viscosity solutions. In: Kalise, Dante, Kunisch, Karl and Rao, Zhiping (eds.) HamiltonJacobiBellman Equations: Numerical Methods and Applications in Optimal Control. Radon Series on Computational and Applied Mathematics . De Gruyter, Berlin, pp. 143154. ISBN 9783110543599
Deckelnick, Klaus and Styles, Vanessa (2018) Stability and error analysis for a diffuse interface approach to an advectiondiffusion equation on a moving surface. Numerische Mathematik, 139 (3). pp. 709741. ISSN 0029599X
Carrillo, José A, Düring, Bertram, Matthes, Daniel and McCormick, David S (2018) A Lagrangian scheme for the solution of nonlinear diffusion equations using moving simplex meshes. Journal of Scientific Computing, 75 (3). pp. 14631499. ISSN 08857474
Burger, Martin, Düring, Bertram, Kreusser, Lisa Maria, Markowich, Peter A and Schönlieb, CarolaBibiane (2018) Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models and Methods in Applied Sciences, 28 (3). pp. 409451. ISSN 02182025
2017
During, Bertram, Georgiou, Nicos and Scalas, Enrico (2017) A stylised model for wealth distribution. In: Akura, Yuji and Kirman, Alan (eds.) Economic Foundations of Social Complexity Science. Springer Singapore, Singapore, pp. 95117. ISBN 9789811057045
Dűring, Bertram and Heuer, Christof (2017) Essentially highorder compact schemes with application to stochastic volatility models on nonuniform grids. 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. 313319. ISBN 9783319612829
Düring, Bertram, Miles, James and Hendricks, Christian (2017) Sparse grid highorder ADI scheme for option pricing in stochastic volatility models. 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. 295312. ISBN 9783319612829
Barrett, John W, Deckelnick, Klaus and Styles, Vanessa (2017) Numerical analysis for a system coupling curve evolution to reactiondiffusion on the curve. SIAM Journal on Numerical Analysis (SINUM), 55 (2). pp. 10801100. ISSN 00361429
Feng, Xiaobing and Jensen, Max (2017) Convergent semiLagrangian methods for the MongeAmpère equation on unstructured grids. SIAM Journal on Numerical Analysis, 55 (2). pp. 691712. ISSN 00361429
Düring, Bertram, Jüngel, Ansgar and Trussardi, Lara (2017) A kinetic equation for economic value estimation with irrationality and herding. Kinetic and Related Models, 10 (1). pp. 239261. ISSN 19375093
Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2017) Local existence for the nonresistive MHD equations in nearly optimal Sobolev spaces. Archive for Rational Mechanics and Analysis, 223 (2). pp. 677691. ISSN 00039527
Richardson, G, Please, C and Styles, V (2017) Derivation and solution of effectivemedium equations for bulk heterojunction organic solar cells. European Journal of Applied Mathematics, 28 (6). pp. 9731014. ISSN 09567925
Yang, Feng Wei, Venkataraman, Chandrasekhar, Styles, Vanessa and Madzvamuse, Anotida (2017) A robust and efficient adaptive multigrid solver for the optimal control of phase field formulations of geometric evolution laws. Communications in Computational Physics, 21 (1). pp. 6592. ISSN 18152406
2016
Düring, Bertram and Miles, James (2016) Highorder ADI scheme for option pricing in stochastic volatility models. Journal of Computational and Applied Mathematics, 316. pp. 109121. ISSN 03770427
McCormick, David S, Olson, Eric J, Robinson, James C, Rodrigo, Jose L, VidalLópez, Alejandro and Zhou, Yi (2016) Lower bounds on blowingup solutions of the threedimensional Navier–Stokes equations in H˙^{3/2}, H˙^{5/2}, and B˙^{5/2}_{2,1}. SIAM Journal on Mathematical Analysis (SIMA), 48 (3). pp. 21192132. ISSN 00361410
Düring, Bertram, Schönlieb, CarolaBibiane and Wolfram, Wolfram, eds. (2016) Gradient flows: from theory to application. ESAIM: Proceedings and Surveys, 54 . EDP Sciences, SMAI.
Chemin, JeanYves, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2016) Local existence for the nonresistive MHD equations in Besov spaces. Advances in Mathematics, 286. pp. 131. ISSN 00018708
2015
McCormick, David S, Robinson, James C and Rodrigo, Jose L (2015) Existence and uniqueness for a coupled parabolicelliptic model with applications to magnetic relaxation. Archive for Rational Mechanics and Analysis, 214 (2). pp. 503523. ISSN 00039527
Düring, Bertram and Heuer, Christof (2015) Highorder compact schemes for parabolic problems with mixed derivatives in multiple space dimensions. SIAM Journal on Numerical Analysis, 53 (5). pp. 21132134. ISSN 00361429
Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2015) Higher order commutator estimates and local existence for the nonresistive MHD equations and related models. Journal of Functional Analysis, 267 (4). pp. 10351056. ISSN 00221236
Düring, Bertram and Heuer, Christof (2015) Highorder compact schemes for BlackScholes basket options. In: Progress in industrial mathematics at ECMI 2014. Mathematics in industry . Springer. ISBN 9783642427596
2014
Düring, Bertram, Fournié, Michel and Heuer, Christof (2014) Highorder compact finite difference schemes for option pricing in stochastic volatility models on nonuniform grids. Journal of Computational and Applied Mathematics, 271. pp. 247266. ISSN 03770427
Karakashian, Ohannes and Makridakis, Charalambos (2014) A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Kortewegde Vries Equation. Mathematics of Computation, 84. pp. 11451167. ISSN 00255718
Calatroni, Luca, Düring, Bertram and Schönlieb, CarolaBibiane (2014) ADI splitting schemes for a fourthorder nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems  Series A, 34 (3). pp. 931957. ISSN 10780947
Giesselmann, Jan, Makridakis, Charalambos and Pryer, Tristan (2014) Energy consistent DG methods for the NavierStokesKorteweg system. Mathematics of Computation, 83 (289). pp. 20712099. ISSN 00255718
2013
McCormick, David S, Robinson, James C and Rodrigo, Jose L (2013) Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO. Milan Journal of Mathematics, 81 (2). pp. 265289. ISSN 14249286
Brett, C E A, Lam, K F, Law, K J H, McCormick, D S, Scott, M R and Stuart, A M (2013) Accuracy and stability of filters for dissipative PDEs. Physica D: Nonlinear Phenomena, 245 (1). pp. 3445. ISSN 01672789