Numerical Analysis and Scientific Computing

Research

 Examples of current research projects

Structure-preserving variational methods for nonlinear partial differential equations with an underlying gradient flow structure

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

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 [arXiv:1602.04758] 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 [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, Omar Lakkis, Tristan Pryer (Reading)

Optimal control

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 Bertram Düring, 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 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 Bertram Düring, 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, Carola-Bibiane (2019) Stability analysis of line patterns of an anisotropic interaction model. SIAM Journal on Applied Dynamical Systems, 18 (4). pp. 1798-1845. ISSN 1536-0040

Düring, Bertram and Pitkin, Alexander (2019) High-order compact finite difference scheme for option pricing in stochastic volatility jump models. Journal of Computational and Applied Mathematics, 355. pp. 201-217. 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. 351-375. ISSN 0885-7474

Düring, Bertram, Torregrossa, Marco and Wolfram, Marie-Therese (2019) Boltzmann and Fokker-Planck equations modelling the Elo rating system with learning effects. 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) An anisotropic interaction model for simulating fingerprints. Journal of Mathematical Biology, 78 (7). pp. 2171-2206. ISSN 0303-6812

Deckelnick, Klaus, Elliott, Charles M, Miura, Tatsu-Hiko and Styles, Vanessa (2019) Hamilton-Jacobi equations on an evolving surface. Mathematics of Computation. ISSN 0025-5718

Düring, Bertram and Pitkin, Alexander (2019) High-order 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 high-order compact finite differences. 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) Kinetic models for optimal control of wealth inequalities. 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) Cahn--Hilliard inpainting with the double obstacle potential. SIAM Journal on Imaging Sciences, 11 (3). pp. 2064-2089. ISSN 1936-4954

Jensen, Max (2018) Numerical solution of the simple Monge–Ampère equation with nonconvex dirichlet data on non-convex domains. 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) On the notion of boundary conditions in comparison principles for viscosity solutions. 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) Stability and error analysis for a diffuse interface approach to an advection-diffusion equation on a moving surface. Numerische Mathematik, 139 (3). pp. 709-741. ISSN 0029-599X

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. 1463-1499. ISSN 0885-7474

Burger, Martin, Düring, Bertram, Kreusser, Lisa Maria, Markowich, Peter A and Schönlieb, Carola-Bibiane (2018) Pattern formation of a nonlocal, anisotropic interaction model. Mathematical Models and Methods in Applied Sciences, 28 (3). pp. 409-451. ISSN 0218-2025

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. 95-117. ISBN 9789811057045

Dűring, Bertram and Heuer, Christof (2017) Essentially high-order compact schemes with application to stochastic volatility models on non-uniform 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. 313-319. ISBN 9783319612829

Düring, Bertram, Miles, James and Hendricks, Christian (2017) Sparse grid high-order 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. 295-312. ISBN 9783319612829

Barrett, John W, Deckelnick, Klaus and Styles, Vanessa (2017) Numerical analysis for a system coupling curve evolution to reaction-diffusion on the curve. SIAM Journal on Numerical Analysis (SINUM), 55 (2). pp. 1080-1100. ISSN 0036-1429

Feng, Xiaobing and Jensen, Max (2017) Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids. SIAM Journal on Numerical Analysis, 55 (2). pp. 691-712. ISSN 0036-1429

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. 239-261. ISSN 1937-5093

Fefferman, Charles L, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2017) Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces. Archive for Rational Mechanics and Analysis, 223 (2). pp. 677-691. ISSN 0003-9527

Richardson, G, Please, C and Styles, V (2017) Derivation and solution of effective-medium equations for bulk heterojunction organic solar cells. European Journal of Applied Mathematics, 28 (6). pp. 973-1014. ISSN 0956-7925

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. 65-92. ISSN 1815-2406

2016

Düring, Bertram and Miles, James (2016) High-order ADI scheme for option pricing in stochastic volatility models. 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) Lower bounds on blowing-up solutions of the three-dimensional 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. 2119-2132. ISSN 0036-1410

Düring, Bertram, Schönlieb, Carola-Bibiane and Wolfram, Wolfram, eds. (2016) Gradient flows: from theory to application. ESAIM: Proceedings and Surveys, 54 . EDP Sciences, SMAI.

Chemin, Jean-Yves, McCormick, David S, Robinson, James C and Rodrigo, Jose L (2016) Local existence for the non-resistive MHD equations in Besov spaces. Advances in Mathematics, 286. pp. 1-31. ISSN 0001-8708

2015

McCormick, David S, Robinson, James C and Rodrigo, Jose L (2015) Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation. Archive for Rational Mechanics and Analysis, 214 (2). pp. 503-523. ISSN 0003-9527

Düring, Bertram and Heuer, Christof (2015) High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions. 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) Higher order commutator estimates and local existence for the non-resistive MHD equations and related models. Journal of Functional Analysis, 267 (4). pp. 1035-1056. ISSN 0022-1236

Düring, Bertram and Heuer, Christof (2015) High-order compact schemes for Black-Scholes 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) High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. Journal of Computational and Applied Mathematics, 271. pp. 247-266. ISSN 0377-0427

Karakashian, Ohannes and Makridakis, Charalambos (2014) A posteriori error estimates for discontinuous Galerkin Methods for the Generalised Korteweg-de Vries Equation. Mathematics of Computation, 84. pp. 1145-1167. ISSN 0025-5718

Calatroni, Luca, Düring, Bertram and Schönlieb, Carola-Bibiane (2014) ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing. Discrete and Continuous Dynamical Systems - Series A, 34 (3). pp. 931-957. ISSN 1078-0947

Giesselmann, Jan, Makridakis, Charalambos and Pryer, Tristan (2014) Energy consistent DG methods for the Navier-Stokes-Korteweg system. Mathematics of Computation, 83 (289). pp. 2071-2099. ISSN 0025-5718

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. 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) Accuracy and stability of filters for dissipative PDEs. Physica D: Nonlinear Phenomena, 245 (1). pp. 34-45. ISSN 0167-2789