Research

For my publications see Sussex Research Online. Preprints can be found on arXiv.org.

At the centre of my research there are fully non-linear differential equations, Hamilton-Jacobi-Bellman equations, stochastic optimal control as well as a priori and a posteriori analysis of finite element methods. My interests include functional analysis, stochastic processes, numerical analysis and scientific computing, with applications to engineering and science.

  • Fully nonlinear equations in non-divergence form are very difficult to discretize, especially with finite element methods. The more common variational notion of a weak solution is replaced with that of a viscosity solution (given certain monotonicity conditions). Viscosity solutions are defined in terms of pointwise inequalities – testing with finite element hat functions, however, corresponds to a local averaging. Thus the numerical method needs to be constructed so that, in essence, the averaged values have in the limit the correct pointwise behaviour. My work with I. Smears [10] gave the first convergence proofs of a finite element method to viscosity solutions of Hamilton-Jacobi-Bellman equations of optimal control. The L result extends the setting of (Barles, Souganidis; 1991): Using alternative projection operators we overcame the long-standing issue of the lack of pointwise consistency of finite element methods. We also provided a proof of strong H1 convergence based on energy arguments, which uses an entirely novel approach. In [4] this approach was extended with weighted Sobolev spaces to include degenerate Bellman equations. See also [3] on the numerical approximation of viscosity boundary conditions and [12] on the choice of the artificial diffusion parameter.
  • Monge-Ampère equations are an important type of fully nonlinear equations arising for example in optimal mass transport, differential geometry and astrophysics. In [5] X. Feng and I established an in the viscosity sense equivalent Bellman formulation of the Monge-Ampère equation, extending Krylov’s work on classical solutions. We designed semi-Lagrangian methods on general triangular grids, for we which showed global convergence of a semismooth Newton solver. This enables to robustly compute numerical approximations on very fine meshes of nonsmooth viscosity solutions, including the challenging degenerate case. An advantage of our construction is that the comparison principle for the Bellman operator extends to nonconvex functions as well. A particular challenge was posed through combination of a lack of consistency near the boundary due to the Wasow-Motzkin theorem and the failure of the comparison principle under viscosity boundary conditions. Interestingly, the method shines light on the open question of well-posedness of the simple Monge-Ampère equation on non-convex domains [2].
  • Ferromagnetic materials in a heat bath are at the nanoscale described by stochastic Landau-Lifschitz-Gilbert equations. In this context [1] we analysed optimal control for switching systems, which arise for instance in data storage devices. With our approach of dynamic programming we could prove the existence of a unique strong solution of the optimal control problem. Prior literature gave only the existence of a weak solution. The crucial and surprising insight permitting this stronger result was a reformulation of the nonlinearity of the linked Bellman equation into an isotropic operator on the state manifold, which gave access to a Hopf-Cole transformation. The resulting numerical scheme allowed us to numerically solve the control problem on a 20 dimensional state manifold (=10 spins), while corresponding prior computations with the Pontryagin approach could only scaled up to 3 stochastic spins (6 dimensions) due to a curse of dimensionality linked to a backward SDE.
  • Typical for equations characterising subsurface flows is the low regularity of the PDE coefficients. S. Bartels, R. Müller and I analysed in [14] the convergence of discontinuous Galerkin (DG) approximations to the solution of the incompressible miscible displacement equations, relevant for instance for enhanced oil recovery and CO2 storage. We only assumed minimal regularity, necessary to define weak solutions and in line with realistic data. In this case the Darcy velocity is in general unbounded. For conforming finite element methods, i.e. if the approximation space belongs to H1(Ω), convergence can be established via the Aubin-Lions lemma from the compact embedding H1(Ω)L2(Ω). But since DG approximations do in general not lie in H1(Ω), an alternative reflexive separable space needs to be identified which contains DG solutions and embeds compactly into L2(Ω). Extensions to higher-order time discretizations are outlined in [13].
  • Discontinuous Galerkin techniques form a family of methods consisting of a number of individual schemes. This has lead to a larger quantity of literature on a posteriori error bounds for such schemes. In [15] C. Carstensen, T. Gudi and I developed a posteriori bounds which rely on the common features of DG methods, thereby offering a unified take on the subject. We apply our technique to 16 DG schemes for the Poisson, Stokes and Lamé equations. We not only recovered several residual-based error bounds derived by other authors separately, but make previously unknown bounds available. Independently from this result, we derived in [18] explicit a posteriori error bounds (i.e. with fully computable constants) for averaging-based estimators for the Poisson, Stokes and Lamé equations.
  • A. Cangiani, E. Georgoulis and I formulated DG methods for mass transfer across semi-permeable membranes for the application of (RAN-driven) transport of molecules across the nucleus wall in living cells. The model is posed as a time-dependent semi-linear convection-diffusion-reaction system on multi-component computational domains. A key element of the analysis is the treatment of the non-monotone, non-Lipschitzian transmission taking place on the domain interfaces. Our analysis establishes optimal a priori bounds for both the semi-discrete and the fully discrete numerical schemes [9]. In [6] the case of fast reactions is incorporated, where the nonlinear reaction terms in each compartment only admit local Lipschitz conditions. The nonlinear interface conditions model selective permeability, congestion and partial reflection.
  • Joule heating models the generation of heat in semiconductors and micro-mechanical devices. A. Målqvist and I analysed the effect of Joule heating with mixed boundary conditions, as they typically arise in applications. A primary motivation for our work was the positioning of micro-optical devices with Joule heating (Henneken, Tichem, Sarro; 2006). Our bounds transfer the recent regularity estimates in Besov spaces on creased domains from (Mitrea, Mitrea; 2007) to a non-linear setting. We proved the convergence of Galerkin methods on general Lipschitz domains with mixed boundary conditions exhibiting severe corner singularities, extending a weak convergence technique by (Browder; 1968). In this way our method avoids a discrete maximum principle, which has in the prior literature lead to very restrictive mesh conditions in 3D.
  • Friedrichs systems are a framework to treat hyperbolic systems and equations of mixed type in a variational setting. One of the main challenges arises where boundary conditions change type as there loss of regularity may break integration-by-parts formulas, making in turn the variational approach more difficult to handle. I provided in [20], also [1617], a density result of smooth functions in the domain of the differential operator (in L2(Ω) in the sense of unbounded operators). This lead to a description of the space of traces, which established the basis for the proof of convergence of DG approximations, only assuming weak differentiability of the exact solution along the characteristics of the differential operator. Such solutions may be of unbounded variation.
  • While DG finite element methods are one of the most successful approaches to treat convection-dominated second-order elliptic differential equations, they have been criticised for their use of ‘additional’ degrees of freedom which do not improve the approximation quality of the finite element space. A. Cangiani, J. Chapman, E. Georgoulis and I proved in the linear setting that discretizations with the ‘additional’ degrees of freedom of DG methods only in the interior and boundary layers exhibit the same stability bound as used for the full DG approximation space [87]. A separate study with P. Houston and E. Süli showed that a strengthened stability bound for the original DG method can be obtained through the use of least-squares flux terms [19].

 

 

References

[1]   M. Jensen, A. Majee, A. Prohl, Dynamic programming for finite ensembles of nanomagnetic particles, Journal of Scientific Computing, 80(1):351-375, 2019.

[2]   M. Jensen, Numerical solution of the simple Monge-Ampère equation with nonconvex Dirichlet data on non-convex domains, Hamilton-Jacobi-Bellman Equations; Kalise, Kunisch, Rao (eds.); De Gruyter, 2018.

[3]   M. Jensen, I. Smears, On the notion of boundary conditions in comparison principles for viscosity solutions, Hamilton-Jacobi-Bellman Equations; Kalise, Kunisch, Rao (eds.); De Gruyter, 2018.

[4]   M. Jensen, L2(Hγ2) finite element convergence for degenerate isotropic Hamilton-Jacobi-Bellman equations, IMA Journal of Numerical Analysis, 37(3):1300-1316, 2017.

[5]   X. Feng, M. Jensen, Convergent semi-Lagrangian methods for the Monge-Ampère equation, SIAM Journal on Numerical Analysis, 55(2):691-712, 2017.

[6]   A. Cangiani, E. Georgoulis, M. Jensen, Discontinuous Galerkin methods for fast reactive mass transfer through semi-permeable membranes, Applied Numerical Mathematics, 104:3-14, 2016.

[7]   A. Cangiani, J. Chapman, E. Georgoulis, M. Jensen, On local super-penalization of interior penalty discontinuous Galerkin methods, International Journal of Numerical Analysis & Modeling, 11(3):478-495, 2014.

[8]   A. Cangiani, J. Chapman, E. Georgoulis, M. Jensen, On the Stability of Continuous-Discontinuous Galerkin Finite Element Methods for Singularly-Perturbed Advection-Diffusion-Reaction Equations, Journal of Scientific Computing, 57(2):313-330, 2013.

[9]   A. Cangiani, E. Georgoulis, M. Jensen, Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes, SIAM Journal on Numerical Analysis, 51(5):2911-2934, 2013.

[10]   M. Jensen, I. Smears, On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations, SIAM Journal on Numerical Analysis, 51(1):137-162, 2013.

[11]   M. Jensen, A. Målqvist, Finite Element Convergence for the Joule Heating Problem with Mixed Boundary Conditions, BIT Numerical Mathematics, 53(2):475-496, 2013.

[12]   M. Jensen, I. Smears, Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations, Proceedings of ENUMATH 2011, 267-274, 2013.

[13]   M. Jensen, R. Müller, Stable Crank-Nicolson Discretisation for Incompressible Miscible Displacement Problems of Low Regularity, Proceeding of ENUMATH 2009, 469-477,2010.

[14]   S. Bartels, M. Jensen, R. Müller, Discontinuous Galerkin Finite Element Convergence for Incompressible Miscible Displacement Problems of Low Regularity, SIAM Journal on Numerical Mathematics, 2009, 47(5):3720-3743.

[15]   C. Carstensen, T. Gudi, M. Jensen, A Unifying Theory of A Posteriori Error Control for Discontinuous Galerkin FEM, Numerische Mathematik, 2009, 112(3):363-379.

[16]   M. Jensen, Remarks on Duality in Graph Spaces of First-Order Linear Operators, PAMM 6:31-34, 2006.

[17]   M. Jensen, On the Discontinuous Galerkin Method for Friedrichs Systems in Graph Spaces, Lecture Notes in Computer Science, Vol. 3743, 2006.

[18]   C. Carstensen, M. Jensen, Averaging Techniques for Reliable and Efficient A Posteriori Finite Element Error Control, Contemporary Mathematics, 383:15-34, 2006.

[19]   P. Houston, M. Jensen and E. Süli, hp-Discontinuous Galerkin FEM with Least-Squares Stabilization, Journal of Scientific Computing, 17 (1-4):3-25, 2002.

[20]   M. Jensen, Discontinuous Galerkin Methods for Friedrichs Systems with Irregular Solutions, D.Phil. thesis, University of Oxford, 2005.