Research interests

tl;dr

geometric variational problems and calculus of variations
finite- and infinite-dimensional differential geometry
variational convergence of discretized problems
geometric curvature functionals
self-avoiding energies
fractional Sobolev spaces
numerical optimization
optimization in Riemannian manifolds

My research covers both theoretical and numerical aspects of geometric variational problems for curves and surfaces. This encompasses classical problems from the calculus of variations, such as Euler elasticae, minimal surfaces, and Willmore surfaces, as well as more recent self-avoiding energy models for embedded manifolds. Quite naturally, this brings me in contact with various fields in mathematics, such as differential geometry, topology, functional analysis, partial differential equations, variational analysis, finite element analysis, and numerical optimization. An important inspiration for my work is the shear beauty of the minimizers of these geometric problems. And indeed, the objective functions in this field typically encode regularity (in the sense of absence of curvature). Hence they lend themselves quite naturally as operationalizations of “beauty”. So besides their physical interpretation as deformation or bending energies, they may also be employed as regularizers for all kinds of design, fitting, and inverse problems in, e.g., engineering, computer graphics, optimal control, and medical imaging.

Discretization and convergence

These variational problems are infinite-dimensional, so in order to make them amenable to computational methods, these problems have to be discretized first, i.e., approximated by finite-dimensional optimization problems. Ideally, such a discretization satisfies the following two conditions:

The set of discrete solutions converges under refinement to the set of smooth solutions in a meaningful and interpretable way, e.g., as in following sequence of images that depict discrete minimal surfaces as various mesh resolutions:
The resulting discrete problems can be efficiently solved by computational methods.

In practice, polyhedral discretizations are often preferred for the sake of the latter, even if this leads to nonconforming discretizations. Nonconforming means that the objects of the discrete model do not reside in the domain of the objective functional of the smooth problem. In particular, this occurs if this objective functional depends on derivatives of order above 1, as it is the case, e.g., for curvature energies. Nonetheless, carefully selected polyhedral models produce accurate results. And this is not yet fully understood. Together with the fact that variational problems are typically nonconvex, that they allow for multiple solutions, and that they feature fully nonlinear Euler-Lagrange equations, this makes the convergence analysis very challenging.

Repulsive energies and their computional treatment

More recently, I focused my research efforts on the discretization and optimization of various self-avoiding energies such as O'Hara's Möbius energy and the integral Menger curvatures of curves or the tangent-point energies of curves and surfaces. For an $n$-dimensional submanifold $M \subset\mathbb{R}^m$, these energies are typically multiple integrals of the form \[ \mathcal{E}(M) = \int_M \int_M K(x,y) \, \mathrm{d} \mathcal{H}^n(x) \, \mathrm{d} \mathcal{H}^n(y) , \] where $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff measure and $K$ is a suitable interaction kernel. For example, in the case of the tangent-point energy, $K$ is given by \[ K(x,y) = \frac{\lvert(1-P(x)) \, (x-y) \rvert^p}{\lvert x-y\rvert^{2p}}, \] where $P(x) \colon \mathbb{R}^m \to \mathbb{R}^m$ is the orthogonal projector onto the tangent space $T_xM$ and $p \geq 2 \, \operatorname{dim}(M)$. If $K$ is appropriately chosen, then the energy $\mathcal{E}$ is well-defined and penalizes self-contact: If $(M_k)_{k \in \mathbb{N}}$ is a sequence of embedded manifolds that converges, e.g., in $C^1$ to an immersed submanifold with nontrivial self-intersections, then the energy $\mathcal{E}(M_k)$ diverges to $\infty$ for $k \to \infty$. Moreover, these energies also penalize roughness or curvature of submanifolds (albeit in a rather subtle way). These properties would make them excellent barrier functions for contact problems. However, there are two principal issues that have to be addressed in order to turn these energies into powerful tools for applications:

As it is typical for shape optimization problems, the discrete problems become increasingly ill-conditioned under mesh refinement. That means that out-of-the-box optimization methods typically perform badly, in particular for fine meshes.
The energies consist of all-pairs interactions. Hence a naive discretization (by replacing the integrals by weighted sums) would lead to the complexity $O(N^2)$ for evaluating the energy and its derivative, where $N$ is the number of degrees of freedom. So merely the evaluation of the energy becomes prohibitively expensive already when $N$ is greater than a few thousand. While this might be sufficient to work with curves, the curse of dimensionality renders this rather impractical for the treatment of surfaces.

The first issue can be resolved by preconditioning with an appropriate Sobolev inner product (see this). Typically, these inner products are related to local and elliptic differential operators of integer order.

The second issue can be overcome by hierarchical techniques such as Barnes-Hut or fast multipole approximations of the energy and hierarchical matrix compression for the Sobolev inner product (see this and this). See also this teaser.