Julius LohmannWorkgroup Prof. Dr. Benedikt WirthInstitute for Computational and Applied Mathematics University of Münster Orléansring 10 D48149 Münster

Research Interests
 Inverse Problems
 Optimal Transport
 Shape Optimization
Ongoing Projects

I study a variant of the multimaterial transport problem, which was introduced by Andrea Marchese, Annalisa Massaccesi, and Riccardo Tione [2,3].
 The multimaterial transport problem in [3] (with cost per distance function ℤ^{m}➝[0,∞), which only allows for rectifiable F [Cor. on p. 265, 4]) was shown to be equivalent to a mass minimization problem on rectifiable 1currents with coefficients in ℤ^{m} [Theorem 2.4, 3]. The authors of [3] introduced socalled calibrations as smooth ℝ^{m}valued differential 1forms that have to satisfy certain conditions related to the problem; these calibrations can be used to validate the optimality of candidate minimizers. Instead, we use a novel approach and introduce calibrations via primaldual optimality conditions. More precisely, we study the case where H is induced (in a suitable sense) by a norm h:ℝ^{m}➝[0,∞) and consider the convex optimization problem (as in [2]) min_{F} 𝕄_{h}(F), where the minimization is over flat1 chains F (with coefficients in ℝ^{m}) that satisfy ∂F=νμ. The functional 𝕄_{h} denotes the hmass. Using Fenchelduality the optimality conditions of this problem and its dual yield our definition of calibrations. As a byproduct we get exactly the true regularity of calibrations (neither too weak nor too strong). Moreover, the optimality conditions can be used to prove properties of optimal multimaterial mass fluxes F, and to provide an integral representation of 𝕄_{h}(F) involving the cost function H (for rectifiable F this representation follows from [p. 268, 4]).
Joint work with: Bernhard Schmitzer and Benedikt Wirth.  We provide a dynamic formulation of (P) (as time evolution of (multi)mass and momentum) as well as some other properties of the induced distance d_{H}. Moreover, we study a novel multiparticle tracking problem in which the trajectories of particles of m different types are to be determined. In particular, we aim to prove a sparsity property for the special case of Wasserstein1 transport using the superposition principle in [1].
Joint work with: Marcello Carioni.
Optimization Problem (P): Let H:ℝ^{m×n}➝[0,∞) be a norm. Further, assume that μ=(μ_{1},...,μ_{m}) and ν=(ν_{1},...,ν_{m}) are the initial and final distributions (supported in some compact domain Ω in ℝ^{n}) of m different materials (with mass(μ_{i})=mass(ν_{i}) for all i). Find an optimal multimaterial mass flux F between μ and ν with respect to H.
A multimaterial mass flux F between μ and ν can be modelled as a ℝ^{m×n}valued Radon measure on Ω (with div(F)=μν). The cost per distance to transport a material vector θ=(θ_{1},...,θ_{m}) in unit direction e is given by H(θ⊗e).
[1] Ehsan Abedi, Zhenhao Li, and Timo Schultz. Absolutely continuous and BVcurves in 1Wasserstein spaces. arXiv:2209.04268 [math.MG]. 2022.
[2] Andrea Marchese, Annalisa Massaccesi, Salvatore Stuvard, and Riccardo Tione. A multimaterial transport problem with arbitrary marginals. Calculus of Variations and Partial Differential Equations 60 (2021) Article 88.
[3] Andrea Marchese, Annalisa Massaccesi, and Riccardo Tione. A Multimaterial Transport Problem and its Convex Relaxation via Rectifiable Gcurrents. SIAM Journal on Mathematical Analysis 51:3 (2019) 19651998.
[4] Brian White. The deformation theorem for flat chains. Acta Mathematica 183:2 (1999) 255271.
Publications
 Julius Lohmann, Bernhard Schmitzer, and Benedikt Wirth. Duality in branched transport and urban planning. Applied Mathematics and Optimization 86:3 (2022) Article 45. [Preprint] [Version of Record]
 Julius Lohmann, Bernhard Schmitzer, and Benedikt Wirth. Formulation of branched transport as geometry optimization. Journal de Mathématiques Pures et Appliquées 163 (2022) 739779. [Preprint] [Version of Record]
Education
 20192023 PhD student at University of Münster
 20182019 Research assistant at FAU ErlangenNürnberg
 20152016 Semester abroad at Stockholm University
 20142017 MSc Mathematics (minor Computer Science) at FAU ErlangenNürnberg
 20112014 BSc Mathematics (minor Computer Science) at FAU ErlangenNürnberg