SRB Measures of Anosov Actions - Weich, Tobias; Guedes Bonthonneau, Yannick; Guillarmou, Colin (Journal of Differential Geometry (to appear) -- arXiv:2103.12127)

Learning of discrete models of variational PDEs from data - Offen, Christian; Ober-Blöbaum, Sina (Chaos)
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation.

A note on the convergence of deterministic gradient sampling in nonsmooth optimization - Gebken, Bennet (Computational Optimization and Applications)
Abstract Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.
Keywords: Applied Mathematics; Computational Mathematics; Control and Optimization

Non-Fock ground states in the translation-invariant Nelson model revisited non-perturbatively - Hasler, David; Hinrichs, Benjamin; Siebert, Oliver (Journal of Functional Analysis)
Keywords: Analysis

Ruelle-Taylor resonaces of Anosov actions - Weich, Tobias; Guedes Bonthonneau, Yannick; Guillarmou, Colin; Hilgert, Joachim (J. Europ. Math. Soc.)

Quantum-Classical Correspondences for Locally Symmetric Spaces - Hilgert, Joachim

Fast Multiobjective Gradient Methods with Nesterov Acceleration via Inertial Gradient-Like Systems - Sonntag, Konstantin; Peitz, Sebastian (Journal of Optimization Theory and Applications)
We derive efficient algorithms to compute weakly Pareto optimal solutions for smooth, convex and unconstrained multiobjective optimization problems in general Hilbert spaces. To this end, we define a novel inertial gradient-like dynamical system in the multiobjective setting, which trajectories converge weakly to Pareto optimal solutions. Discretization of this system yields an inertial multiobjective algorithm which generates sequences that converge weakly to Pareto optimal solutions. We employ Nesterov acceleration to define an algorithm with an improved convergence rate compared to the plain multiobjective steepest descent method (Algorithm 1). A further improvement in terms of efficiency is achieved by avoiding the solution of a quadratic subproblem to compute a common step direction for all objective functions, which is usually required in first-order methods. Using a different discretization of our inertial gradient-like dynamical system, we obtain an accelerated multiobjective gradient method that does not require the solution of a subproblem in each step (Algorithm 2). While this algorithm does not converge in general, it yields good results on test problems while being faster than standard steepest descent.

A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces - Sonntag, Konstantin; Gebken, Bennet; Müller, Georg; Peitz, Sebastian; Volkwein, Stefan (arXiv:2402.06376)
The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

On token signed graphs - Dalfó, C.; Fiol, M. A.; Steffen, Eckhard (arXiv:2403.02924)
We introduce the concept of a k-token signed graph and study some of its combinatorial and algebraic properties. We prove that two switching isomorphic signed graphs have switching isomorphic token graphs. Moreover, we show that the Laplacian spectrum of a balanced signed graph is contained in the Laplacian spectra of its k-token signed graph. Besides, we introduce and study the unbalance level of a signed graph, which is a new parameter that measures how far a signed graph is from being balanced. Moreover, we study the relation between the frustration index and the unbalance level of signed graphs and their token signed graphs.

Feynman-Kac formulas for semigroups generated by multi-polaron Hamiltonians in magnetic fields and on general domains - Hinrichs, Benjamin; Matte, Oliver (arXiv:2403.12147)
We prove Feynman-Kac formulas for the semigroups generated by selfadjoint operators in a class containing Fr\"ohlich Hamiltonians known from solid state physics. The latter model multi-polarons, i.e., a fixed number of quantum mechanical electrons moving in a polarizable crystal and interacting with the quantized phonon field generated by the crystal's vibrational modes. Both the electrons and phonons can be confined to suitable open subsets of Euclidean space. We also include possibly very singular magnetic vector potentials and electrostatic potentials. Our Feynman-Kac formulas comprise Fock space operator-valued multiplicative functionals and can be applied to every vector in the underlying Hilbert space. In comparison to the renormalized Nelson model, for which analogous Feynman-Kac formulas are known, the analysis of the creation and annihilation terms in the multiplicative functionals requires novel ideas to overcome difficulties caused by the phonon dispersion relation being constant. Getting these terms under control and generalizing other construction steps so as to cover confined systems are the main achievements of this article.

Heteroclinic dynamics in network dynamical systems with higher-order interactions - Bick, Christian; von der Gracht, Sören (Journal of Complex Networks)
Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.
Keywords: Applied Mathematics; Computational Mathematics; Control and Optimization; Management Science and Operations Research; Computer Networks and Communications

Edge-Connectivity and Pairwise Disjoint Perfect Matchings in Regular Graphs - Ma, Yulai; Mattiolo, Davide; Steffen, Eckhard; Wolf, Isaak Hieronymus (Combinatorica)
For 0 ≤ t ≤ r let m(t, r) be the maximum number s such that every t-edge-connected r-graph has s pairwise disjoint perfect matchings. There are only a few values of m(t, r) known, for instance m(3, 3) = m(4, r) = 1, and m(t, r) ≤ r − 2 for all t = 5, and m(t, r) ≤ r − 3 if r is even. We prove that m(2l, r) ≤ 3l − 6 for every l ≥ 3 and r ≥ 2l.
Keywords: Computational Mathematics; Discrete Mathematics and Combinatorics

A new Lagrangian approach to control affine systems with a quadratic Lagrange term - Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Almagro, Rodrigo T. Sato Martín de; Szemenyei, Flóra Orsolya (Journal of Computational Dynamics)
In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretisation of the optimal control problem via variational integrators in a straightforward way.
Keywords: Optimal control problem; Lagrangian system; Hamiltonian system; Variations; Pontryagin's maximum principle.

Numerical analysis of an evolving bulk--surface model of tumour growth - Edelmann, Dominik; Kovács, Balázs; Lubich, Christian (arXiv)

Locally homogeneous Axiom A flows I: projective Anosov subgroups and exponential mixing - Delarue, Benjamin; Monclair, Daniel; Sanders, Andrew (arXiv:2403.14257)
By constructing a non-empty domain of discontinuity in a suitable homogeneous space, we prove that every torsion-free projective Anosov subgroup is the monodromy group of a locally homogeneous contact Axiom A dynamical system with a unique basic hyperbolic set on which the flow is conjugate to the refraction flow of Sambarino. Under the assumption of irreducibility, we utilize the work of Stoyanov to establish spectral estimates for the associated complex Ruelle transfer operators, and by way of corollary: exponential mixing, exponentially decaying error term in the prime orbit theorem, and a spectral gap for the Ruelle zeta function. With no irreducibility assumption, results of Dyatlov-Guillarmou imply the global meromorphic continuation of zeta functions with smooth weights, as well as the existence of a discrete spectrum of Ruelle-Pollicott resonances and (co)-resonant states. We apply our results to space-like geodesic flows for the convex cocompact pseudo-Riemannian manifolds of Danciger-Gu\'eritaud-Kassel, and the Benoist-Hilbert geodesic flow for strictly convex real projective manifolds.

Quantum resonances and scattering poles of classical rank one locally symmetric spaces - Delarue, Benjamin; Hilgert, Joachim (arXiv:2403.14426)
For negatively curved symmetric spaces it is known that the poles of the scattering matrices defined via the standard intertwining operators for the spherical principal representations of the isometry group are either given as poles of the intertwining operators or as quantum resonances, i.e. poles of the meromorphically continued resolvents of the Laplace-Beltrami operator. We extend this result to classical locally symmetric spaces of negative curvature with convex-cocompact fundamental group using results of Bunke and Olbrich. The method of proof forces us to exclude the spectral parameters corresponding to singular Poisson transforms.

Numerical surgery for mean curvature flow of surfaces - Kovács, Balázs (SIAM Journal on Scientific Computing)

Hankel transform, K-Bessel functions and zeta distributions in the Dunkl setting - Brennecken, Dominik (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces - Papageorgiou, Efthymia (Journal of Evolution Equations)
Abstract This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G / K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi- K -invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi- K -invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G / K . In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi- K -invariance.
Keywords: Mathematics (miscellaneous)

Research on curriculum resources in mathematics education: a survey of the field - Rezat, Sebastian (ZDM – Mathematics Education)