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

Construction of A-stable explicit last-stage diagonal implicit Runge–Kutta (ELDIRK) methods - Mahnken, Rolf; Westermann, Hendrik (Computational Mechanics)
ELDIRK methods are defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge-Kutta methods, with the consequence, that no additional system of equations must be solved, compared to the embedded RK method. Two general formulations for second- and third-order ELDIRK methods have been obtained recently in Mahnken [21] with specific schemes, e.g. for the embedded implicit Euler method, the embedded trapezoidal-rule and the embedded Ellsiepen method. In the first part of this paper, we investigate some general stability characteristics of ELDIRK methods, and it will be shown that the above specific RK schemes are not A-stable. Therefore, in the second part, the above-mentioned general formulations are used for further stability investigations, with the aim to construct new second- and third-order ELDIRK methods which simultaneously are A-stable. Two numerical examples are concerned with the curing for a thermosetting material and phase-field RVE modeling for crystallinity and orientation. The numerical results confirm the theoretical results on convergence order and stability.
Keywords: Applied Mathematics; Computational Mathematics; Computational Theory and Mathematics; Mechanical Engineering; Ocean Engineering; Computational Mechanics

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

TS-SEP: Joint Diarization and Separation Conditioned on Estimated Speaker Embeddings - Boeddeker, Christoph; Subramanian, Aswin Shanmugam; Wichern, Gordon; Haeb-Umbach, Reinhold; Le Roux, Jonathan (IEEE/ACM Transactions on Audio, Speech, and Language Processing)
Keywords: Electrical and Electronic Engineering; Acoustics and Ultrasonics; Computer Science (miscellaneous); Computational Mathematics

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints - Bartels, Sören; Kovács, Balázs; Wang, Zhangxian (IMA Journal of Numerical Analysis)
Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Analysis of a chemotaxis-SIS epidemic model with unbounded infection force - Tao, Youshan; Winkler, Michael (Nonlinear Analysis: Real World Applications)
Keywords: Applied Mathematics; Computational Mathematics; General Economics; Econometrics and Finance; General Engineering; General Medicine; Analysis

Possible points of blow-up in chemotaxis systems with spatially heterogeneous logistic source - Black, Tobias; Fuest, Mario; Lankeit, Johannes; Mizukami, Masaaki (Nonlinear Analysis: Real World Applications)
Keywords: Applied Mathematics; Computational Mathematics; General Economics; Econometrics and Finance; General Engineering; General Medicine; Analysis

Derivation of third order Runge–Kutta methods (ELDIRK) by embedding of lower order implicit time integration schemes for local and global error estimation - Mahnken, Rolf (Computational Mechanics)
Abstract Three prominent low order implicit time integration schemes are the first order implicit Euler-method, the second order trapezoidal rule and the second order Ellsiepen method. Its advantages are stability and comparatively low computational cost, however, they require the solution of a nonlinear system of equations. This paper presents a general approach for the construction of third order Runge–Kutta methods by embedding the above mentioned implicit schemes into the class of ELDIRK-methods. These will be defined to have an Explicit Last stage in the general Butcher array of Diagonal Implicit Runge–Kutta (DIRK) methods, with the consequence, that no additional system of equations must be solved. The main results—valid also for non-linear ordinary differential equations—are as follows: Two extra function calculations are required in order to embed the implicit Euler-method and one extra function calculation is required for the trapezoidal-rule and the Ellsiepen method, in order to obtain the third order properties, respectively. Two numerical examples are concerned with a parachute with viscous damping and a two-dimensional laser beam simulation. Here, we verify the higher order convergence behaviours of the proposed new ELDIRK-methods, and its successful performances for asymptotically exact global error estimation of so-called reversed embedded RK-method are shown.
Keywords: Applied Mathematics; Computational Mathematics; Computational Theory and Mathematics; Mechanical Engineering; Ocean Engineering; Computational Mechanics

L_2 error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data - Hesse, Kerstin; Le Gia, Quoc Thong (Journal of Computational and Applied Mathematics)
Keywords: Applied Mathematics; Computational Mathematics

Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces - Elliott, Charles M.; Garcke, Harald; Kovács, Balázs (Numerische Mathematik)
Abstract An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $$H^1$$ H1 norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
Keywords: Applied Mathematics; Computational Mathematics

Time-dependent electromagnetic scattering from thin layers - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Abstract The scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems - Kovács, Balázs; Li, Buyang (IMA Journal of Numerical Analysis)
Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^∞ (0,T;W^{1,∞ })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^∞ (0,T;W^{1,∞ })$ norm is obtained by combining the stability analysis with the consistency estimates.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions - Altmann, Robert; Kovács, Balázs; Zimmer, Christoph (IMA Journal of Numerical Analysis)
Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $τ \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for a splitting integrator for abstract semilinear boundary coupled systems - Csomós, Petra; Farkas, Bálint; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces - Beschle, Cedric Aaron; Kovács, Balázs (Numerische Mathematik)
Abstract In this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics

FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation - Bohn, Jan; Feischl, Michael; Kovács, Balázs (Computational Methods in Applied Mathematics)
Abstract The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis

Goal-oriented error estimation and h-adaptive finite elements for hyperelastic micromorphic continua - Ju, Xiaozhe; Mahnken, Rolf; Xu, Yangjian; Liang, Lihua (Computational Mechanics)
Keywords: Applied Mathematics; Computational Mathematics; Computational Theory and Mathematics; Mechanical Engineering; Ocean Engineering; Computational Mechanics

Quantum generalizations of the polynomial hierarchy with applications to QMA(2) - Gharibian, Sevag; Santha, Miklos; Sikora, Jamie; Sundaram, Aarthi; Yirka, Justin (Computational Complexity)
Keywords: Computational Mathematics; Computational Theory and Mathematics; General Mathematics; Theoretical Computer Science

A polymorphic uncertainty model for the curing process of transversely fiber-reinforced plastics - Penner, Eduard; Caylak, Ismail; Mahnken, Rolf (Mathematics and Mechanics of Complex Systems)
Keywords: Computational Mathematics; Numerical Analysis; Civil and Structural Engineering