TY - JOUR A1 - Lie, Han Cheng A1 - Stahn, Martin A1 - Sullivan, Tim J. T1 - Randomised one-step time integration methods for deterministic operator differential equations JF - Calcolo N2 - Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065-1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings. KW - Time integration KW - Operator differential equations KW - Randomisation KW - Uncertainty quantification Y1 - 2022 U6 - https://doi.org/10.1007/s10092-022-00457-6 SN - 0008-0624 SN - 1126-5434 VL - 59 IS - 1 PB - Springer CY - Milano ER - TY - JOUR A1 - Malem-Shinitski, Noa A1 - Ojeda, Cesar A1 - Opper, Manfred T1 - Variational bayesian inference for nonlinear hawkes process with gaussian process self-effects JF - Entropy N2 - Traditionally, Hawkes processes are used to model time-continuous point processes with history dependence. Here, we propose an extended model where the self-effects are of both excitatory and inhibitory types and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and derive an inference algorithm by performing inference on an aggregated sum of Gaussian Processes. Approximate Bayesian inference is achieved via data augmentation, and we describe a mean-field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from different domains and compare it to previously reported results. KW - Bayesian inference KW - point process KW - Gaussian process Y1 - 2022 U6 - https://doi.org/10.3390/e24030356 SN - 1099-4300 VL - 24 IS - 3 PB - MDPI CY - Basel ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, Tim J. T1 - Gamma-convergence of Onsager-Machlup functionals BT - II. Infinite product measures on Banach spaces JF - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data N2 - We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory. KW - Bayesian inverse problems KW - Gamma-convergence KW - maximum a posteriori KW - estimation KW - Onsager-Machlup functional KW - small ball probabilities KW - transition path theory Y1 - 2021 U6 - https://doi.org/10.1088/1361-6420/ac3f82 SN - 0266-5611 SN - 1361-6420 VL - 38 IS - 2 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Lazarides, Rebecca A1 - Rubach, Charlott T1 - Instructional characteristics in mathematics classrooms BT - relationships to achievement goal orientation and student engagement JF - Mathematics Education Research Journal N2 - This longitudinal study examined relationships between student-perceived teaching for meaning, support for autonomy, and competence in mathematic classrooms (Time 1), and students’ achievement goal orientations and engagement in mathematics 6 months later (Time 2). We tested whether student-perceived instructional characteristics at Time 1 indirectly related to student engagement at Time 2, via their achievement goal orientations (Time 2), and, whether student gender moderated these relationships. Participants were ninth and tenth graders (55.2% girls) from 46 classrooms in ten secondary schools in Berlin, Germany. Only data from students who participated at both timepoints were included (N = 746 out of total at Time 1 1118; dropout 33.27%). Longitudinal structural equation modeling showed that student-perceived teaching for meaning and support for competence indirectly predicted intrinsic motivation and effort, via students’ mastery goal orientation. These paths were equivalent for girls and boys. The findings are significant for mathematics education, in identifying motivational processes that partly explain the relationships between student-perceived teaching for meaning and competence support and intrinsic motivation and effort in mathematics. KW - Mathematics classrooms KW - Instruction KW - Intrinsicmotivation KW - Achievement goal orientation KW - Effort KW - Gender Y1 - 2017 U6 - https://doi.org/10.1007/s13394-017-0196-4 SN - 1033-2170 SN - 2211-050X VL - 29 SP - 201 EP - 217 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Eshghi, Nasim A1 - Mach, Thomas A1 - Reichel, Lothar T1 - New matrix function approximations and quadrature rules based on the Arnoldi process JF - Journal of computational and applied mathematics N2 - The Arnoldi process can be applied to inexpensively approximate matrix functions of the form f (A)v and matrix functionals of the form v*(f (A))*g(A)v, where A is a large square non-Hermitian matrix, v is a vector, and the superscript * denotes transposition and complex conjugation. Here f and g are analytic functions that are defined in suitable regions in the complex plane. This paper reviews available approximation methods and describes new ones that provide higher accuracy for essentially the same computational effort by exploiting available, but generally not used, moment information. Numerical experiments show that in some cases the modifications of the Arnoldi decompositions proposed can improve the accuracy of v*(f (A))*g(A)v about as much as performing an additional step of the Arnoldi process. KW - Arnoldi process KW - Matrix function approximation KW - Quadrature rule Y1 - 2021 U6 - https://doi.org/10.1016/j.cam.2021.113442 SN - 0377-0427 SN - 1879-1778 VL - 391 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Omel'chenko, Oleh A1 - Laing, Carlo R. T1 - Collective states in a ring network of theta neurons JF - Proceedings of the Royal Society of London. Series A, Mathematical, physical and engineering sciences N2 - We consider a ring network of theta neurons with non-local homogeneous coupling. We analyse the corresponding continuum evolution equation, analytically describing all possible steady states and their stability. By considering a number of different parameter sets, we determine the typical bifurcation scenarios of the network, and put on a rigorous footing some previously observed numerical results. KW - theta neurons KW - neural networks KW - bumps Y1 - 2022 U6 - https://doi.org/10.1098/rspa.2021.0817 SN - 1364-5021 SN - 1471-2946 VL - 478 IS - 2259 PB - Royal Society CY - London ER - TY - JOUR A1 - Kolbe, Benedikt Maximilian A1 - Evans, Myfanwy E. T1 - Enumerating isotopy classes of tilings guided by the symmetry of triply JF - Siam journal on applied algebra and geometry N2 - We present a technique for the enumeration of all isotopically distinct ways of tiling a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This generalizes the enumeration using Delaney--Dress combinatorial tiling theory of combinatorial classes of tilings to isotopy classes of tilings. To accomplish this, we derive an action of the mapping class group of the orbifold associated to the symmetry group of a tiling on the set of tilings. We explicitly give descriptions and presentations of semipure mapping class groups and of tilings as decorations on orbifolds. We apply this enumerative result to generate an array of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the threedimensional symmetries of the primitive, diamond, and gyroid triply periodic minimal surfaces, which have relevance to a variety of physical systems. KW - isotopic tiling theory KW - mapping class group KW - orbifolds KW - group KW - presentations KW - representations of groups as automorphism groups of KW - algebraic systems KW - triply periodic minimal surface KW - Delaney--Dress KW - tiling theory KW - hyperbolic tilings KW - two-dimensional topology Y1 - 2022 U6 - https://doi.org/10.1137/20M1358943 SN - 2470-6566 VL - 6 IS - 1 SP - 1 EP - 40 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Pasemann, Gregor A1 - Flemming, Sven A1 - Alonso, Sergio A1 - Beta, Carsten A1 - Stannat, Wilhelm T1 - Diffusivity estimation for activator-inhibitor models BT - theory and application to intracellular dynamics of the actin cytoskeleton JF - Journal of nonlinear science N2 - A theory for diffusivity estimation for spatially extended activator-inhibitor dynamics modeling the evolution of intracellular signaling networks is developed in the mathematical framework of stochastic reaction-diffusion systems. In order to account for model uncertainties, we extend the results for parameter estimation for semilinear stochastic partial differential equations, as developed in Pasemann and Stannat (Electron J Stat 14(1):547-579, 2020), to the problem of joint estimation of diffusivity and parametrized reaction terms. Our theoretical findings are applied to the estimation of effective diffusivity of signaling components contributing to intracellular dynamics of the actin cytoskeleton in the model organism Dictyostelium discoideum. KW - Parametric drift estimation KW - Stochastic reaction– diffusion KW - systems KW - Maximum likelihood estimation KW - Actin cytoskeleton dynamics Y1 - 2021 U6 - https://doi.org/10.1007/s00332-021-09714-4 SN - 0938-8974 SN - 1432-1467 VL - 31 IS - 3 PB - Springer CY - New York ER - TY - JOUR A1 - Kürschner, Patrick A1 - Freitag, Melina A. T1 - Inexact methods for the low rank solution to large scale Lyapunov equations JF - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians N2 - The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration (and therefore to "inexact methods"). We will show that one can terminate this inner iteration before full precision has been reached and still obtain very good accuracy in the final solution to the Lyapunov equation. In particular, for both the RKSM and the LR-ADI method we derive theory for a relaxation strategy (e.g. increasing the solve tolerance of the inner iteration, as the outer iteration proceeds) within the iterative methods for solving the large linear systems. These theoretical choices involve unknown quantities, therefore practical criteria for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples, which show that the total amount of work for solving Lyapunov equations can be reduced significantly. KW - Lyapunov equation KW - alternating direction implicit KW - rational Krylov KW - subspaces KW - low-rank approximations Y1 - 2019 U6 - https://doi.org/10.1007/s10543-020-00813-4 SN - 0006-3835 SN - 1572-9125 VL - 60 IS - 4 SP - 1221 EP - 1259 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Hehn, Jennifer A1 - Mendez, Daniel A1 - Uebernickel, Falk A1 - Brenner, Walter A1 - Broy, Manfred T1 - On integrating design thinking for human-centered requirements engineering JF - IEEE software N2 - We elaborate on the possibilities and needs to integrate design thinking into requirements engineering, drawing from our research and project experiences. We suggest three approaches for tailoring and integrating design thinking and requirements engineering with complementary synergies and point at open challenges for research and practice. KW - requirements engineering KW - prototypes KW - software KW - electronic mail KW - tools KW - organizations KW - design thinking Y1 - 2019 U6 - https://doi.org/10.1109/MS.2019.2957715 SN - 0740-7459 SN - 1937-4194 VL - 37 IS - 2 SP - 25 EP - 31 PB - Inst. of Electr. and Electronics Engineers CY - Los Alamitos ER - TY - JOUR A1 - Rosenau, Philip A1 - Pikovskij, Arkadij T1 - Waves in strongly nonlinear Gardner-like equations on a lattice JF - Nonlinearity / the Institute of Physics and the London Mathematical Society N2 - We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm. KW - nonlinear lattice KW - solitary wave KW - Gardner equation KW - compacton Y1 - 2021 U6 - https://doi.org/10.1088/1361-6544/ac0f51 SN - 0951-7715 SN - 1361-6544 VL - 34 IS - 8 SP - 5872 EP - 5896 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Cvetković, Nada A1 - Conrad, Tim A1 - Lie, Han Cheng T1 - A convergent discretization method for transition path theory for diffusion processes JF - Multiscale modeling & simulation : a SIAM interdisciplinary journal N2 - Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential. KW - ergodic diffusion processes KW - transition paths KW - rare events KW - Monte Carlo KW - methods Y1 - 2021 U6 - https://doi.org/10.1137/20M1329354 SN - 1540-3459 SN - 1540-3467 VL - 19 IS - 1 SP - 242 EP - 266 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - JOUR A1 - Perera, Upeksha A1 - Böckmann, Christine T1 - Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion JF - Mathematics N2 - In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively. KW - higher-order Sturm–Liouville problems KW - inverse Sturm–Liouville problems KW - Magnus expansion Y1 - 2019 U6 - https://doi.org/10.3390/math7060544 SN - 2227-7390 VL - 7 IS - 6 PB - MDPI CY - Basel, Schweiz ER - TY - JOUR A1 - Cseh, Ágnes A1 - Kavitha, Telikepalli T1 - Popular matchings in complete graphs JF - Algorithmica : an international journal in computer science N2 - Our input is a complete graph G on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is popular. A matching M is popular if M does not lose a head-to-head election against any matching M ': here each vertex casts a vote for the matching in {M,M '} in which it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-complete for even n, as we show here. This is one of the few graph theoretic problems efficiently solvable when n has one parity and NP-complete when n has the other parity. KW - Popular matching KW - Complexity KW - Stable matching Y1 - 2021 U6 - https://doi.org/10.1007/s00453-020-00791-7 SN - 0178-4617 SN - 1432-0541 VL - 83 IS - 5 SP - 1493 EP - 1523 PB - Springer CY - New York ER - TY - JOUR A1 - Chang, Der-Chen A1 - Mahmoudi, Mahdi Hedayat A1 - Schulze, Bert-Wolfgang T1 - Volterra operators in the edge-calculus JF - Analysis and Mathematical Physics N2 - We study the Volterra property of a class of anisotropic pseudo-differential operators on R x B for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable t of t extends to symbolswith respect to t to the lower complex v half-plane. The resulting space ofVolterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph. D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph. D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113-137, 1997; Osaka J Math 37: 221-260, 2000). KW - Volterra operator KW - Anisotropic pseudo-differential operators KW - Edge calculus KW - Operator-valued symbols of Mellin type Y1 - 2018 U6 - https://doi.org/10.1007/s13324-018-0238-4 SN - 1664-2368 SN - 1664-235X VL - 8 IS - 4 SP - 551 EP - 570 PB - Springer CY - Basel ER - TY - JOUR A1 - Flad, Heinz-Jürgen A1 - Flad-Harutyunyan, Gohar A1 - Schulze, Bert-Wolfgang T1 - Explicit Green operators for quantum mechanical Hamiltonians BT - II. edge-type singularities of the helium atom JF - Asian-European journal of mathematics : AEJM N2 - We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second-order, we have calculated the symbols of an asymptotic parametrix of the nonrelativistic Hamiltonian of the helium atom within the Born-Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behavior of the eigenfunctions near an edge. KW - Singular analysis KW - Schrodinger equation KW - many-electron systems KW - asymptotic properties of eigenfunctions Y1 - 2020 U6 - https://doi.org/10.1142/S1793557120501223 SN - 1793-5571 SN - 1793-7183 VL - 13 IS - 7 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Rungrottheera, Wannarut A1 - Lyu, Xiaojing A1 - Schulze, Bert-Wolfgang T1 - Parameter-dependent edge calculus and corner parametrices JF - Journal of nonlinear and convex analysis : an international journal N2 - Let B be a compact manifold with smooth edge of dimension > 0. We study the interplay between parameter-dependent edge algebra algebra on B and operator families belonging to the corner calculus, and we characterize parametrices in the corner case. KW - Edge calculus KW - corner parametrices Y1 - 2018 SN - 1345-4773 SN - 1880-5221 VL - 19 IS - 12 SP - 2021 EP - 2051 PB - Yokohama Publishers CY - Yokohama ER - TY - JOUR A1 - Stauffer, Maxime A1 - Mengesha, Isaak A1 - Seifert, Konrad A1 - Krawczuk, Igor A1 - Fischer, Jens A1 - Serugendo, Giovanna Di Marzo T1 - A computational turn in policy process studies BT - coevolving network dynamics of policy change JF - Complexity N2 - The past three decades of policy process studies have seen the emergence of a clear intellectual lineage with regard to complexity. Implicitly or explicitly, scholars have employed complexity theory to examine the intricate dynamics of collective action in political contexts. However, the methodological counterparts to complexity theory, such as computational methods, are rarely used and, even if they are, they are often detached from established policy process theory. Building on a critical review of the application of complexity theory to policy process studies, we present and implement a baseline model of policy processes using the logic of coevolving networks. Our model suggests that an actor's influence depends on their environment and on exogenous events facilitating dialogue and consensus-building. Our results validate previous opinion dynamics models and generate novel patterns. Our discussion provides ground for further research and outlines the path for the field to achieve a computational turn. Y1 - 2022 U6 - https://doi.org/10.1155/2022/8210732 SN - 1076-2787 SN - 1099-0526 VL - 2022 PB - Wiley-Hindawi CY - London ER - TY - JOUR A1 - Hedayat Mahmoudi, Mahdi A1 - Schulze, Bert-Wolfgang T1 - Corner boundary value problems JF - Asian-European journal of mathematics N2 - The paper develops some crucial steps in extending the first-order cone or edge calculus to higher singularity orders. We focus here on order 2, but the ideas are motivated by an iterative approach for higher singularities. KW - Mellin operators KW - Mellin oscillatory integrals KW - exit calculus KW - weighted Sobolev spaces Y1 - 2016 U6 - https://doi.org/10.1142/S1793557117500541 SN - 1793-5571 SN - 1793-7183 VL - 10 IS - 1 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Cseh, Agnes A1 - Faenza, Yuri A1 - Kavitha, Telikepalli A1 - Powers, Vladlena T1 - Understanding popular matchings via stable matchings JF - SIAM journal on discrete mathematics N2 - An instance of the marriage problem is given by a graph G = (A boolean OR B, E), together with, for each vertex of G, a strict preference order over its neighbors. A matching M of G is popular in the marriage instance if M does not lose a head-to-head election against any matching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exists and can be easily computed is the set of dominant matchings. A popular matching M is dominant if M wins the head-to-head election against any larger matching. Thus, every dominant matching is a max-size popular matching, and it is known that the set of dominant matchings is the linear image of the set of stable matchings in an auxiliary graph. Results from the literature seem to suggest that stable and dominant matchings behave, from a complexity theory point of view, in a very similar manner within the class of popular matchings. The goal of this paper is to show that there are instead differences in the tractability of stable and dominant matchings and to investigate further their importance for popular matchings. First, we show that it is easy to check if all popular matchings are also stable; however, it is co-NP hard to check if all popular matchings are also dominant. Second, we show how some new and recent hardness results on popular matching problems can be deduced from the NP-hardness of certain problems on stable matchings, also studied in this paper, thus showing that stable matchings can be employed to show not only positive results on popular matchings (as is known) but also most negative ones. Problems for which we show new hardness results include finding a min-size (resp., max-size) popular matching that is not stable (resp., dominant). A known result for which we give a new and simple proof is the NP-hardness of finding a popular matching when G is nonbipartite. KW - popular matching KW - stable matching KW - complexity KW - dominant matching Y1 - 2022 U6 - https://doi.org/10.1137/19M124770X SN - 0895-4801 SN - 1095-7146 VL - 36 IS - 1 SP - 188 EP - 213 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER -