Space-time Cylinder Research Articles

Reconstruction of the initial data from the trace of the solutions on an infinite time cylinder of damped wave equations *

In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation. We recover the initial velocity from the trace of the solution on a space-time cylinder. This inverse problem is related to photoacoustic tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.

Hyperbolic problems with totally characteristic boundary

We study first-order symmetrizable hyperbolic N×N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\ imes N$$\\end{document} systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=0$$\\end{document}, these systems take the form ∂tu+A(t,x,y,xDx,Dy)u=f(t,x,y),(t,x,y)∈(0,T)×R+×Rd,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u + {{\\mathcal {A}}}(t,x,y,xD_x,D_y) u = f(t,x,y), \\quad (t,x,y)\\in (0,T)\ imes {{\\mathbb {R}}}_+\ imes {{\\mathbb {R}}}^d, \\end{aligned}$$\\end{document}where A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is a first-order differential operator with coefficients smooth up to x=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=0$$\\end{document} and the derivative with respect to x appears in the combination xDx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$xD_x$$\\end{document}. No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator ∂t+A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t + {{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is well-posed in that scale. More specifically, solutions u exhibit formal asymptotic expansions of the form u(t,x,y)∼∑(p,k)(-1)kk!x-plogkxupk(t,y)asx→+0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u(t,x,y) \\sim \\sum _{(p,k)} \\frac{(-1)^k}{k!}x^{-p} \\log ^k \\!x \\, u_{pk}(t,y) \\quad \\hbox { as}\\ x\\rightarrow +0 \\end{aligned}$$\\end{document}where (p,k)∈C×N0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p,k)\\in {{\\mathbb {C}}}\ imes {{\\mathbb {N}}}_0$$\\end{document} and ℜp→-∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re p\\rightarrow -\\infty $$\\end{document} as |p|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|p|\\rightarrow \\infty $$\\end{document}, provided that the right-hand side f and the initial data u|t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u|_{t=0}$$\\end{document} admit asymptotic expansions as x→+0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x \\rightarrow +0$$\\end{document} of a similar form, with the singular exponents p and their multiplicities unchanged. In fact, the coefficients upk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{pk}$$\\end{document} are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients upk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{pk}$$\\end{document} solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator ∂t+A(t,x,y,xDx,Dy)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t+{{\\mathcal {A}}}(t,x,y,xD_x,D_y)$$\\end{document} is well-posed in the scale of standard Sobolev spaces Hs((0,T)×R+1+d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s((0,T)\ imes {{\\mathbb {R}}}_+^{1+d})$$\\end{document}.

New second order sufficient optimality conditions for state constrained parabolic control problems

We study a control problem governed by a semilinear parabolic equation with pointwise control and state constraints imposed at every point of the space-time cylinder. We obtain second order sufficient optimality conditions for local optimality in the sense of L 2 ( Q ) . Our results are valid for spatial domains of dimension less than or equal to three.

Second-Order Sufficient Conditions in the Sparse Optimal Control of a Phase Field Tumor Growth Model with Logarithmic Potential

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn-Hilliard type. The evolution of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the L1-norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish. In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions have been established for the case without sparsity, the case with sparsity has not been treated before.

Improved rates for a space–time FOSLS of parabolic PDEs

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components (u1,u2)=(u,-∇xu)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(u_1,\ extbf{u}_2)=(u,-\ abla _\ extbf{x} u)$$\\end{document}. The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norms of ∇xu1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ abla _\ extbf{x} u_1$$\\end{document} and u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}, the (graph) norm of U contains the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2$$\\end{document}-norm of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}. When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of u2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{u}_2$$\\end{document}. In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of ∂tu1+divxu2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u_1 +{{\\,\ extrm{div}\\,}}_\ extbf{x} \ extbf{u}_2$$\\end{document}, i.e., of the forcing term f=(∂t-Δx)u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=(\\partial _t-\\Delta _x)u$$\\end{document}. Numerical results show significantly improved convergence rates.

Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.

An Anisotropic hp-mesh Adaptation Method for Time-Dependent Problems Based on Interpolation Error Control

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder varOmega times (0,T). We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L^infty (0,T; L^q(varOmega ))-norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method.

Local bounds of the gradient of weak solutions to the porous medium equation

Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[ a{\buildrel\mbox{def}\over{=}}\frac1{|B_r(x_o)|}\int_{B_r(x_o)}u(x,t_o)\,dx>0, \] then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}r^2$. This implies that in the same cylinder the solution $u$ is H\"older continuous with exponent $\alpha=\frac1{m-1}$, which is known to be optimal. Moreover, $u$ presents a sort of instantaneous regularisation, which we quantify.

Regularity results and numerical solution by the discontinuous Galerkin method to semilinear parabolic initial boundary value problems with nonlinear Newton boundary conditions in a polygonal space-time cylinder

Abstract In this note we consider a parabolic evolution equation in a polygonal space-time cylinder. We show, that the elliptic part is given by a m-accretive mapping from L q (Ω) → L q (Ω). Therefore we can apply the theory of nonlinear semigroups in Banach spaces in order to get regularity results in time and space. The second part of the paper deals with the numerical solution of the problem. It is dedicated to the application of the space-time discontinuous Galerkin method (STDGM). It means that both in space as well as in time discontinuous piecewise polynomial approximations of the solution are used. We concentrate to the theoretical analysis of the error estimation.

A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations

In this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our theoretical findings.

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation

We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.

Calculating heat and wave propagation from lateral Cauchy data

UDC 519.6 We give an overview of recent methods based on semi-discretisation in time for the inverse ill-posed problem of calculating the solution of evolution equations from time-like Cauchy data. Specifically, the function value and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data is to be generated on the remaining lateral part of the cylinder for either the heat or wave equation. The semi-discretisation in time constitutes of applying the Laguerre transform or the Rothe method (finite difference approximation), and has the feature that the similar sequence of elliptic problems is obtained for both the heat and wave equation, only the values of certain parameters change. The elliptic equations are solved numerically by either a boundary integral approach involving the Nystreom method or a method of fundamental solutions (MFS). Theoretical properties are stated together with discretisation strategies in space. Systems of linear equations are obtained for finding values of densities or coefficients. Tikhonov regularization is incorporated for the stable solution of the linear equations. Numerical results included show that the proposed strategies give good accuracy with an economical computational cost.

Estimates of the Distance to the Solution of an Evolutionary Problem Obtained by Linearization of the Navier–Stokes Equation

The paper is concerned with linearization of the Navier–Stokes equation in the space-time cylinder QT . The main goal is to deduce computable estimates of the distance between the exact solution and a function in the energy admissible class of vector-valued functions. First, the estimates are derived for the case, where this class contains only divergence free (solenoidal) functions. In the next section, estimates of the distance to sets of divergence free functions depending on the space and time variables are considered. These results are used to extend earlier derived estimates to non-solenoidal approximations. The corresponding estimates contain an additional term which can be viewed as a penalty for the violation of the divergence free condition.

Mass propagation for electromagnetic Schrödinger evolutions

We investigate the validity of Gaussian lower bounds for solutions to an electromagnetic Schrödinger equation with a bounded time-dependent complex electric potential and a time-independent vector magnetic potential. We prove that, if a suitable geometric condition is satisfied by the vector potential, then positive masses inside of a bounded region at a single time propagate outside the region, provided a suitable average in space–time cylinders is taken.

On non-denseness for a method of fundamental solutions with source points fixed in time for parabolic equations

Linear combinations of fundamental solutions to the parabolic heat equation with source points fixed in time is investigated. The open problem whether these linear combinations generate a dense set in the space of square integrable functions on the lateral boundary of a space-time cylinder, is settled in the negative. Linear independence of the set of fundamental solutions is shown to hold. It is outlined at the end, for a particular example, that such linear combinations constitute a linearly independent and dense set in the space of square integrable functions on the upper top part (where time is fixed) of the boundary of this space-time cylinder.

On the space-time discretization of variational retarded potential boundary integral equations

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.

The obstacle problem for singular doubly nonlinear equations of porous medium type

In this paper we prove the existence of variational solutions to the obstacle problem associated with doubly nonlinear equations \partial_t (|u|^{m-1}u) - \mathrm {div}(D_\xi f(Du)) = 0 with m > 1 and a convex function f satisfying a standard p -growth condition for an exponent p \in (1,\infty) in a bounded space-time cylinder \Omega_T := \Omega \times (0,T) . The obstacle function \psi and the boundary values g are time dependent. The proof relies on a nonlinear version of the method of minimizing movements.

UNCOVERING THE AGGREGATION PATTERN OF GPS TRAJECTORY BASED ON SPATIOTEMPORAL CLUSTERING AND 3D VISUALIZATION

Abstract. Constrained by road network structure, travel choice and city function zoning, GPS trajectory data exhibits significant spatiotemporal correlation. Unveiling the clustering and distribution patterns of GPS trajectory can help to better understand the travel behaviour as well as the corresponding spatial and temporal characteristics. This paper proposes an approach to identify and visualize the aggregation pattern from GPS trajectory data. Firstly, slow feature trajectory sequences are extracted from raw taxi trajectory data. Together with taxi states information, these sequences are processed as shorter length tracks for faster discovery of cluster similarity. Thereafter, the temporal and spatial similarity and dissimilarity metrics between the trajectories are established, and the temporal and spatial distances between the trajectories are defined to form a space-time cylinder model. Next, based on the idea of density clustering, the DBSCAN spatiotemporal expansion of trajectory data is proposed. Feature trajectory sequences are then clustered into groups with high similarity. Finally, for a more intuitive understanding of the trajectory aggregate distribution, time dimension info of each point in the sequences is used as Z axis, thus the sequences are stretched on the map in different colour for 3D visualization. The proposed method is validated by a case study of taxi trajectory data analysis in Wuhan City, China.

Operator-Algebraic Construction of Gauge Theories and Jones’ Actions of Thompson’s Groups

Using ideas from Jones, lattice gauge theory and loop quantum gravity, we construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a separable compact group $G$, we construct localized time-zero fields on the spatial torus as a net of C*-algebras together with an action of the gauge group that is an infinite product of $G$ over the dyadic rationals and, using a recent machinery of Jones, an action of Thompson's group $T$ as a replacement of the spatial diffeomorphism group. Adding a family of probability measures on the unitary dual of $G$ we construct a state and obtain a net of von Neumann algebras carrying a state-preserving gauge group action. For abelian $G$, we provide a very explicit description of our algebras. For a single measure on the dual of $G$, we have a state-preserving action of Thompson's group and semi-finite von Neumann algebras. For $G=\mathbf{S}$ the circle group together with a certain family of heat-kernel states providing the measures, we obtain hyperfinite type III factors with a normal faithful state providing a nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the latter case, we additionally have a non-singular action of the group of rotations with dyadic angles, as a subgroup of Thompson's group $T$, for geometrically motivated choices of families of heat-kernel states.

Space-Time Discontinuous Petrov–Galerkin Methods for Linear Wave Equations in Heterogeneous Media

Abstract We establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves of Gopalakrishnan–Sepulveda [J. Gopalakrishnan and P. Sepulveda, A space-time DPG method for the wave equation in multiple dimensions, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 129–154] and Ernesti–Wieners [J. Ernesti and C. Wieners, A space-time discontinuous Petrov–Galerkin method for acoustic waves, Space-Time Methods. Applications to Partial Differential Equations, Radon Ser. Comput. Appl. Math. 21, Walter de Gruyter, Berlin 2019, 99–127] and is based on the abstract definition of traces on the skeleton of the time-space substructuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder.