Reconstructions of sea level prior to the satellite altimeter era are usually derived from tide gauge records; however most algorithms for this assume that modes of sea level variability are ...stationary which is not true over several decades. Here we suggest a method that is based on optimized data-dependent triangulations of the network of gauge stations. Data-dependent triangulations are triangulations of point sets that rely not only on 2D point positions but also on additional data (here: sea surface anomalies). In particular, min-error criteria have been suggested to construct triangulations that approximate a given surface. In this article, we show how data-dependent triangulations with min-error criteria can be used to reconstruct 2D maps of the sea surface anomaly over a longer time period, assuming that anomalies are continuously monitored at a sparse set of stations and, in addition, observations of a control surface is provided over a shorter time period. At the heart of our method is the idea to learn a min-error triangulation based on the control data that is available, and to use the learned triangulation subsequently to compute piece-wise linear surface models for epochs in which only observations from monitoring stations are given. Moreover, we combine our approach of min-error triangulation with k-order Delaunay triangulation to stabilize the triangles geometrically. We show that this approach is in particular advantageous for the reconstruction of the sea surface by combining tide gauge measurements (which are sparse in space but cover a long period back in time) with data of modern satellite altimetry (which have a high spatial resolution but cover only the last decades). We show how to learn a min-error triangulation and a min-error k-order Delaunay triangulation using an exact algorithm based on integer linear programming. We confront our reconstructions against the Delaunay triangulation which had been proposed earlier for sea-surface modeling and find superior quality. With real data for the North Sea we show that the min-error triangulation outperforms the Delaunay method substantially for reconstructions back in time up to 18 years, and the k-order Delaunay min-error triangulation even up to 21 years for k=2. With a running time of less than one second our approach would be applicable to areas with far greater extent than the North Sea.
•Historical evolution of sea surfaces by combining tide gauges and satellite altimetry.•Learning a surface representation using optimal data-dependent triangulations.•Min-error triangulation using an exact integer linear programming approach.•Better reconstruction than Delaunay triangulation for time spans up to 20 years.
This topical review gives a comprehensive overview and assessment of recent results in causal dynamical triangulations, a modern formulation of lattice gravity, whose aim is to obtain a theory of ...quantum gravity nonperturbatively from a scaling limit of the lattice-regularized theory. In this manifestly diffeomorphism-invariant approach one has direct, computational access to a Planckian spacetime regime, which is explored with the help of invariant quantum observables. During the last few years, there have been numerous new and important developments and insights concerning the theory's phase structure, the roles of time, causality, diffeomorphisms and global topology, the application of renormalization group methods and new observables. We will focus on these new results, primarily in four spacetime dimensions, and discuss some of their geometric and physical implications.
The paper generalizes the classical C1 cubic Clough–Tocher spline space over a triangulation to C1 spaces of any degree higher that three. It shows that the considered spaces can be equipped with a ...basis consisting of non-negative locally supported functions forming a partition of unity and demonstrates the applicability of the basis in the context of the finite element method. The studied spaces have optimal approximation power and are defined by enforcing additional smoothness inside the triangles of the triangulation where the Clough–Tocher splitting is used. Locally, over each triangle of the triangulation, the splines are expressed in the Bernstein–Bézier form, which enables one to take the full advantage of the geometric properties and computational techniques that come with such a representation. Solving boundary problems with Galerkin discretization is thus relatively straightforward and is illustrated with several examples.
Not so HOT Triangulations Mitchell, Scott A.; Knupp, Patrick; Mackay, Sarah ...
Computer aided design,
02/2023, Letnik:
158, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Here, we propose primal–dual mesh optimization algorithms that overcome shortcomings of the standard algorithm while retaining some of its desirable features. “Hodge-Optimized Triangulations” defines ...the “HOT energy” as a bound on the discretization error of the diagonalized Delaunay Hodge star operator. HOT energy is a natural choice for an objective function, but unstable for both mathematical and algorithmic reasons: it has minima for collapsed edges, and its extrapolation to non-regular triangulations is inaccurate and has unbounded minima. We propose a different extrapolation with a stronger theoretical foundation, and avoid extrapolation by recalculating the objective just beyond the flip threshold. We propose new objectives, based on normalizations of the HOT energy, with barriers to edge collapses and other undesirable configurations. We propose mesh improvement algorithms coupling these. When HOT optimization nearly collapses an edge, we actually collapse the edge. Otherwise, we use the barrier objective to update positions and weights and remove vertices. By combining discrete connectivity changes with continuous optimization, we more fully explore the space of possible meshes and obtain higher quality solutions.
We compare two constructions of exact Lagrangian fillings of Legendrian positive braid closures, the Legendrian weaves of Casals--Zaslow, and the decomposable Lagrangian fillings of ...Ekholm--Honda--Kalman, and show that they coincide for large families of Lagrangian fillings. As a corollary, we obtain an explicit correspondence between Hamiltonian isotopy classes of decomposable Lagrangian fillings of Legendrian (2, n) torus links described by Ekholm--Honda--Kalman and the weave fillings constructed by Treumann and Zaslow. We apply this result to describe the orbital structure of the Kalman loop and give a combinatorial criteria to determine the orbit size of a filling. We follow our geometric discussion with a Floer-theoretic proof of the orbital structure, where an identity studied by Euler in the context of continued fractions makes a surprise appearance. We conclude by giving a purely combinatorial description of the Kalman loop action on the fillings discussed above in terms of edge flips of triangulations. Keywords: Lagrangian fillings, Legendrian knots, triangulations, Euler's continuants, Euler's identity.
The diameter of associahedra Pournin, Lionel
Advances in mathematics (New York. 1965),
07/2014, Letnik:
259
Journal Article
Recenzirano
Odprti dostop
It is proven here that the diameter of the d-dimensional associahedron is 2d−4 when d is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of ...a convex polygon, and their distance is obtained using combinatorial arguments. This settles two problems posed about twenty-five years ago by Daniel Sleator, Robert Tarjan, and William Thurston.
Recently, the first named author defined a 2-parametric family of groups
G
n
k
V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint ...(2015), arXiv:1501.05208. Those groups may be regarded as analogues of braid groups.
Study of the connection between the groups
G
n
k
and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of
n
particles possess a nice codimension one property governed by exactly
k
particles, then these dynamical systems admit a topological invariant valued in
G
n
k
”.
The
G
n
k
groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the
G
n
k
groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the
G
n
k
groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance
G
n
k
groups to get rid of this
2
-torsion.
Later the first and the fourth named authors introduced and studied the second family of groups, denoted by
Γ
n
k
, which are closely related to triangulations of manifolds.
The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129–145 says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257–1270; the
Γ
n
k
naturally appear when considering the set of triangulations with the fixed number of points.
There are two ways of introducing the groups
Γ
n
k
: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a “braid group” of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version.
In this paper, we give a survey of the ideas lying in the foundation of the
G
n
k
and
Γ
n
k
theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.
PDF
