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1 [CT066]

[CT066]

[00629] Stability Analysis of Split Equality and Split Feasibility Problems

Session Date & Time : 2E (Aug.22, 17:40-19:20)
Type : Contributed Talk
Abstract : In this talk, the stability of solutions to parametric split equality and split feasibility problems is addressed for the first time. Characterizations for the Lipschitz-likeness of solution maps are obtained by exploiting special structures of the problems and by using an advanced result of B.S. Mordukhovich on parametric generalized equations. Examples are presented to illustrate how the obtained results work in practice and to show that extra mild assumptions made cannot be omitted.
Classification : 49J53, 49K40, 65K10, 90C25, 90C31
Author(s) :
- Huong Thi Vu (Institute of Mathematics, Vietnam Academy of Science and Technology)
- Yen Dong Nguyen (Institute of Mathematics, Vietnam Academy of Science and Technology)

[02374] Embarrassingly-parallel optimization algorithms for high-dimensional optimal control

Session Date & Time : 2E (Aug.22, 17:40-19:20)
Type : Contributed Talk
Abstract : Developing efficient algorithms for Hamilton--Jacobi partial differential equations $(\text{HJ PDEs})$ is crucial for solving high-dimensional optimal control problems in real time but notoriously tricky due to the so-called curse of dimensionality. In this talk, we present novel grid-free and embarrassingly-parallel optimization algorithms for solving a broad class of HJ PDEs relevant to high-dimensional state-dependent optimal control problems. We illustrate their performance and efficiency on large-scale multi-agent path planning problems.
Classification : 49L12, 65K10, 90C30, 49M29, 49M37
Author(s) :
- Gabriel Provencher Langlois (New York University)
- Jerome Darbon (Brown University)

[00238] Numerical Schemes for Generalized Isoperimetric Constraint Fractional Variational Problem

Session Date & Time : 2E (Aug.22, 17:40-19:20)
Type : Contributed Talk
Abstract : This paper discusses three numerical schemes for Generalized Isoperimetric Constraint Fractional Variational Problems (GICFVPs) defined using generalized fractional derivatives. Three Numerical schemes, i.e. linear, quadratic, and quadratic-linear schemes, are used to get numerical solutions of a GICFVP. The convergence rate of the linear and quadratic schemes for $\alpha\in(0,1)$ are $2-\alpha$ and $3-\alpha$. It is observed that the presented schemes perform well, and when the step size $\mathrm{h}$ is decreased, the desired solution is attained.
Classification : 49R99, 65K10, 65L60, 65L70
Author(s) :
- DIVYANSH PANDEY (IIT (BHU), Varanasi)
- Rajesh Kumar Pandey (IIT (BHU), Varanasi)

[00796] Universal Formula for Area of all Polygons

Session Date & Time : 2E (Aug.22, 17:40-19:20)
Type : Contributed Talk
Abstract : It is of great relevance to measuring the areas of sites, volumes, and area-to-volume ratios of buildings. But what if these lands and structures come in variable and complex shapes? How will these measurements be carried out accurately? Measurements of the areas and volumes of more varieties of beautifully shaped Structures accurately are difficult as there is no well-known formula to make this calculation when they are not rectangular or in the form of quadrilaterals. I have formulated a formula that can calculate the area of any regular or irregular Polygon of any number of sides and shapes. This formula can help students in schools and colleges better solve for Area of Polygons and Volumes of any structure. Review of submission Other formulas exist to roughly calculate the area of a Polygon. However, the difference between my Universal formula and the advantage is identified as follows. The shoelace formula or algorithm This formula was formulated by Albrecht Ludwig Friedrich Meister in 1769. It uses a mathematical algorithm to determine the area of a simple polygon whose vertices are described by the Cartesian coordinates in the plane. My Universal formulas for the area of Polygons don't require the Cartesian coordinates of the polygons to find the area of Polygons. The advantage of this is that there's no need for plotting points and constant multiplication of the coordinates making up the Polygon. All you need is the length of the sides of the Polygon and the interior angles, especially for the irregular Polygons. Additionally, in the shoelace formula, the signed area depends on the ordering of the vertices and the orientation of the plane, including a mapping of the positive x-axis to the positive y-axis. In contrast, using the universal formula, knowledge of the orientation and axis of the plane is not necessary or needed. Pick's Theorem Pick's theorem involves simple Polygons with integer vertex coordinates in terms of the number of integer points within it and on its boundary. In contrast, for the universal formula, you only need the length of the sides be it an integer or not. Hence, it is not limited to integer vertex coordinates. Also, despite being proved using Euler's formula which involves dividing the polygon into several grid points, and Minkowski's theorem which involves lattice points in symmetric convex sets, users of Pick's theorem or formula would acknowledge the fact that it is cumbersome and requires a lot of information about the polygon including the plane of graphs, subdivisions, number of vertices, edges, faces of the planar graph and so on. In comparison, the universal Formula requires just a little information to find the area of the polygon. Other Methods Other methods exist for roughly calculating the area of Polygons though most times inaccurately. The most accurate method includes the well-known formulas for finding the area of triangles and quadrilaterals like squares, rectangles, rhomboids, trapeziums, etc. Other less accurate methods include: 1) Triangle method: involves decomposing the polygon into triangles. The inaccuracy of this method lies in the lack of knowledge of the dimensions of the triangles like height, the total number of triangles to be formed, and the additions and approximations of the total area. 2) Trapezoid method: involves decomposing the Polygons into shapes with trapeziums and adding the resultant areas. It shares similar problems with the Triangle method. 3) Lopshits in 1963 formulated a method similar to the universal formula. However, it differs to a large extent and is cumbersome. 4) Grid squares: Polygons can also be divided into grid squares having their vertices at their grid points. This method is limited. 5) The area can also be approximated using the radius of an inscribed circle (or Apothem) and its perimeter, including some trigonometric ratio. This method involves practical errors and errors due to approximations. Summary of advantages of the Universal formula and its difference from others 1) most of the formulas listed require coordinates or Visual representation for Polygons with more than four sides unlike the universal formula 2) Also, the other formulas are cumbersome to learn, remember, and use 3) other formulas like Pick's theorem have limitations in the applications because they require integers, graphs, coordinates, vertex points, matrices, and inscribed or circumscribed circles 4) A lot of formulas require the knowledge of Apothem which is not easily derived theoretically and involves approximations. 5) The universal formula accounts for the area of all Polygons from triangles, quadrilaterals, and other Polygons in general. Hence there's no need for a separate formula for triangles, squares, and other Polygons.
Classification : 51E24, 51M30
Author(s) :
- Wisdom Malachy Uke (Geometry)

[00567] Topology-aware algorithm for constructing cartograms from density-equalising map projections

Session Date & Time : 2E (Aug.22, 17:40-19:20)
Type : Contributed Talk
Abstract : Cartograms are maps in which the areas of enumeration units $\text{(}$e.g. administrative divisions$\text{)}$ are proportional to quantitative data $\text{(}$e.g. population$\text{)}$. Generating cartograms with density-equalising map projections guarantees that geographic neighbours remain neighbours in the cartograms if all boundaries are infinitely dense sequences of points. However, computers represent boundaries with only finitely many points, often causing invalid topologies in the cartogram. This talk shows how line densification and topology-aware simplification solve this problem.
Classification : 51M30, 53-08, 68-04
Author(s) :
- Michael T Gastner (Yale-NUS College)
- Nguyen Phong Le (Yale-NUS College)
- Nihal Z Miaji (Yale-NUS College)
- Adi Singhania (Yale-NUS College)