Title: Explicit Constructions of Canonical and Absolute Minimal Degree Liftings of Twisted Edwards Curves

Abstract: Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. We study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmüller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general formulas are given.

Additionally, we define a notion of Minimal and Absolute Minimal Degree Liftings modulo pn and an associated lift of points we call an Edwards Lift. We show how the Canonical Lift may be used to find examples of Absolute Minimal Degree Liftings, at least modulo p^3.

]]>Affiliation: RWTH Aachen

Host: Cory Hauck

Title: Gradient-free Optimization Methods and Its Mean-Field Limits

Abstract: We are interested in the construction of numerical methods for constrained high-dimensional constrained nonlinear optimization problems by gradient free techniques. Gradients are replaced by particle approximations and recently different methods have been proposed, e.g., consensus-based, swarm-based or ensemble Kalman based methods. We discuss recent extensions to the constrained and the parametric case as well as their corresponding mean field descriptions in the many particle limit. Those allow to show convergence as well as the analysis of properties of the new algorithm. Several numerical examples, also in high dimensions, illustrate the theoretical findings as well as the performance of those methods.

Speaker: Diego del-Castillo-Negrete, ORNL

Abstract: The exit time probability, which gives the likelihood that an initial condition leaves a prescribed region of the phase space of a dynamical system at, or before, a given time, is arguably one of the most natural and important transport problems.

In this seminar we present an accurate and efficient numerical method for computing this probability for systems described by Fokker-Planck equations modeling local transport (e.g., diffusion) or nonlocal transport (e.g., fractional diffusion). The method is based on the direct evaluation of the Feynman-Kac formula that establishes a link between the adjoint Fokker-Planck equation and the corresponding forward SDE (stochastic differential equations). In the case of local transport, the SDE are driven by Brownian motion and in case of nonlocal transport, depending on the nonlocal kernel, by compound Poisson processes or alpha-stable processes. We illustrate and benchmark the proposed method with numerical examples of interest to physics applications.

* Work done in collaboration with G. Zhang and M. Yang, ORNL

]]>Title: Point configurations and geometric averaging operators

Abstract:

Two classic questions - the Erdos distinct distance problem, which asks about the least number of distinct distances determined by points in the plane, and its continuous analog, the Falconer distance problem - both focus on distance. Here, distance can be thought of as a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will go through some of the history of such point configuration questions for triangles and end with some exciting recent progress.

Title: Quantitative Homogenization of Elliptic Equations

Abstract:

These lectures provide an introduction to the quantitative homogenization theory for second order elliptic equations and systems in divergence form with periodic coefficients. We begin with the classical qualitative theory of homogenization in Lecture One and give an overview of the quantitative results obtained in recent years. In Lectures Two and Three, we will discuss some of techniques developed to establish the large-scale regularity estimates.

]]>Title: MegaMenger Graphs

Abstract: In October 2014, many faculty and students around the world worked to build models of the Menger sponge, a type of fractal, out of business cards. This model can itself be modeled using graph theory, with each vertex representing a small cube, and an edge between two vertices whenever they share a face. We study graphs representing different steps of building the Menger sponge and Sierpinski carpet to determine their order, size, vertex degrees, and chromatic number, along with the surface area of the Menger sponge. Calculating these quantities requires solving many recurrence relations.

]]>Adaptive and Topological Deep Learning

with Applications to Neuroscience

Title: Computational Aspects of Mixed Characteristic Witt Vectors and Denominators in Canonical Liftings of Elliptic Curves

Abstract: Given an ordinary elliptic curve E over a field k of characteristic p, there is an elliptic curve E over the Witt vectors W(k) for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over F_p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on some conjectures of Finotti, and introduce some conjectures of our own. We also investigate the structure of rings of Witt vectors over arbitrary commutative rings and give a computationally useful isomorphism for Witt vectors over Z/p^a Z.

]]>Affiliation: University of Electronic Science and Technology, Chengdu, China.

Title: Finite Element Discretization for Variable-order Fractional Diffusion Problems

Abstract: We propose a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. A number of challenges are encountered when discretizing such problems. The first is the heterogeneous kernel singularity in the fractional integral operator. The second comes from the dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. To address these challenges, we propose a strategy that decomposes the system matrix into three components. The first is a sparse matrix that handles the singular near-field separately and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far-field and is approximated by a hierarchical $\mathcal{H}^{2}$ matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. In this talk, analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

]]>Title: Doubly Degenerate Cahn-Hilliard Models of Surface Diffusion

Abstract: Motion by surface diffusion is a type of surface-area-diminishing motion such that the enclosed volume is preserved and is important is many physical applications, including solid state de-wetting. In this talk I will describe a relatively recent diffuse interface model for surface diffusion, wherein the sharp-interface surface description is replaced by a diffuse interface, or boundary layer, with respect to some order parameter. One of the nice features of the new doubly degenerate Cahn-Hilliard (DDCH) model is that it permits a hyperbolic tangent description of the diffuse interfaces, in an asymptotic sense, but, at the same time, supports a maximum principle, meaning that the order parameter stays between two predetermined values. Furthermore, numerics show that convergence to the sharp interface solutions for the DDCH model is faster than that of the standard regular Cahn-Hilliard (rCH) model.The down side is that the new DDCH model is singular and much more nonlinear than the rCH model, which makes numerical solution difficult, and it is still only first order accurate asymptotically. We will describe positivity-preserving numerical methods for the new model and review some existing numerics. We will also describe very recent results on the rigorous Gamma convergence of the underlying diffuse interface energy.

]]>Title: Quantitative Homogenization of Elliptic Equations

Abstract: These lectures provide an introduction to the quantitative homogenization theory for second order elliptic equations and systems in divergence form with periodic coefficients. We begin with the classical qualitative theory of homogenization in Lecture One and give an overview of the quantitative results obtained in recent years. In Lectures Two and Three, we will discuss some of techniques developed to establish the large-scale regularity estimates.

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