arXiv research

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

Forward-Euler fails for simulating Wasserstein gradient flows with KL divergence.

The paper defines and calculates Euler characteristics for quandles.

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Leg…

In this paper we will discuss some new developments in the design of numerical methods for optimal control problems of Lagrangian systems on Lie groups. We will construct these geometric integrators using discrete variational calculus on Lie groups, deriving a discrete version of the second-order Euler-Lagrange equatio…

We analyze exponential integrability properties of the Cox-Ingersoll-Ross (CIR) process and its Euler discretizations with various types of truncation and reflection at 0. These properties play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a class of sto…

Corrected samplers reduce discretization error in discrete flow models without additional computational cost.

In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to simultaneously cover several cases of interest in discrete and continuous descri…

Let $e$ denote the Euler class on the space $Hom(Γ_g, PSL(2,\mathbb R))$ of representations of the fundamental group $Γ_g$ of the closed surface $Σ_g$ of genus $g$. Goldman showed that the connected components of $Hom(Γ_g, PSL(2,\mathbb R))$ are precisely the inverse images $e^{-1}(k)$, for $2-2g\leq k\leq 2g-2$, and t…

Novel discretization of Euler equations for incompressible fluids.

EuSN uses Euler discretization for stable, non-dissipative reservoir computing.

In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by Alan Weinstein in [19]. To do this, we will need some results on the geometry of …

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDiff^δ_ω(S^1);\mathbb{Q})$ is zero where $Diff^δ_ω(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler clas…

New methods solve complex financial equations.

Study on stability in discretized hydrodynamics model.

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme.…

Develops multifactor approximations for SVEs with completely monotone kernels.

In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho…

Simplified analysis of diffusion models using discrete random variables.

We introduce the $Γ$-Euler-Satake characteristics of a general orbifold $Q$ presented by an orbifold groupoid $\mathcal{G}$, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…

Higher-order optimization problems naturally appear when investigating the effects of a patent with finite length, as in the pioneering work of Futagami and Iwaisako (2007). In this paper, we establish the Euler equations and transversality conditions necessary for analyzing such higher-order optimization problems. We …

Defines discrete differential geometry concepts in homotopy type theory.

We consider the stochastic volatility model $dS_t = σ_t S_t dW_t,dσ_t = ωσ_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $β=1$ (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in …

The paper proves a new discrete Laplacian for 3D meshes and shows its superiority over primal construction.

Asymptotic error distribution for approximation of a stochastic integral with respect to continuous semimartingale by Riemann sum with general stochastic partition is studied. Effective discretization schemes of which asymptotic conditional mean-squared error attains a lower bound are constructed. Two applications are …

A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying …

We determine the extent to which the collection of $Γ$-Euler-Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the collection of $Γ$-Euler-Satake characteristics corresponding to free or free abelian $Γ$ and are not…

New Euler characteristics for groupoids generalize orbifold Euler characteristics.

This work proves a strong convergence result for a geometric EM scheme on Riemannian manifolds.

The paper introduces a new discretization of Gaussian curvature on surfaces.

In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular rea…

RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.

We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

Paper proposes an algorithm for sampling from complex mixture distributions without requiring smoothness.

We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discretization leads to factorizable discrete nets with a novel discrete analog of the orthogonality property. A discrete confocal coordinate system…

Motivated by decompositions of spaces that arise in continuous and discrete Morse theory, we describe a so called fibrous decomposition Z = X_0(Y_1)X_1 ... X_{n-1}(Y_n)X_n of a space Z. Among the applications is a succinct formula for the Euler-Poincare characteristic of Z, e(Z) = e(X_0) - e(Y_1) + e(X_1) - ... + e(X_{…

Improved KLMC for sampling under various conditions.

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a con…

New method improves Euler approximation for local stochastic volatility models.

The paper proves a theorem for discretizing Gaussian curvature on surfaces.

Study on invariants of complex hyperbolic disc bundles over surfaces, proving a conjecture.

Derives Mirror Descent from gradient flow on a Riemannian manifold.

Discrete Lagrange problems solved with Lie group constraints.

The paper constructs 3-manifolds with co-orientable taut foliations but no foliations with vanishing Euler class.

Study vector fields on non-compact manifolds with group action.

We consider a discrete-time approximation of paths of an Ornstein--Uhlenbeck process as a mean for estimation of a price of European call option in the model of financial market with stochastic volatility. The Euler--Maruyama approximation scheme is implemented. We determine the estimates for the option price for prede…

We present a dynamical system framework for understanding Nesterov's accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integrat…