arXiv research

A recent paper of Arnold, Falk, and Winther [Bull AMS, 47 (2010)] showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direc…

The paper analyzes finite element methods on manifolds with approximate metrics.

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochast…

Finite element method approximates scalar curvature in arbitrary dimensions.

Alternative wavelet analysis method for financial signals.

A new method uses SPDEs to efficiently model random fields on complex domains.

The paper solves a complex option pricing model using finite elements.

Paper solves convertible bond valuation using finite elements with penalty method.

New finite element method for complex forms in any dimension.

Finite element method applied to Leland's model for option pricing with transaction costs.

NP-ODE models FEA simulations with uncertainty, improving accuracy and efficiency.

The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.

The paper improves convergence rates of curvature approximations using Regge elements.

Study finds minimal length networks connecting three points in Heisenberg group.

Constructs finite element spaces for $(p,q)$-forms, excluding one subspace.

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de …

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We s…

The paper analyzes how Gaussian kernel parameters affect posterior covariance in Gaussian processes.

Characterizes periodic elements in Artin-Tits groups via stability conditions.

Two elements generate extended mapping class groups of certain surfaces.

A neural atlas simplifies 3D geometry simulation by avoiding meshing.

A positive integer $m$ will be called a {\it finitistic order} for an element $γ$ of a group $Γ$ if there exist a finite group $G$ and a homomorphism $h:Γ\to G$ such that $h(γ)$ has order $m$ in $G$. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian gr…

Extends gradient-based optimization to spline functions.

Variational approximations for curve flows on Riemannian manifolds.

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundar…

Let G be a finitely generated group with a given word metric. The asymptotic density of elements in G that have a particular property P is defined to be the limit, as r goes to infinity, of the proportion of elements in the ball of radius r which have the property P. We obtain a formula to compute the asymptotic densit…

Rust library solves complex equations on abstract simplicial complexes.

Statistical finite elements use Langevin dynamics to efficiently handle uncertainty quantification.

If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .

Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to…

Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if $(g, r) \neq (2, 5k+4)$ for some integer $k$.

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the proces…

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension $n$, arbitrary polynomial degree $r$, and arbitrary differential form degree $k$. The study of finite …

The paper constructs discrete Hessian and divdiv complexes on triangulations and proves their cohomology isomorphic to continuous versions.

Using piezoelectric impedance/admittance sensing for structural health monitoring is promising, owing to the simplicity in circuitry design as well as the high-frequency interrogation capability. The actual identification of fault location and severity using impedance/admittance measurements, nevertheless, remains to b…

Erdős introduced the noncommuting graph, in order to study the number of commuting elements in a finite group. Despite the use of combinatorial ideas, his methods involved several techniques of classical analysis. The interest for this graph is becoming relevant in the last years for various reasons. Here we deal with …

We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…

EPGP surrogate outperforms finite elements in solving wave equations.

Develops unisolvent weights for Nédélec second family finite elements in 2D.

We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset F_n$ which is contained in the union of finitely many $Aut(F_n)$-orbits, we construct finite-index normal subgroups of $F_n$ wh…

Study shows Morse elements are common in acylindrically hyperbolic groups.

Constructs CAT(0) actions for certain groups without unipotent elements.

The paper approximates Einstein tensor using finite elements.

Paper uses IGA for efficient pricing of financial derivatives, comparing it to FDM and FEM.

We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.

Whether you trade futures for yourself or a hedge fund, your strategy is counted. Long and short position limits make the number of unique strategies finite. Formulas of the numbers of strategies, transactions, do nothing actions are derived. A discrete distribution of actions, corresponding probability mass, cumulativ…

The paper constructs hyperbolic elements in multiple spaces.

Study on curvature in finitely generated groups, showing positive curvature in specific cases.