The paper solves a complex option pricing model using finite elements.
problem Risk-Adjusted Pricing Methodology (RAPM) Black-Scholes model with transaction costs.
method Spatial finite element models based on P1 and/or P2 elements, combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
Paper solves convertible bond valuation using finite elements with penalty method.
problem Valuation of convertible bonds under penalty TF model.
method Solves TF system of equations using P1 and P2 finite elements with penalty method.
result Numerical solutions compare favorably with finite difference method.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ-form-valued k-forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
Finite element method applied to Leland's model for option pricing with transaction costs.
problem Option pricing with transaction costs using Leland's model.
method Spatial finite element models based on P1 and/or P2 elements combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
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…
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
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→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…
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.
problem Solving partial differential equations on abstract simplicial complexes.
method Finite Element Exterior Calculus, intrinsic Riemannian metric, first-order Whitney basis functions.
result Verification through convergence studies on elliptic Hodge-Laplace eigenvalue and source problems.
Statistical finite elements use Langevin dynamics to efficiently handle uncertainty quantification.
problem Uncertainty quantification in finite element models with observed data.
method Langevin dynamics, unadjusted Langevin algorithm (ULA), for sampling posterior distributions.
result ULA provides a scalable and efficient method for characterizing the posterior distribution of statFEM models.
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 .
The paper analyzes finite element methods on manifolds with approximate metrics.
problem Analyzing finite element methods on manifolds with approximate metrics.
method Intrinsic finite element exterior calculus applied to manifolds with Regge metrics.
result Analysis and implementation of a method for computing an approximate Levi-Civita connection form.
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)=(2,5k+4) for some integer k.
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.
problem Discrete construction of Hessian and divdiv complexes on triangulations.
method Construction of discrete Hessian and divdiv complexes using finite elements and Dirac measures on triangulations.
result The cohomology of the constructed complexes is isomorphic to the continuous de Rham cohomology.
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.
problem Benchmarking Gaussian Process surrogates vs. finite elements for wave equation solutions.
method EPGP uses penalized least squares and exponential-polynomial bases; CN-FEM employs Crank--Nicolson time stepping.
result EPGP achieves lower error than CN-FEM under matched degrees-of-freedom.
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
Develops unisolvent weights for Nédélec second family finite elements in 2D.
problem Finding efficient degrees of freedom for Nédélec second family finite elements.
method Uses techniques of homological algebra to obtain degrees of freedom for differential forms.
result Provides a family of unisolvent and minimal physical degrees of freedom for Nédélec second family finite elements.
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 O⊂Fn which is contained in the union of finitely many Aut(Fn)-orbits, we construct finite-index normal subgroups of Fn wh…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Finite element method approximates scalar curvature in arbitrary dimensions.
problem Approximating scalar curvature using finite elements in arbitrary dimensions.
method Piecewise polynomial interpolants of a smooth Riemannian metric on a triangulated polyhedral domain.
result Finite element interpolants converge to scalar curvature with rate O(hr+1) in H−2(Ω) norm. Constructs CAT(0) actions for certain groups without unipotent elements.
problem Understanding actions of certain groups on CAT(0) spaces.
method Constructs an isometric action of a group on a CAT(0) space.
result Fundamental groups of certain 3-manifolds do not admit faithful finite-dimensional unitary representations.
The paper approximates Einstein tensor using finite elements.
problem Approximating Einstein tensor for piecewise polynomial metrics.
method Finite element method applied to Riemannian metrics.
result Convergence rate of O(hr+1) in H−2(Ω)-norm. We prove that for genus g=3,4, the extended mapping class group Mod±(Sg) can be generated by two elements of finite orders. But for g=1, Mod±(S1) cannot be generated by two elements of finite orders.
The paper constructs hyperbolic elements in multiple spaces.
problem Constructing hyperbolic elements in multiple Gromov-hyperbolic spaces.
method Explicit construction under minimal conditions.
result Set of simultaneously hyperbolic elements has strictly positive density.
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.
Study on mapping class groups of infinite type surfaces.
problem Property Pextnaive for mapping class groups of infinite type surfaces. method Analyzes the existence of elements g and hi satisfying specific group properties. result Establishes the existence of g for any finite collection of non-trivial elements hi. Characterizes when geodesics in groups are generic.
problem Understanding genericity of geodesics in groups.
method Characterizes Gromov hyperbolicity via contracting elements.
result Genericity of contracting geodesics in groups.
Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface Σg, which are rotations of Σg through 2π/p, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.
Study shows how numerical discretization affects reconstructions and parameter distributions in nano metrology.
problem Impact of numerical discretization on parameter reconstructions and model parameter distributions.
method Bayesian target vector optimization, finite element model, Gaussian process, stochastic machine learning surrogate models, Markov chain Monte Carlo sampler.
result Numerical discretization parameters impact the accuracy and distribution of reconstructed model parameters.
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admi…
The paper studies cohomology of groups with contracting elements.
problem Understanding the cohomology of groups with specific elements.
method Proving infinite-dimensional relative bounded cohomology for groups with contracting elements.
result The cohomology is infinite-dimensional for groups with contracting elements.
New method approximates anisotropic curve shortening flow.
problem Approximating anisotropic curve shortening flow.
method Weak formulation and finite element approximation.
result Optimal H1-error bound for approximation. Finite order elements with infinite centralizers in 3-manifold groups imply specific structure.
problem Understanding centralizers of torsion elements in 3-manifold groups.
method Analyzing PD3-complexes and applying the Projective Plane Theorem. result 3-manifolds with specific properties (e.g., RP2imesS1) arise from certain centralizer conditions. This article presents a finite element method (FEM) for a partial integro-differential equation (PIDE) to price two-asset options with underlying price processes modeled by an exponential Levy process. We provide a variational formulation in a weighted Sobolev space, and establish existence and uniqueness of the FEM-ba…
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…
Geometric model for a specific group in Artin groups.
problem Understanding the structure of outer automorphism groups in Artin groups.
method Proving proper cocompact action on a subcomplex of the spine of outer space.
result U(A; G, Ht) acts properly cocompactly on a finite-dimensional subcomplex.
In order to generalize finite element methods to differential forms, Arnold, Falk, and Winther constructed two families of spaces of polynomial differential forms on a simplex T, the PrΛk(T) spaces and the Pr−Λk(T) spaces, where k is the degree of the form and r is the degree of its coe…
A new method uses SPDEs to efficiently model random fields on complex domains.
problem Efficient representation of random fields on complex domains for engineering and machine learning.
method Uses SPDEs to develop a scalable framework for statFEM and GP regression.
result Can model anisotropic, non-stationary random fields with arbitrary smoothness.
We present a local formulation for 2D Discrete Exterior Calculus (DEC) similar to that of the Finite Element Method (FEM), which allows a natural treatment of material heterogeneity (element by element). It also allows us to deduce, in a robust manner, anisotropic fluxes and the DEC discretization of the pullback of 1-…
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…