arXiv research

New integrators for mechanical systems on Lie groups simplify based on group properties.

In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijectiv…

The paper addresses numerical integration issues in SV models, proposing a fast regime switching algorithm.

Develops integrators for nonholonomic systems on Lie groups.

Bayesian neural networks speed up numerical integration.

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numeri…

Neural dynamical systems are dynamical systems that are described at least in part by neural networks. The class of continuous-time neural dynamical systems must, however, be numerically integrated for simulation and learning. Here, we present a compact neural circuit for two common numerical integrators: the explicit …

This paper deals with the evaluation of double line integrals of the squared exponential covariance function. We propose a new approach in which the double integral is reduced to a single integral using the error function. This single integral is then computed with efficiently implemented numerical techniques. The perf…

The paper compares numerical schemes for nonholonomic systems using retraction maps.

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…

A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly dete…

We discuss two numerical methods, based on a path integral approach described in a previous paper (I), for solving the stochastic equations underlying the financial markets: the Monte Carlo approach, and the Green function deterministic numerical method. Then, we apply the latter to some specific financial problems. In…

Bayesian quadrature uses probabilistic models for estimating intractable integrals.

Bayesian Probabilistic Integration uses BART for high-dimensional, non-smooth functions.

A new method simulates square-root processes efficiently.

Bayesian quadrature improves integration efficiency with invariant priors.

A new method for estimating uncertainties in neural ODEs without numerical integration.

Geometric integrator preserves coadjoint orbits in dissipative systems.

Probabilistic solvers improve stability for stiff systems.

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation …

This paper tackles Bayesian system identification with probabilistic numerical methods.

Integration of the form $\int_a^\infty {f(x)w(x)dx}$, where $w(x)$ is either $\sin (ω{\kern 1pt} x)$ or $\cos (ω{\kern 1pt} x)$, is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration…

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…

DNA-SE uses deep learning to solve semiparametric problems efficiently.

Gradient flow preserves speed for integral Menger curvature curves.

The paper efficiently solves a complex option valuation equation for two assets.

Efficiently approximates integrals using a subset of samples from a target distribution in RKHS.

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic dif…

The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.

Bayesian quadrature improves integration on Riemannian manifolds.

Paper proposes efficient method to calculate Fisher-Bingham distribution normalizing constant.

Median-of-means sampling outperforms mean-of-means for large sample sizes in numerical integration.

This paper proposes a new method to learn integration schemes for complex ODEs.

New method for pricing American options in time-dependent models, improving accuracy and efficiency.

In the framework of path integral the evolution operator kernel for the Merton-Garman Hamiltonian is constructed. Based on this kernel option formula is obtained, which generalizes the well-known Black-Scholes result. Possible approximation numerical schemes for path integral calculations are proposed.

There is renewed interest in formulating integration as an inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. A…

Fenrir uses probabilistic numerics to simplify solving initial value problems.

This paper develops a novel numerical method for pricing American options in a two-asset jump-diffusion model.

Method calculates function integrals on complex manifolds.

Research covers geometry, analysis, and integration on infinite-dimensional spaces.

The paper finds new metrics for geodesic flows with rational integrals.

In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall…

The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable…

Study discretizes Dirac and port-Hamiltonian systems using manifolds.

We consider an application involving a financial quadratic portfolio of options, when the joint underlying log-returns changes with multivariate elliptic distribution. This motivates the needs for methods for the approximation of multiple integrals over hyperboloids. A transformation is used to reduce the hyperboloid i…

Paper solves stock loan pricing with finite maturity using integral equations.

Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time…

Combines normalizing flows and quasi-Monte Carlo for improved numerical integration.