arXiv research

New estimator for SDEs is shown to be an adjoint state method.

Two of the most important areas in computational finance: Greeks and, respectively, calibration, are based on efficient and accurate computation of a large number of sensitivities. This paper gives an overview of adjoint and automatic differentiation (AD), also known as algorithmic differentiation, techniques to calcul…

Proves formal self-adjointness of certain differential operators.

The study confirms essential self-adjointness for certain differential operators on manifolds.

We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.

Framework for pricing waterfall structures using simulation and uncertainty modeling.

Proposes DAM for optimizing discrete generative models.

In this note we derive the backward (automatic) differentiation (adjoint [automatic] differentiation) for an algorithm containing a conditional expectation operator. As an example we consider the backward algorithm as it is used in Bermudan product valuation, but the method is applicable in full generality. The method …

Derives adjoint formulas for matrix operations and applies them to specific cases.

The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.

We use AD to compute gradients for complex functionals in stochastic model calibration.

Let $Δ$ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil of operators $\hPi(Δ)=\{Δ_ł\}$ passing through the operator $Δ$ such that any $Δ_ł$ is a linear differential operator acting on densities of weight $ł$. This pencil can be iden…

Let $M$ be a complete Riemannian manifold and let $Ω^*(M)$ denote the space of differential forms on $M$. Let $d:Ω^*(M) \to Ω^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V…

We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely efficient calculation of correlation Risk of option prices computed with Monte Carlo simulations. A key point in the construction is the use of binning to simultaneously achieve computational efficiency and accurate confidence intervals. We illus…

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator…

Abstracts a construction of boundary triplets for self-adjoint elliptic problems.

The paper presents a method to infer unknown forcing functions in differential equations using Gaussian processes and adjoints.

Differential ML combines AAD with ML for fast, accurate financial derivatives pricing and risk management.

The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential eq…

Efficiently differentiate functions of large matrices using new adjoint systems.

The well known conformal covariance of the Dirac operator acting on spinor fields over a semi Riemannian spin manifold does not extend to powers thereof in general. For odd powers one has to add lower order curvature correction terms in order to obtain conformal covariance. We derive an algorithmic construction in term…

In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form $G=\frac{1}{2}\sum_1^m (Ey_i-C_i)^2$, which often appear in the calibration of stochastic models. { We demonstrate that it allows a perfect SIMD\footnote{Single Input Multiple Data} parallelization and provide its relative co…

SDE Matching eliminates simulation for training Latent SDEs, achieving similar performance.

New method efficiently computes gradients for stochastic differential equations.

SNAPO optimizes policies for complex sequential decisions using differentiable simulation.

New method reduces errors in pricing and sensitivities for discontinuous payoffs.

Study essential spectrum of differential operators on geometrically finite orbifolds.

Researchers create a parametrix for resolvents on manifolds with ends.

Essential self-adjointness and spectrum of CR GJMS operator proved.

Paper proves index theorem for self-adjoint elliptic boundary problems.

This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.

Investigates the relationship between ResNets and Neural ODEs, quantifying their closeness and providing training methods.

The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.

Neural controlled DEs model irregular time series by adjusting based on observations.

For a Riemannian covering $p \colon M_{2} \to M_{1}$, we compare the spectrum of an essentially self-adjoint differential operator $D_{1}$ on a bundle $E_{1} \to M_{1}$ with the spectrum of its lift $D_{2}$ on $p^{*}E_{1} \to M_{2}$. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $…

The paper studies Lie n-algebroids and their representations up to homotopy.

The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.

Based on operator identities and their formal adjoints, we derive two symmetry operators for the linearized Einstein operator on vacuum backgrounds of Petrov type D and in particular the Kerr spacetime. One of them is of differential order four and coincides with a result of Cohen and Kegeles. The other one is a new op…

ACA method improves gradient estimation for neural ODEs, reducing error and training time.

We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables, together with higher order derivatives. In particular, we show that the second varia…

Framework calculates positional influence in causal residual Transformers.

Adjoint sampler targets infinite-dimensional function spaces for efficient sampling.

We develop a categorical index calculus for elliptic symbol families. The categorified index problems we consider are a secondary version of the traditional problem of expressing the index class in K-theory in terms of differential-topological data. They include orientation problems for moduli spaces as well as similar…

Graph neural networks learn PDEs from sparse, irregular data.

We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint representations of the orthogonal and symplectic Lie algebras $so(n)$ and $sp(n)$.

We provide criteria for self-adjointness and τ-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of…

Motivated by a problem in local differential geometry of Cauchy--Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or, equivalently, consisting of a nondegenerate Hermitian form and a symmetric bilinear f…

Consider a formally self-adjoint first order linear differential operator acting on pairs (2-columns) of complex-valued scalar fields over a 4-manifold without boundary. We examine the geometric content of such an operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients…