We use AD to compute gradients for complex functionals in stochastic model calibration.
problem Computing gradients for functions involving expectations in stochastic models.
method Automatic Adjoint Differentiation and parallelization.
result Faster and easier to implement approaches for gradient computation.
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…
In this work, we discuss the Automatic Adjoint Differentiation (AAD) for functions of the form G=21∑1m(Eyi−Ci)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…
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 …
Framework for pricing waterfall structures using simulation and uncertainty modeling.
problem Pricing complex structured finance instruments under uncertainty.
method Simulation-based uncertainty modeling, calibrated probability distributions, PyTorch implementation, Adjoint Algorithmic Differentiation (AAD).
result Efficient gradient computation for risk sensitivity analysis and optimization.
Teaches matrix calculus for machine learning and optimization.
problem Computing derivatives of functions involving matrices.
method Extends differential calculus to vector spaces, focusing on practical applications in machine learning.
result Introduction of adjoint and reverse-mode differentiation for efficient computation.
Differential ML combines AAD with ML for fast, accurate financial derivatives pricing and risk management.
problem Computational bottlenecks in financial derivatives risk management.
method Novel algorithms using automatic adjoint differentiation (AAD) for training fast, accurate approximations in real-time.
result Convergence guarantees for fast, accurate pricing and risk approximations for arbitrary derivatives instruments.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Automatic differentiation is involved for long in applied mathematics as an alternative to finite difference to improve the accuracy of numerical computation of derivatives. Each time a numerical minimization is involved, automatic differentiation can be used. In between formal derivation and standard numerical schemes…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
ACA method improves gradient estimation for neural ODEs, reducing error and training time.
problem Inaccurate gradient estimation methods hinder the performance of neural ODEs on benchmark tasks.
method Adaptive Checkpoint Adjoint (ACA) method that applies trajectory checkpointing, deletes redundant components, and supports adaptive solvers.
result ACA reduces error rate by half and training time by half compared to adjoint and naive methods on image classification tasks.
We describe a canonical form for linear differential operators that are formally self-adjoint or formally skew-adjoint.
We review the current state of automatic differentiation (AD) for array programming in machine learning (ML), including the different approaches such as operator overloading (OO) and source transformation (ST) used for AD, graph-based intermediate representations for programs, and source languages. Based on these insig…
Derives adjoint formulas for matrix operations and applies them to specific cases.
problem Computing adjoints for matrix operations and specific matrix types.
method Derives adjoint formulas for matrix operations and applies them to specific cases.
result Closed-form expressions for adjoints in specific matrix types.
The paper explores self-adjointness of Laplace-Beltrami operator on special geometric manifolds.
problem Characterizing self-adjoint extensions of the Laplace-Beltrami operator on α-Grushin manifolds. method Introducing an exotic calculus of pseudodifferential operators adapted to the geometry of the singularity.
result Criterion for essential self-adjointness and determination of several self-adjoint extensions.
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 Δł 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)→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…
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
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.
problem Computing the index of families of self-adjoint elliptic boundary problems.
method Abstract axiomatic version of boundary triplets and their applications.
result Analytic proof of index theorem and computation of index differences.
The paper presents a method to infer unknown forcing functions in differential equations using Gaussian processes and adjoints.
problem Inferring unknown forcing functions in differential equations from noisy observations.
method Using adjoint methods to efficiently infer Gaussian process (GP) driven differential equations, with truncated basis expansions of the GP kernel.
result Efficient Bayesian inference of forcing functions modeled as GPs using adjoints, with lower computation than MCMC methods.
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.
problem Differentiating functions of large matrices in scientific and probabilistic machine learning models.
method Deriving and implementing new adjoint systems for Lanczos and Arnoldi iterations in JAX.
result Efficient differentiation of PDEs, Gaussian process models, and Bayesian neural networks.
Proposes DAM for optimizing discrete generative models.
problem Challenges in optimizing discrete generative models.
method Discrete Adjoint Matching (DAM) for discrete state spaces.
result Demonstrates effectiveness on synthetic and mathematical reasoning tasks.
Proposes a new model for RANS simulations with uncertainty.
problem Uncertainty in Reynolds-averaged Navier-Stokes simulations.
method Data-driven closure model with aleatoric uncertainty, Bayesian formulation, sparse indirect data.
result Accurate probabilistic predictions, even in regions of model error.
New method reduces errors in pricing and sensitivities for discontinuous payoffs.
problem Errors in pricing and sensitivities for discontinuous payoffs in digital and barrier options.
method Alternative methods for estimating sensitivities, including likelihood ratio and hybrid methods.
result New methods substantially reduce test errors in prices and sensitivities.
SNAPO optimizes policies for complex sequential decisions using differentiable simulation.
problem Optimizing policies for high-dimensional, sequential decisions under uncertainty.
method Embeds neural policy in a differentiable simulator, computes gradients efficiently.
result Produces sensitivities at a cost proportional to one reverse pass, regardless of sensitivity count.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
Researchers create a parametrix for resolvents on manifolds with ends.
problem Essential self-adjointness of elliptic symmetric differential operators on manifolds with ends.
method Introduced semiclasical pseudodifferential operators compatible with the end structure.
result Essential self-adjointness of elliptic symmetric differential operators proved.
Essential self-adjointness and spectrum of CR GJMS operator proved.
problem Characterizing the spectrum of CR GJMS operator.
method Proving essential self-adjointness and closed range, analyzing spectrum.
result CR GJMS operator has discrete spectrum with finite-dimensional eigenspaces.
Paper proves index theorem for self-adjoint elliptic boundary problems.
problem Proving index theorem for self-adjoint elliptic boundary problems.
method Topological and pseudo-differential methods, generalized Atiyah-Singer approach.
result Removed technical assumption to prove index theorem.
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.
problem Quantifying the distance between ResNet dynamics and Neural ODE solutions.
method Bounding the distance between hidden state trajectories and Neural ODE solutions, using gradient descent and Heun's method.
result Gradient descent and Heun's method can implicitly regularize ResNets towards Neural ODEs, especially for smooth residual functions.
The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.
Automatic differentiation---the mechanical transformation of numeric computer programs to calculate derivatives efficiently and accurately---dates to the origin of the computer age. Reverse mode automatic differentiation both antedates and generalizes the method of backwards propagation of errors used in machine learni…
A new method for solving complex inverse problems using deep learning.
problem Estimating complex spatially-varying parameters in high-dimensional Bayesian inverse problems.
method A variational inference method with a deep generative prior to approximate the posterior distribution.
result The method improves estimation accuracy and efficiency for solving high-dimensional inverse problems.
Neural controlled DEs model irregular time series by adjusting based on observations.
problem Modeling irregularly sampled multivariate time series with memory-efficient adjoint-based backpropagation.
method Neural controlled differential equations (CDEs) that adjust based on subsequent observations.
result Achieves state-of-the-art performance on various datasets.
For a Riemannian covering p:M2→M1, we compare the spectrum of an essentially self-adjoint differential operator D1 on a bundle E1→M1 with the spectrum of its lift D2 on p∗E1→M2. We prove that if the covering is infinite sheeted and amenable, then the spectrum of $…
Develops a mathematical model for automatic differentiation in machine learning.
problem Current automatic differentiation lacks a simple mathematical model for machine learning.
method Articulates relationships between program differentiation and nonsmooth functions, provides a class of functions and nonsmooth calculus.
result Shows how nonsmooth calculus applies to stochastic approximation methods and evidence of artificial critical points.
The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.
problem The Plebański complex studies the linearization of equations for hyper-Kähler manifolds.
method Defined and studied properties of the Plebański complex, showing it fits into the elliptic complex framework.
result The Plebański complex is an elliptic differential operator that squares to the Laplacian and is composed of two Dirac operators.
Gradient-enhanced deep GPs improve multifidelity model accuracy.
problem Improving accuracy in multifidelity models using gradient data.
method Extending deep Gaussian processes to incorporate gradient data.
result Gradient-enhanced deep GP outperforms other models in predicting aerodynamic coefficients.
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…
We present a system for the automatic differentiation of a higher-order functional array-processing language. The core functional language underlying this system simultaneously supports both source-to-source automatic differentiation and global optimizations such as loop transformations. Thanks to this feature, we demo…
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…
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…
SDE Matching eliminates simulation for training Latent SDEs, achieving similar performance.
problem Training Latent SDEs with adjoint sensitivity methods is computationally expensive and limited.
method SDE Matching, inspired by Score- and Flow Matching, eliminates simulation for training Latent SDEs.
result SDE Matching achieves performance comparable to adjoint sensitivity methods while reducing computational complexity.
This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general n∈N. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case …
Develops algorithm to differentiate Metropolis-Hastings for optimization.
problem Optimizing intractable densities with discrete components.
method Fuses stochastic automatic differentiation with Markov chain coupling schemes.
result Unbiased and low-variance gradient estimator for intractable densities.