**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Concept# Automatic differentiation

Summary

In mathematics and computer algebra, automatic differentiation (auto-differentiation, autodiff, or AD), also called algorithmic differentiation, computational differentiation, is a set of techniques to evaluate the partial derivative of a function specified by a computer program.
Automatic differentiation exploits the fact that every computer calculation, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, partial derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor of more arithmetic operations than the original program.
Difference from other differentiation methods
Automatic differentiation is distinct from symbolic differentiation and numerical differentiation.
Symbolic differentiation fac

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications

Loading

Related people

Loading

Related units

Loading

Related concepts

Loading

Related courses

Loading

Related lectures

Loading

Related publications (14)

Loading

Loading

Loading

Related people (4)

Related concepts (8)

Machine learning (ML) is an umbrella term for solving problems for which development of algorithms by human programmers would be cost-prohibitive, and instead the problems are solved by helping machin

In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinag

Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solv

Related units (2)

Physically based rendering methods can create photorealistic images by simulating the propagation and interaction of light in a virtual scene. Given a scene description including the shape of objects, participating media, material properties, etc., the simulation computes an image representing the radiance reaching the sensor.This thesis, however, pursues the corresponding inverse problem: given observations such as pictures of a scene, we want to recover a plausible description of its components. Unfortunately, the rendering process is typically far too complex to invert analytically. We therefore turn to iterative gradient-based approximations of the inverse, that require efficiently estimating gradients of an objective function with respect to the scene parameters of interest.The first part of this thesis is dedicated to algorithms for gradient estimation. We consider the usage of automatic differentiation and examine the associated tradeoffs. Next, we introduce radiative backpropagation, an adjoint method that casts the gradient estimation problem into a modified light transport problem, unlocking vastly more efficient implementations. For the case of participating media, we propose differential ratio tracking, a sampling technique that addresses the bias and high variance found in existing gradient estimators.In the second part, we focus on the design of systems to support effective differentiable rendering research, including the efficient implementation of the algorithms above. We describe the architecture and features of Mitsuba 2, an open-source retargetable physically based renderer. Mitsuba 2 supports different representations of colors (RGB, spectral, polarized), computing platforms (scalar, CPU vectorized, GPU), and numerical precision, within a single codebase. Importantly, automatic differentiation can be applied throughout the system. Then, we extend this design by applying an automatic conversion from wavefront-style rendering to a megakernel-based approach, leveraging just-in-time compilation. We obtain a fast, flexible and memory-efficient framework for primal and differentiable physically based rendering.Finally, the third part showcases three applications of differentiable physically based rendering: caustic design, inverse volume rendering, and material & lighting estimation in real indoor scenes. In all cases, special care is taken to avoid sub-optimal local minima due to the ambiguous and non-convex nature of the reconstruction problems.

In this thesis, I define and explain the notion of punctual analytical uniform development (DUAP) versus moments or cumulants approximation of punctual uniforms analyticals 's class statistics. Hence, I derive "truncated" DUAP's version for numerical computation and implementation which called finite DUAP (F-DUAP). Using F-DUAP approximation lead to an error which was estimated. Due to nature's one of axiom of UAP statistics, the concept extension of DUAP method, to an other class statistic is limited. So, a new local theoretical concept was defined named analytical uniform development (DUA). This generalization let all derived DUAP's theorems become more general. Automatic differentiation and F-DUAP allow the implementation of DUAP or DUA method's on computer: I write the CUMAD and CUMADG codes that make the methods of a practical use. By, the programme CUMAD, I valid the utility of DUAP method, when I applied it to the approximation to moments of "weighted sum of squares statistic".

Related courses (27)

This course offers an introduction to numerical methods for the solution of mathematical problems as: solution of systems of linear and non-linear equations, functions approximation, integration and differentiation and solution of differential equations.

Mettre en pratique les bases de la programmation vues au semestre précédent. Développer un logiciel structuré. Méthode de debug d'un logiciel. Introduction à la programmation scientifique. Introduction à l'instrumentation virtuelle.

Since 2010 approaches in deep learning have revolutionized fields as diverse as computer vision, machine learning, or artificial intelligence. This course gives a systematic introduction into influential models of deep artificial neural networks, with a focus on Reinforcement Learning.

, , ,

Dr.Jit is a new just-in-time compiler for physically based rendering and its derivative. Dr.Jit expedites research on these topics in two ways: first, it traces high-level simulation code (e.g., written in Python) and aggressively simplifies and specializes the resulting program representation, producing data-parallel kernels with state-of-the-art performance on CPUs and GPUs. Second, it simplifies the development of differentiable rendering algorithms. Efficient methods in this area turn the derivative of a simulation into a simulation of the derivative. Dr.Jit provides fine-grained control over the process of automatic differentiation to help with this transformation. Specialization is particularly helpful in the context of differentiation, since large parts of the simulation ultimately do not influence the computed gradients. Dr.Jit tracks data dependencies globally to find and remove redundant computation.

Related lectures (52)