Symbolic Differentiation Research Articles

Clad — Automatic Differentiation Using Clang and LLVM

Differentiation is ubiquitous in high energy physics, for instance in minimization algorithms and statistical analysis, in detector alignment and calibration, and in theory. Automatic differentiation (AD) avoids well-known limitations in round-offs and speed, which symbolic and numerical differentiation suffer from, by transforming the source code of functions. We will present how AD can be used to compute the gradient of multi-variate functions and functor objects. We will explain approaches to implement an AD tool. We will show how LLVM, Clang and Cling (ROOT's C++11 interpreter) simplifies creation of such a tool. We describe how the tool could be integrated within any framework. We will demonstrate a simple proof-of-concept prototype, called Clad, which is able to generate n-th order derivatives of C++ functions and other language constructs. We also demonstrate how Clad can offload laborious computations from the CPU using OpenCL.

Analiza dynamiki modeli wzrostu gospodarczego za pomocą środowiska obliczeniowego Mathematica

Differential equations and optimal control theory are the basic tools of the mathematical theory of economic growth. The aim of this paper is to show (using the example of the Mankiw-Romer-Weil and Lucas-Uzawa growth models) some capabilities of Mathematica in the symbolic and numerical solving of differential equations, calculating optimal values of integral functional (discounted lifetime utility) when only numerical optimal solutions are known, the symbolic differentiation of solutions given in terms of hypergeometric Gauss functions and the graphical presentation of solutions to differential equations.

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The Personality Stereotypes of Three Cohabiting Ethnic Groups

The personality stereotypes of two Finno-Ugric ethnic groups, Erzians and Mokshans, who live together with Russians in the Republic of Mordovia, Russian Federation, were studied. As in previous studies, we found little support for the idea that ethnic personality stereotypes describe accurately, or even in an exaggerated manner, actual personality dispositions to think, feel, and behave in a characteristic way. The mean self-reported personality traits of participants were mostly identical, irrespective of the ethnic group they belong to. Erzians’ personality stereotypes about a typical Mokshan and, in turn, Mokshans’ stereotypes about a typical Erzian were socially less favorable compared with the way in which both ethnic groups perceived a typical Russian and, especially, a representative of their own ethnic group. As both in-group and out-group personality stereotypes were largely unfounded, there was no support for the primordial or essentialist account of ethnic stereotypes. The main function of the polarization of personality stereotypes may, thus, be symbolic self-identification and differentiation from other very similar ethnic groups.

Applications of algorithmic differentiation to phase retrieval algorithms.

In this paper, we generalize the techniques of reverse-mode algorithmic differentiation to include elementary operations on multidimensional arrays of complex numbers. We explore the application of the algorithmic differentiation to phase retrieval error metrics and show that reverse-mode algorithmic differentiation provides a framework for straightforward calculation of gradients of complicated error metrics without resorting to finite differences or laborious symbolic differentiation.

Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models

This paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework. This framework hinges on a new equation, the unifying chain rule, from which all the methods can be derived. The computation of derivatives is described as a two-step process: the evaluation of the partial derivatives and the computation of the total derivatives, which are dependent on the partial derivatives. Finite differences, the complex-step method, and symbolic differentiation are discussed as options for computing the partial derivatives. It is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation, the direct and adjoint methods, and coupled analytic methods for multidisciplinary systems. Each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches. The paper ends with a discussion of current challenges and possible future research directions in this field.

Dirty work and the construction of identity. An ethnographic study of management accounting practices

This paper examines the processes by which identity work influences accounting and organisational practices. Analysing ethnographic material, we study how accountants engage in a struggle for recognition in a context where tensions emerge from the confrontation between idealised occupational aspirations and situated possibilities. To theorise this struggle we draw on Everett Hughes’s conceptualisation of a moral division of labour. Building on his concept of “dirty work”, we differentiate between the “unclean” and the “polluted”. Accountants have to perform tasks that are incompatible with the aspirational identities they claim; more than “boring”, these tasks become symbols of misrecognition. We call these unclean tasks. Yet even tasks that, in a more favourable context, would be associated with prestigious aspects of the job, can become degrading in specific situations. We call them polluted work. We highlight how trying to comply with a positively-anticipated role transition can help avoid unclean work yet generate more polluted work. Our analysis suggests that paying greater attention to symbolic differentiations between prestigious and shameful aspects of work can improve our understanding of accounting, identity work and organisational practices.

Nonlinear system identification employing automatic differentiation

An optimization based state and parameter estimation method is presented where the required Jacobian matrix of the cost function is computed via automatic differentiation. Automatic differentiation evaluates the programming code of the cost function and provides exact values of the derivatives. In contrast to numerical differentiation it is not suffering from approximation errors and compared to symbolic differentiation it is more convenient to use, because no closed analytic expressions are required. Furthermore, we demonstrate how to generalize the parameter estimation scheme to delay differential equations, where estimating the delay time requires attention.

Reliability-based design optimization using local sensitivity with application to magnetic nano switch

This paper presents a method and a tool for solving the reliability-based design optimization (RBDO). The RBDO aims to minimize a cost function by changing the value of design parameters while ensuring a level of reliability. Uncertainties propagation is a key concept in reliable studies and it can be associated with a sensitivity analysis in order to sort parameter influences. Global and local sensitivities are compared in this study in order to keep a reasonable cost versus accuracy ratio. A software tool has been also developed to automate reliable studies. It is applied to the reliable optimization of a magnetic nano switch with SQP algorithm using Jacobian calculated by composition of automatic and symbolic differentiation.

Symbolic generation of the kinematics of multibody systems in EasyDyn: From MuPAD to Xcas/Giac

In the EasyDyn multibody open source project, computer algebra has been used from the beginning to generate the expressions of velocities and accelerations of the bodies, by symbolic differentiation of their position. Originally, the MuPAD computer algebra system had been retained because it was freely available for non commercial purposes and showed very good technical features. Unfortunately, MuPAD is nowadays only available through commercial channels and needs to be replaced to keep EasyDyn publicly available. This paper presents why Xcas/Giac is finally selected, among other long-term promising projects like Axiom, Maxima, Sage or Yacas. Among the choice criteria, the accessibility, the portability, the ease of use, the automatic export to C language, and the similarity with the MuPAD language are all considered. The performances of the MuPAD and Xcas/Giac implementations are also compared on some examples.

Manifold mapping optimization with or without true gradients

This paper deals with the space mapping optimization algorithms in general and with the manifold mapping technique in particular. The idea of such algorithms is to optimize a model with a minimum number of each objective function evaluations using a less accurate but faster model. In this optimization procedure, fine and coarse models interact at each iteration in order to adjust themselves in order to converge to the real optimum. The manifold mapping technique guarantees mathematically this convergence but requires gradients of both fine and coarse model. Approximated gradients can be used for some cases but are subject to divergence. True gradients can be obtained for many numerical model using adjoint techniques, symbolic or automatic differentiation. In this context, we have tested several manifold mapping variants and compared their convergence in the case of real magnetic device optimization.

Symbolic differentiation of factorized polynomials with repeated roots and the identification of their loci

The mathematical apparatus of algebraic combinatorics is excellently suited to the investigation of symbolic differentiation of analytic functions. In particular, the relationship between the algebraic roots of polynomials and those of their derivatives remains an important topic with wide applications to various branches of pure and applied mathematics. In this paper, a methodology inspired by combinatorial theory is employed to derive analytic expressions for the k-th derivative q(k) of factorized polynomial functions with repeated roots \((x^{\xi} - a_{1})^{\alpha_{1}} (x^{\xi} - a_{2})^{\alpha_{2}} \cdots (x^{\xi} - a_{n})^{\alpha_{n}}\), where ξ∈ℝ∗, ai, αi∈ℝ and i=1,…,n. It is shown that these derivatives are generating functions for classes of integer sequences whose properties are employed to develop a binary tree algorithm that is suitable for the symbolic evaluation of q(k). Compared to the application of Faa di Bruno’s famous differentiation formula for composite functions and to other existing methods for symbolic differentiation, the algorithm is superior because it does not involve finding integer partitions, which is an NP-complete problem. Mathematical identities that relate this topic to other branches of mathematics (e.g. to statistics via the multinomial distribution and multinomial coefficients) are derived and, in addition, a method for identifying the loci of the roots of polynomial derivatives is outlined. The practical significance of these contributions lies in their applicability to various areas of engineering and physics.

Network-based Modeling and Index Analysis of Coupled Electro-Mechanical Systems

AbstractWe consider the modeling of multi-physics dynamical systems, in particular the coupling of mechanical and electrical subsystems. The current state of the art in modeling and simulation tools like Dymola or Matlab/Simulink is a network-based modeling via interconnection of several standardized subcomponents. This approach of modeling dynamical systems leads to differential-algebraic equations (DAEs) that can easily be of very high index. In these tools the usual way to deal with high index DAEs is to use computer-algebra packages and symbolic differentiation to identify and resolve the algebraic constraints to obtain a system in minimal coordinates. This approach has several disadvantages. In this contribution we propose a new remodeling approach for coupled electro-mechanical systems. The first step is based on an index reduction formalism for each uni-physics subcomponent by using so-called minimal extension yielding a minimally extended index 1 formulation for each subcomponent. In a second step the index of the overall coupled system is analyzed. We provide conditions for the coupled system to be of index 1.

First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation

Design sensitivity analysis of flexible multibody systems is important in optimizing the performance of mechanical systems. The choice of coordinates to describe the motion of multibody systems has a great influence on the efficiency and accuracy of both the dynamic and sensitivity analysis. In the flexible multibody system dynamics, both the floating frame of reference formulation (FFRF) and absolute nodal coordinate formulation (ANCF) are frequently utilized to describe flexibility, however, only the former has been used in design sensitivity analysis. In this article, ANCF, which has been recently developed and focuses on modeling of beams and plates in large deformation problems, is extended into design sensitivity analysis of flexible multibody systems. The Motion equations of a constrained flexible multibody system are expressed as a set of index-3 differential algebraic equations (DAEs), in which the element elastic forces are defined using nonlinear strain-displacement relations. Both the direct differentiation method and adjoint variable method are performed to do sensitivity analysis and the related dynamic and sensitivity equations are integrated with HHT-I3 algorithm. In this paper, a new method to deduce system sensitivity equations is proposed. With this approach, the system sensitivity equations are constructed by assembling the element sensitivity equations with the help of invariant matrices, which results in the advantage that the complex symbolic differentiation of the dynamic equations is avoided when the flexible multibody system model is changed. Besides that, the dynamic and sensitivity equations formed with the proposed method can be efficiently integrated using HHT-I3 method, which makes the efficiency of the direct differentiation method comparable to that of the adjoint variable method when the number of design variables is not extremely large. All these improvements greatly enhance the application value of the direct differentiation method in the engineering optimization of the ANCF-based flexible multibody systems.

Inversion of first-arrival seismic traveltimes without rays, implemented on unstructured grids

SUMMARY We develop a method for inverting first-arrival seismic traveltimes without using ray tracing in the forward solution, nor to calculate sensitivity information. We consider unstructured 2-D triangular and 3-D tetrahedral grids for discretizing the subsurface velocity distribution. Unstructured grids provide some computational advantages when dealing with complicated shapes that are difficult to represent with rectilinear grids. However, we stress that the key details of our inversion approach that avoid ray tracing are equally relevant to rectilinear grids. The forward problem is solved using the Fast Marching Method, an efficient numerical algorithm that can be used for propagating first-arrival seismic wave fronts through a velocity distribution. Our minimum structure inversion algorithm uses sensitivity information calculated directly during the forward solution. This calculation relies on explicit symbolic differentiation of the forward modelling equations and, as such, our inversion approach does not require ray tracing and uses sensitivity information that is numerically consistent with the forward solution. Our method is applied to 2-D and 3-D geologically realistic synthetic scenarios based on the Voisey's Bay massive sulfide deposit in Labrador, Canada.

Symbolic dynamics and geometry: using D* in graphics and game programming

This book explains how to use the new symbolic differentiation system D* for applications in computer games and engineering simulation.The development of D* is significant because it is a faster way to render high-resultion graphics in real time-thousands of times faster than, for example, Mathematica or automatic differentiation. The book covers many applications but looks at two specific ones in detail: real-time Lagrangian physics simulation and procedural 3D geometric modeling.

Computation of normal form coefficients of cycle bifurcations of maps by algorithmic differentiation

As an alternative to symbolic differentiation (SD) and finite differences (FD) for computing partial derivatives, we have implemented algorithmic differentiation (AD) techniques into the Matlab bifurcation software Cl_MatcontM, http://sourceforge.net/projects/matcont, where we need to compute derivatives of an iterated map, with respect to state variables. We use derivatives up to the fifth order, of the iteration of a map to arbitrary order. The multilinear forms are needed to compute the normal form coefficients of codimension-1 and -2 bifurcation points. Methods based on finite differences are inaccurate for such computations. Computation of the normal form coefficients confirms that AD is as accurate as SD. Moreover, elapsed time in computations using AD grows linearly with the iteration number J, but more like J d for d th derivatives with SD. For small J, SD is still faster than AD.

Salvaging Decency: Mobile Home Residents’ Strategies of Managing the Stigma of “Trailer” Living

This paper is based on 45 ethnographic interviews conducted with residents of mobile home communities in West Central Florida between 2005 and 2008. It investigates their strategies of managing the stigma that is commonly associated with living in a mobile home. Informants routinely encounter negative stereotypes regarding their “trailer” home, community, and lifestyle in public discourse and personal interactions, and consequently have developed ways of salvaging their decency. My analysis of these strategies particularly emphasizes two versions of distancing, here called “bordering” and “fencing,” as examples of symbolic boundary work. Other techniques discussed include ignoring, passing, humoring, resisting, normalizing, upstaging, and blaming. Throughout the paper, I argue that mobile home residents’ ways of salvaging decency are both similar and different compared to how other disparaged groups deal with stigmatization. The conclusion discusses broader sociological implications of the research in enhancing our understanding of the experience of stigmatization, folk conceptions of decency, symbolic and social differentiation, as well as race and class dynamics.

Structural design sensitivity using boundary elements and polynomial response function

This main issue of this paper is a conjunction of the structural design sensitivity analysis using the Boundary Element Method with the polynomial response function determination. The procedure is so general that it enables sensitivity analysis for potential and elasticity problems within both homogeneous and heterogeneous plane and 3D problems. The essential difference with respect to the previous approaches like the Direct Differentiation Method or the Adjoint Variable Method is in discrete evaluation of the structural response using the response polynomials of some state parameters and design variable as the independent parameter. Such a determination is carried out via the several solutions of the given boundary value problem, where design parameter mean value is regularly perturbed in each of the solutions to cover the closest neighborhood of this mean value. Those few solutions make it possible to recover the polynomial response function from node-to node within the boundary elements, so that further symbolic differentiation using MAPLE returns the sensitivity gradients particular values. The entire procedure is tested here twice—first example deals with the homogeneous cantilever beam, where comparison against pure analytical differentiation is done and, separately, for two-component composite cantilever, where such a comparison is made against the central difference method linked with the same BEM solution.

A Esfera do “Humano”: Uma Reflexão sobre o Caráter Cultural do Psiquismo do Homem

The bonds established between a social process marked by complexity and ample symbolic differentiation and the impacts this process arouses in the individual's existential dimension are clear outlined when the trajectory of individual conception until post-modernity is followed, a perspective that inevitably leads us to discuss the concepts of subject, subjectivity and representation. It is necessary, A EsfErA Do “HumAno”: umA rEflExao sobrE o CAratEr CulturAl Do Psiquismo Do HomEm revista multiplas leituras 170 Revista Multiplas Leituras, v.1, n. 2, p. 169-182, jul. / dez. 2008 therefore, reflect upon the way we build our reality, our universe of significations, and how language exerts a role of utmost omportance in our re-elaboration of the world, of life and our subjectivy, that is, reflect upon the cultural character of our psychism.