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Following the seminal 'Smoking Adjoint' paper by Giles and Glasserman [Smoking adjoints: Fast monte carlo greeks. Risk, 2006, 19, 88–92], the development of Adjoint Algorithmic Differentiation (AAD) has revolutionized the way risk is computed in the financial industry. In this paper, we provide a tutorial of this technique, illustrate how it is immediately applicable for Monte Carlo and Partial Differential Equations applications, the two main numerical techniques used for option pricing, and review the most significant literature in quantitative finance of the past fifteen years. [ABSTRACT FROM AUTHOR]

Credit Valuation Adjustment is a balance sheet item which is nowadays subject to active risk management by specialized traders. However, the most important risk factors, which are the default intensities of the counterparties, affect in a nondifferentiable way the most general Monte Carlo estimator of the adjustment, through simulation of default times. Thus the computation of first- and second-order (pure and mixed) sensitivities with respect to inputs affecting these risk factors cannot rely on direct path-wise differentiation, while any approach involving finite differences is slow and shows very high statistical noise. We present ad hoc estimators which have empirically a much smaller variance while offering very low runtime overheads over the baseline computation of the price adjustment, regardless of the number of sensitivities of interest, by leveraging adjoint (i.e., backward) algorithmic differentiation in their implementation. These estimators allow for a generic copula-based dependence structure among the default events, and can be applied beyond our main application, to payoffs depending on more than two of them. We also discuss the conversion of the so-obtained sensitivities to model parameters (e.g., default intensities) into sensitivities to the market quotes used in calibration (e.g., Credit Default Swap spreads) by generalizations of an existing implicit-function based first-order algorithm. [ABSTRACT FROM AUTHOR]

Algorithmic differentiation (AD) is a promising tool to accurately and efficiently compute sensitivities of several code outputs w.r.t. the code input parameters for the complex Monte Carlo code EIRENE. We illustrate the procedure by assessing the sensitivity of the total atom content w.r.t. the atomic and molecular cross‐sections and rate coefficients. AD does not suffer from decorrelation between primal and perturbed particle trajectories, which easily happens for finite difference (FD) sensitivity calculations. Consequently, the resulting statistical errors on AD sensitivities are up to five orders of magnitude smaller than the statistical errors on FD sensitivities for a JET low‐recycling background plasma. Although this statistical error reduction is promising for the application of the AD sensitivities in a future optimization or uncertainty quantification framework, we show that the AD sensitivities can blow up for a minority of long‐lived particles, especially occurring in detachment. Finally, we conclude that the Monte Carlo simulation and estimator type possibly need to be adapted to improve the accuracy of the sensitivities. [ABSTRACT FROM AUTHOR]

A modification of Newton's method for solving systems of n > 0 nonlinear equations is presented. The new matrix-free method is exact as opposed to a range of inexact Newton methods in the sense that both the Jacobians and the solutions to the linear Newton systems are computed without truncation. It relies on a given decomposition of a structurally dense invertible Jacobian of the residual into a product of q > 0 structurally sparse invertible elemental Jacobians according to the chain rule of differentiation. Inspired by the adjoint mode of algorithmic differentiation, explicit accumulation of the Jacobian of the residual is avoided. Prospective, generally applicable implementations of the new method can be based on similar ideas. Sparsity is exploited for the direct solution of the linear Newton systems. Optimal exploitation of sparsity yields various well-known computationally intractable combinatorial optimization problems in sparse linear algebra such as BANDWIDTH or DIRECTED ELIMINATION ORDERING. The method is motivated in the context of a decomposition into elemental Jacobians with bandwidth 2µ + 1 for 0 ≤ µ << n. In the likely scenario of q > n, the computational cost of the standard Newton algorithm is dominated by the cost of accumulating the Jacobian of the residual. It can be estimated as O(qµn²), thus exceeding the cost of O(n³) for the direct solution of the linear Newton system. The new method reduces this cost to O(qnµ²), yielding a potential improvement by a factor of O(n/µ). Supporting run time measurements are presented for the tridiagonal case showing a reduction of the computational cost by O(n). Generalization yields the combinatorial Matrix-Free Exact Newton Step problem. We prove NP-completeness, and we present algorithmic components for building methods for the approximate solution. Potential applications of the matrix-free exact Newton method in machine learning of surrogates for computationally expensive nonlinear residuals are touched on briefly as part of various conclusions to be drawn. [ABSTRACT FROM AUTHOR]

Electron Probe Microanalysis (EPMA) is a nondestructive technique to determine the chemical composition of material samples in the micro- to nanometer range. Based on intensity measurements of x-radiation, the reconstruction of the material composition and structure poses an inverse problem. The reconstruction methods currently applied are based on models that assume a homogeneous or a layered structure of the studied material. To increase the spatial resolution of reconstruction in EPMA the combination of a more sophisticated reconstruction method, which is based on a model that allows complex material structure, together with multiple measurements with varying beam configurations is required. We present a deterministic k-ratio model that is based on the PN model, an approximation of the radiative transfer equation for electron transport. Our goal is to approximate a maximum likelihood solution of the inverse problem using gradient-based optimization. We detail the application of the model in the context of algorithmic differentiation, in particular by deriving its continuous adjoint formulation. Algorithmic differentiation provides the flexibility to adapt the reconstruction method to various material parametrizations and thus to regularize and take into account prior knowledge. Through examples, we verify our implementation and demonstrate the flexibility of the reconstruction/differentiation method. [ABSTRACT FROM AUTHOR]

Algorithmic differentiation is a tool used in several branches of computational science, both in conjunction with the interval calculus, and apart of it. This paper presents the ADHC library [14] developed by the author, and identified as package "na60" by Numerical Algorithms journal. This library makes intensive use of the C++ template meta-programming, and it has several unique features. ADHC seems particularly useful for interval-related applications. The library has been used by some solvers, also developed by the author, including HIBA_USNE. The paper describes the library, presenting its features, focusing on the new ones, added in version 2.0; in particular, we describe bounding subdifferentials of non-smooth functions and computing derivatives over various datatypes. Efficiency comparison with respect to other packages is also presented. Then some examples of ADHC applications are given, involving the use of HIBA_USNE and standard benchmark problems for solving nonlinear systems. A particular emphasis is put on examples related to modern machine learning, but not limited to them. Planned extensions and possible directions for future development of ADHC are also outlined and discussed. [ABSTRACT FROM AUTHOR]

We generalize the proof of NP-completeness of Jacobian accumulation using a given number of floating-point operations to arbitrary order. [ABSTRACT FROM AUTHOR]

We show that the derivatives of the Sinkhorn-Knopp algorithm, or iterative proportional fitting procedure, converge towards the derivatives of the entropic regularization of the optimal transport problem with a locally uniform linear convergence rate. [ABSTRACT FROM AUTHOR]

We introduce a generalization of separability for global optimization, presented in the context of a simple branch and bound method. Our results apply to continuously differentiable objective functions implemented as computer programs. A significant search space reduction can be expected to yield an acceleration of any global optimization method. We show how to utilize interval derivatives calculated by adjoint algorithmic differentiation to examine the monotonicity of the objective with respect to so called structural separators and how to verify the latter automatically. [ABSTRACT FROM AUTHOR]

We present the new software OpDiLib, a universal add-on for classical operator overloading AD tools that enables the automatic differentiation (AD) of OpenMP parallelized code. With it, we establish support for OpenMP features in a reverse mode operator overloading AD tool to an extent that was previously only reported on in source transformation tools. We achieve this with an event-based implementation ansatz that is unprecedented in AD. Combined with modern OpenMP features around OMPT, we demonstrate how it can be used to achieve differentiation without any additional modifications of the source code; neither do we impose a priori restrictions on the data access patterns, which makes OpDiLib highly applicable. For further performance optimizations, restrictions like atomic updates on adjoint variables can be lifted in a fine-grained manner. OpDiLib can also be applied in a semi-automatic fashion via a macro interface, which supports compilers that do not implement OMPT. We demonstrate the applicability of OpDiLib for a pure operator overloading approach in a hybrid parallel environment. We quantify the cost of atomic updates on adjoint variables and showcase the speedup and scaling that can be achieved with the different configurations of OpDiLib in both the forward and the reverse pass. [ABSTRACT FROM AUTHOR]

Algorithmic Differentiation (AD) has become a powerful tool to improve our understanding of the Earth System, because it can generate adjoint code which permits efficient calculation of gradients that are essential to sensitivity studies, inverse problems, parameter estimation and data assimilation. Most source-to-source transformation tools, however, have been designed for FORTRAN and support for C remains limited. Here we use the Adjoinable Land Ice Flow model (ALIF), a C clone of the C++ Ice Sheet System Model (ISSM) and employ source-to-source AD to produce its adjoint code. We present the first running source-to-source adjoint of ALIF, and its application to basal drag inversion under Pine Island Glacier, West Antarctica. ALIF brought several challenges to AD tool development, such as the correct treatment of the context code, which does not compute the differentiable function, but controls this computation through the setup of data structures, including possible aliasing, as well as data-flow reversal in the presence of pointers and dynamic memory, which are ubiquitous in codes such as ISSM and ALIF. We present the strategies we have developed to overcome these challenges.

The role of clouds for radiative transfer, precipitation formation, and their interaction with atmospheric dynamics depends strongly on cloud microphysics. The parameterization of cloud microphysical processes in weather and climate models is a well‐known source of uncertainties. Hence, robust quantification of this uncertainty is mandatory. Sensitivity analysis to date has typically investigated only a few model parameters. We propose algorithmic differentiation (AD) as a tool to detect the magnitude and timing at which a model state variable is sensitive to any of the hundreds of uncertain model parameters in the cloud microphysics parameterization. AD increases the computational cost by roughly a third in our simulations. We explore this methodology as the example of warm conveyor belt trajectories, that is, air parcels rising rapidly from the planetary boundary layer to the upper troposphere in the vicinity of an extratropical cyclone. Based on the information of derivatives with respect to the uncertain parameters, the ten parameters contributing most to uncertainty are selected. These uncertain parameters are mostly related to the representation of hydrometeor diameter and fall velocity, the activation of cloud condensation nuclei, and heterogeneous freezing. We demonstrate the meaningfulness of the AD‐estimated sensitivities by comparing the AD results with ensemble simulations spawned at different points along the trajectories, where different parameter settings are used in the various ensemble members. The ranking of the most important parameters from these ensemble simulations is consistent with the results from AD. Thus, AD is a helpful tool for selecting parameters contributing most to cloud microphysics uncertainty. Plain Language Summary: The formation of clouds is determined by processes that act on smaller scales than weather prediction models can resolve. Consequently, a parameterization with typically hundreds of parameters is constructed to determine the effects of these processes on the resolved larger scales. These parameters are a well‐known source of uncertainty in weather and climate models. Classical attempts to quantify this uncertainty are typically limited to a few parameters. We propose algorithmic differentiation (AD) as a tool to detect parameters with the largest impact for any of the hundreds of parameters on multiple model state variables at every time step in our simulation. This increases the computational cost by roughly a third. The relevance of the AD‐estimated impact is demonstrated by comparing the AD results with ensemble simulations, where different parameter settings are used in the various ensemble members. The ranking of the most important parameters from these ensemble simulations is consistent with the results from AD. Thus, AD is a helpful tool to identify parameters objectively that contribute most to uncertainty in cloud parameterizations. Key Points: Quantification of multi‐parameter uncertainty of cloud microphysical evolution of WCB trajectories using algorithmic differentiationUncertainty at every time step derived with algorithmic differentiation representative for key uncertainty over at least 30 min intervalsParameterization of CCN activation, diameter size, and fall velocity of hydrometeors have the largest mean impact on water vapor contents [ABSTRACT FROM AUTHOR]

Abstract The role of clouds for radiative transfer, precipitation formation, and their interaction with atmospheric dynamics depends strongly on cloud microphysics. The parameterization of cloud microphysical processes in weather and climate models is a well‐known source of uncertainties. Hence, robust quantification of this uncertainty is mandatory. Sensitivity analysis to date has typically investigated only a few model parameters. We propose algorithmic differentiation (AD) as a tool to detect the magnitude and timing at which a model state variable is sensitive to any of the hundreds of uncertain model parameters in the cloud microphysics parameterization. AD increases the computational cost by roughly a third in our simulations. We explore this methodology as the example of warm conveyor belt trajectories, that is, air parcels rising rapidly from the planetary boundary layer to the upper troposphere in the vicinity of an extratropical cyclone. Based on the information of derivatives with respect to the uncertain parameters, the ten parameters contributing most to uncertainty are selected. These uncertain parameters are mostly related to the representation of hydrometeor diameter and fall velocity, the activation of cloud condensation nuclei, and heterogeneous freezing. We demonstrate the meaningfulness of the AD‐estimated sensitivities by comparing the AD results with ensemble simulations spawned at different points along the trajectories, where different parameter settings are used in the various ensemble members. The ranking of the most important parameters from these ensemble simulations is consistent with the results from AD. Thus, AD is a helpful tool for selecting parameters contributing most to cloud microphysics uncertainty.

Radial anomalous diffusion coefficients are among the largest sources of uncertainty in mean‐field plasma‐edge codes such as SOLPS‐ITER. These coefficients are machine, scenario, and space‐dependent, thus hampering the predictive and interpretive capability of these codes. In fact, modellers usually adjust these coefficients manually, based on expert judgement or on large, computationally expensive, parameter scans. Matching data from various diagnostics, each with their own experimental uncertainties, additionally complicates the problem of finding a good parameter set. In addition, standard non‐linear regression techniques have shown to become prohibitive for expensive plasma‐edge simulations with many unknown parameters, as gradient calculation was based on finite differences. In this paper, we apply algorithmic differentiation (AD) as an efficient and more accurate alternative for gradient calculation in large, continuously developed codes as SOLPS‐ITER. In addition, the lack of uncertainty information and danger of data overfitting, key limitations of regression techniques, are overcome by combining data and uncertainties from different diagnostics in a Bayesian framework. We implement for the first time such a Bayesian inference framework into SOLPS‐ITER, using gradient‐based optimization with gradients obtained through tangent AD to find the maximum a posteriori (MAP) values of the parameters. The recently developed κ turbulence model is employed to limit the number of unknown parameters compared to full spatial profiles of diffusion coefficients. The Bayesian MAP‐estimation is compared to standard regression techniques on a small‐scale tokamak. We adopt fictitious experimental data obtained from a reference SOLPS‐ITER solution with artificial measurement noise. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for the accurate estimation of the derivative (aka. sensitivity) of expectations of functions involving an indicator function by modifying a (stochastic) algorithmic differentiation, replacing the derivative of the indicator function with a suitable operator. We show that we can split this operator into a conditional expectation operator and the density. This allows using different or improved numerical approximation methods for these operators, e.g. regression. The method is an improvement of the approach presented in C.P. Fries [Automatic backward differentiation for American Monte-Carlo algorithms (conditional expectation). Risk, April, 2018; Stochastic automatic differentiation: Automatic differentiation for Monte-Carlo simulations, Quant. Finance 19(6) (2019), pp. 1043–1059]. The finite difference approximation of a partial derivative of a Monte-Carlo integral of a discontinuous function is known to exhibit a high Monte-Carlo error. The issue is evident since the Monte-Carlo approximation of a discontinuous function is just a finite sum of discontinuous functions and as such, not even differentiable. The algorithmic differentiation of a discontinuous function is problematic. A natural approach is to replace the discontinuity by continuous functions. This is equivalent to replacing a path-wise automatic differentiation by a (local) finite difference approximation. The decoupling of the integration of the Dirac delta and the remaining conditional expectation introduced here results in an improvement in terms of variance reduction and implementation design. [ABSTRACT FROM AUTHOR]

An unexpected path reversal is discovered for a serial robot with an offset-axis wrist over a finite range of proximity to the wrist singularity. This is replicated by a kinematic model. A prior spherical-wrist method transits the singularity with limited joint rate and acceleration under a constant rate of tool traversal. Accurate position is maintained by controlling a small deviation in orientation. Extensions to the method for an offset wrist (1) find the least-maximum deviation, (2) identify and locate where a path reversal occurs, and (3) use this point to control step size in a high-order predictor-correction path following procedure. [ABSTRACT FROM AUTHOR]

This article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account. [ABSTRACT FROM AUTHOR]

Adjusting the displacement path of a serial robot encountering the wrist singularity to pass either through the singularity or around it mitigates its adverse effects. Both such path adjustments are commonly called singularity avoidance and are applied here to either a spherical or an offset wrist. These adjustments avoid high joint rates that can occur at singularity encounter. A recent through-the singularity method limits joint rates and accelerations in the robot with either a spherical or offset wrist when conducting a constant rate of traversal of the tool manipulated by the robot. A kinematic model adding multiple virtual joints allows a modified high-order path-following algorithm to maintain accurate tool position while achieving an optimal level of tool deviation in orientation. Whereas a path reversal resulting from a turning-point type singularity had been revealed for an offset wrist over a finite range of close-approach, these conditions are met when connecting the isolated path segments. Procedures are developed here with this capability for an around-the-singularity path. Choosing between the through and around-singularity alternatives offers the overall optimum. [ABSTRACT FROM AUTHOR]

The abs-normal form (ANF) can be used to represent almost any piecewise linear function. Several of these piecewise linear functions exhibit a certain structure that has an impact on the numerical efficiency of the ANF representation. In this paper, three common structures are investigated that typically arise in applications and require special numerical treatment: the sum, the composition, and the component-wise maximum/minimum of several functions. For these structures, the corresponding expressions of the resulting abs-normal form are provided, as well as some alternatives. The theoretical observations are supported by numerical results. [ABSTRACT FROM AUTHOR]

Penalized least squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise without requiring to invert the covariance matrix. Then, in order to avoid expansive grid search, it is necessary to design algorithmic scheme minimizing the quadratic risk with respect to regularization parameters. This work extends the Stein's Unbiased GrAdient estimator of the Risk of Deledalle et al. (SIAM J Imaging Sci 7(4):2448–2487, 2014) to the case of correlated Gaussian noise, deriving a general automatic tuning of regularization parameters. First, the theoretical asymptotic unbiasedness of the gradient estimator is demonstrated in the case of general correlated Gaussian noise. Then, the proposed parameter selection strategy is particularized to fractal texture segmentation, where problem formulation naturally entails inter-scale and spatially correlated noise. Numerical assessment is provided, as well as discussion of the practical issues. [ABSTRACT FROM AUTHOR]

With more efficient structures, last trends in aeronautics have witnessed an increased flexibility of wings, calling for adequate design and optimization approaches. To correctly model the coupled physics, aerostructural optimization has progressively become more important, being nowadays performed also considering higher-fidelity discipline methods, i.e., CFD for aerodynamics and FEM for structures. In this work a model for high-fidelity gradient-based aerostructural optimization of wings, assisted by algorithmic differentiation and including aerodynamic and structural nonlinearities, is presented. First, the model is illustrated: a key feature lies in its enhanced modularity. Each discipline solver, employing algorithmic differentiation for the evaluation of adjoint-based sensitivities, is interfaced at high-level by means of a wrapper to both solve the aerostructural primal problem and evaluate discrete-consistent gradients of the coupled problem. Second, to demonstrate the feasibility of the method, a framework is ad hoc set up, within the open-source SU2 multiphysics suite, with the inclusion of a geometrically nonlinear beam FE and an interface module to deal with non-matching 3D surfaces. Finally, the framework is applied to perform aerostructural optimization of aeroelastic test cases based on the ONERA M6 and NASA CRM wings. Single-point optimizations, employing Euler or RANS flow models, are carried out to find wing optimal outer mold line in terms of aerodynamic efficiency. Results remark the importance of taking into account the aerostructural coupling when performing wing shape optimization. [ABSTRACT FROM AUTHOR]

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To solve a nonlinear eigenvalue problem we develop algorithms which compute zeros of . We show how to apply third order iteration methods for that purpose. The necessary derivatives of the determinant are computed by algorithmic differentiation. Since many nonlinear eigenvalue problems have banded matrices we also present an algorithm which makes use of their structure. [ABSTRACT FROM AUTHOR]

The paper considers the problem of algorithmic differentiation of information matrix difference equations for calculating the information matrix derivatives in the information Kalman filter. The equations are presented in the form of a matrix MWGS (modified weighted Gram–Schmidt) transformation. The solution is based on the usage of special methods for the algorithmic differentiation of matrix MWGS transformation of two types: forward (MWGS-LD) and backward (MWGS-UD). The main result of the work is a new MWGS-based array algorithm for computing the information matrix sensitivity equations. The algorithm is robust to machine round-off errors due to the application of the MWGS orthogonalization procedure at each step. The obtained results are applied to solve the problem of parameter identification for state-space models of discrete-time linear stochastic systems. Numerical experiments confirm the efficiency of the proposed solution.

Algorithmic differentiation (AD) is a widely-used approach to compute derivatives of numerical models. Many numerical models include an iterative process to solve non-linear systems of equations. To improve efficiency and numerical stability, AD is typically not applied to the linear solvers. Instead, the differentiated linear solver call is replaced with hand-produced derivative code that exploits the linearity of the original call. In practice, the iterative linear solvers are often stopped prematurely to recompute the linearisation of the non-linear outer loop. We show that in the reverse-mode of AD, the derivatives obtained with partial convergence become inconsistent with the original and the tangent-linear models, resulting in inaccurate adjoints. We present a correction term that restores consistency between adjoint and tangent-linear gradients if linear systems are only partially converged. We prove the consistency of this correction term and show in numerical experiments that the accuracy of adjoint gradients of an incompressible flow solver applied to an industrial test case is restored when the correction term is used. [ABSTRACT FROM AUTHOR]

Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous state-of-the-art approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks from deep learning like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency. The methods that are described in this paper for computing derivatives of matrix and tensor expressions have been implemented in the online tool . [ABSTRACT FROM AUTHOR]

Computer Aided Design (CAD) systems and tools are considered essential for industrial design. They construct and manipulate the geometry of a certain component with an arbitrary set of design parameters. However, it is a challenging task to incorporate the parametric definition in a gradient-based shape optimization loop, since the CAD libraries usually do not provide shape sensitivities w.r.t. the design parameters of the model to be optimized. Typically, these derivatives are evaluated with inaccurate finite differences. On the contrary, to obtain the exact derivative information, algorithmic differentiation (AD) can be applied if the CAD sources are available. In this study, the Rolls-Royce in-house airfoil design and blade generation tool Parablading is differentiated using the AD software tools ADOL-C and Tapenade. The differentiated CAD tool is coupled with a discrete adjoint CFD solver that is part of the Rolls-Royce in-house HYDRA suite of codes, also produced by algorithmic differentiation. This differentiated design chain is used to perform gradient-based shape optimization of the TU Berlin TurboLab stator test-case w.r.t. minimize the total pressure loss and exit angle deviation objectives. [ABSTRACT FROM AUTHOR]

Algorithmic differentiation tools can automate the adjoint transformation of parallel message-passing codes [J. Utke, L. Hascoët, P. Heimbach, C. Hill, P. Hovland, and U. Naumann, Toward Adjoinable MPI, in 2009 IEEE International Symposium on Parallel & Distributed Processing, May, IEEE, 2009, pp. 1–8] using the AMPI library. Nevertheless, a non-trivial and manual step after the differentiation is the initialization of the seed and retrieval of the output values from the differentiated code. Ambiguities in seeding occur in programs where the user is unable to expose the complete program flow with a single entry and single exit point to the AD tool. We present the ambiguities associated with seed initialization and output retrieval for adjoint transformation of halo and zero-halo partitioned MPI programs. We introduce a general framework to eliminate ambiguities in seeding and retrieval for shared-node reduction over +, and * operators using a conceptual master-worker model. The model shows the need for new MPI calls for retrieval and eliminate MPI calls for seed initialization. Different implementations for seeding manually assembled adjoints were inferred from the model, namely, partial and unique seeding. We successfully applied the seeding techniques to a 3D zero-halo partitioned unstructured compressible discrete adjoint solver and highlight the merits and demerits of each strategy. [ABSTRACT FROM AUTHOR]

A set of Fortran subroutines for reverse mode algorithmic (or automatic) differentiation of the basic linear algebra subprograms (BLAS) is presented. This is preceded by a description of the mathematical tools used to obtain the formulae of these derivatives, with emphasis on special matrices supported by the BLAS: triangular, symmetric, and band. All single and double precision BLAS derivatives have been implemented, together with the Cholesky factorization from Linear Algebra Package (LAPACK). The subroutines are written in Fortran 2003 with a Fortran 77 interface to allow use from C and C++, as well as dynamic languages such as R, Python, Matlab, and Octave. The subroutines are all implemented by calling BLAS, thereby attaining fast runtime. Timing results show derivative runtimes that are about twice those of the corresponding BLAS, in line with theory. The emphasis is on reverse mode because it is more important for the main application that we have in mind, numerical optimization. Two examples are presented, one dealing with the least squares modeling of groundwater, and the other dealing with the maximum likelihood estimation of the parameters of a vector autoregressive time series. The article contains comprehensive tables of formulae for the BLAS derivatives as well as for several non-BLAS matrix operations commonly used in optimization. [ABSTRACT FROM AUTHOR]

The Natural Coordinates (NCs) method for Lagrangian modelling and simulation of multibody systems is valued for giving simple, sparse models. We describe our version of it and compare with the classical approach of Jalon and Bayo (JBNCs). Our NCs use the high-index differential-algebraic equation solver Daets. Algorithmic differentiation, not symbolic algebra, forms the equations of motion from the Lagrangian. We obtain significantly smaller equation systems than JBNCs, at the cost of a non-constant mass matrix for fully 3D models|a minor downside in the Daets context. Examples in 2D and 3D are presented, with numerical results. [ABSTRACT FROM AUTHOR]