|
Calculation of Machine Precision Second Order Derivatives Using Dual-complex Numbers |
2024 |
H Millwater, M Balcer, N Millwater |
Sensitivity Methods, CTSE, Dual Numbers, Automatic Differentiation, Numerical Methods |
It is well known that both complex and dual numbers can be employed to obtain machine precision first-order derivatives; however, neither, on their own, can compute machine precision second-order derivatives. |
Calculation of Machine Precision Second Order Derivatives Using Dual-complex Numbers |
|
Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element Method |
2024 |
JD Navarro, JC Velasquez-Gonzalez, M Aristizabal, G Jarmer, SS Kessler, A Montoya, HR Millwater, D Restrepo |
FEM, Mathematical Analysis, Finite Element Software, Mechanical Properties, Broyden Fletcher Goldfarb Shanno, Sensitivity Analysis, Complex Algebra, CTSE |
Many modern structural health monitoring (SHM) systems use piezoelectric transducers to induce and measure guided waves propagating in structures for structural damage detection. |
Arbitrary-Order Sensitivity Analysis in Wave Propagation Problems Using Hypercomplex Spectral Finite Element Method |
|
A fast method for computing arbitrary order Stress-Intensity Factor derivatives of 3D finite element simulations using Hypercomplex Automatic Differentiation |
2024 |
Mauricio Aristizabal, Harry Millwater, Arturo Montoya |
FEM, fracture mechanics, WCCM 2024 |
A fast method for computing arbitrary order Stress-Intensity Factor derivatives of 3D finite element simulations using Hypercomplex Automatic Differentiation. |
A fast method for computing arbitrary order Stress-Intensity Factor derivatives of 3D finite element simulations using Hypercomplex Automatic Differentiation |
|
Solution and Sensitivity Analysis of Nonlinear Equations using a Hypercomplex Variable Newton-Raphson Method |
2023 |
M Aristizabal, JL Hernández-Estrada, M García, H Millwater |
Hypercomplex Numbers, Newton-Raphson, Catenary, Elastic Cable, Nonlinear Equations |
The classical Newton-Raphson (NR) method for solving nonlinear equations is enhanced in two ways through the use of hypercomplex variables and algebra. |
Solution and Sensitivity Analysis of Nonlinear Equations using a Hypercomplex Variable Newton-Raphson Method |
|
Efficient Development and Application of Taylor Series Expansions as Surrogate Models for Uncertainty Quantification |
2023 |
M Balcer |
Uncertainty Quantification, Sobol' Indices, FEM |
A derivative-based Uncertainty Quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol’ indices of the model output. |
Efficient Development and Application of Taylor Series Expansions as Surrogate Models for Uncertainty Quantification |
|
Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD) |
2023 |
JC Velasquez-Gonzalez, JD Navarro, M Aristizabal, H Millwater, A Montoya, D Restrepo |
Complex Variable Differentiation, Structural Dynamics, Eigenvalues, Eigenvectors, Sensitivity Analysis, Modal Analysis, Optimization |
The calculation of accurate arbitrary-order sensitivities of eigenvalues and eigenvectors is crucial for structural analysis applications, including topology optimization, system identification, finite element model updating, and fault diagnosis. |
Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD) |
|
HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method |
2023 |
M Balcer, M Aristizabal, JS Rincón-Tabares, A Montoya, D Restrepo, H Millwater |
Uncertainty Quantification, FEM, HYPAD, Sobol' Indices |
A derivative-based uncertainty quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. |
HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method |
|
Sensitivity Analysis for Transient Thermal Problems Using the Complex-Variable Finite Element Method |
2022 |
JS Rincón-Tabares, JC Velasquez-Gonzalez, D Ramirez-Tamayo, A Montoya, H Millwater, D Restrepo |
Finite Differences, Finite Elements, Transient Conduction, Complex-Variable Differentiation, Sensitivity Analysis, Heat Conduction |
Solving transient heat transfer equations is required to understand the evolution of temperature and heat flux. This physics is highly dependent on the materials and environmental conditions. |
Sensitivity Analysis for Transient Thermal Problems Using the Complex-Variable Finite Element Method |
|
Transient Thermomechanical Sensitivity Analysis using a Complex Variable Finite Element Method |
2022 |
GA Rios, JS Rincón-Tabares, A Montoya, D Restrepo, H Millwater |
CTSE, ZFEM, Transient Thermomechanical Sensitivity Analysis, Thermal Stresses |
A complex variable transient thermomechanical element was developed and used to compute highly accurate sen |
Transient Thermomechanical Sensitivity Analysis using a Complex Variable Finite Element Method |
|
Arbitrary-Order Sensitivity Analysis in Phononic Metamaterials using the Multicomplex Taylor Series Expansion Method Coupled With Bloch’s Theorem |
2022 |
JD Navarro, H Millwater, A Montoya, D Restrepo |
Computational Mechanics, Dynamics, Mechanical Properties of Materials, Wave Propagation |
Phononic metamaterials (PMs) exhibit frequency ranges at which elastic waves are attenuated called band gaps. However, this phenomenon is highly sensitive to geometrical variations and the unit cell’s mechanical properties. |
Arbitrary-Order Sensitivity Analysis in Phononic Metamaterials using the Multicomplex Taylor Series Expansion Method Coupled With Bloch’s Theorem |
|
Computation of Two Dimensional Mixed-mode Stress Intensity Factor Rates Using a Complex-variable Interaction Integral |
2022 |
AM Aguirre-Mesa, S Restrepo-Velasquez, D Ramirez-Tamayo, A Montoya, H Millwater |
M-Integral, Interaction Integral, ZFEM, CTSE |
The well-known interaction integral, also known as the M-integral or I-integral, is a method to compute the mixed-mode stress intensity factors (SIFs) for fracture mechanics problems. |
Computation of Two Dimensional Mixed-mode Stress Intensity Factor Rates Using a Complex-variable Interaction Integral |
|
On the Equivalence of Automatic and Symbolic Differentiation |
2022 |
S Laue |
Automatic Differentiation, Symbolic Differentiation, Derivatives |
We show that reverse mode automatic differentiation and symbolic differentiation are equivalent in the sense that they both perform the same operations when computing derivatives. |
On the Equivalence of Automatic and Symbolic Differentiation |
|
Higher-order Sensitivity Analysis for Linear and Nonlinear Problems Using the Multidual Finite Element Method |
2022 |
D Avila |
ZFEM, Mesh Sensitivity Analysis, Uncertainty Quantification |
This research work evaluated the accuracy and efficiency of the multidual complex-valued finite element method (ZFEM) in computing higher order derivatives of output variables with respect to input variables of elastic, up to the third order. |
Higher-order Sensitivity Analysis for Linear and Nonlinear Problems Using the Multidual Finite Element Method |
|
A Complex-variable Finite Element Method-based Inverse Methodology to Extract Constitutive Parameters Using Experimental Data |
2022 |
D Ramirez-Tamayo, A Soulami, V Gupta, D Restrepo, A Montoya, E Nickerson, T Roosendaal, K Simmons, G Petrossian, H Millwater |
ZFEM, Inverse Methods, Abaqus, UEL, Automatic Differentiation |
This paper presents the use of full-field kinematic measurements obtained using the digital image correlation (DIC) procedure and load–displacement data to determine constitutive material properties by solving an inverse finite element opt. problem. |
A Complex-variable Finite Element Method-based Inverse Methodology to Extract Constitutive Parameters Using Experimental Data |
|
Generalizations of Singular Value Decomposition to Dual-numbered Matrices |
2022 |
R Gutin |
Singular Value Decomposition, Linear Algebra, Dual Numbers |
We present two generalizations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. |
Generalizations of Singular Value Decomposition to Dual-numbered Matrices |
|
Tearing Energy Calculation in Hyperelastic Fracture Mechanics using the Local and Global Complex Variable Finite Element Method |
2022 |
E Ytuarte, AM Aguirre-Mesa, D Ramirez-Tamayo, D Avila, H Millwater, D Restrepo, A Montoya |
Hyperelasticity, Fracture, Tearing Energy, Energy Release Rate, High-order Derivatives |
The “local” and “global” complex variable FEM were extended for computing the energy release rate (ERR) of materials undergoing nonlinear elastic deformation at both small and large strains. |
Tearing Energy Calculation in Hyperelastic Fracture Mechanics using the Local and Global Complex Variable Finite Element Method |
|
AutoMat -- Automatic Differentiation for Generalized Standard Materials on GPUs |
2022 |
J Blühdorn, N Gauger, M Kabel |
Automatic Differentiation, Generalized Standard Materials, Numerical Methods, GPU Computing |
We propose a universal method for the evaluation of generalized standard materials that greatly simplifies the material law implementation process. |
AutoMat -- Automatic Differentiation for Generalized Standard Materials on GPUs |
|
Demonstration of Prospective Application of the Dual Number Automatic Differentiation for Uncertainty Propagation in Neutronic Calculations |
2021 |
P Bokov, S Groenewald, D Botes, B Adetula |
Dual Numbers, Automatic Differentiation, Uncertainty Propagation, Sensitivity Analysis, Sandwich Formula, Material Testing Reactor |
Automatic differentiation (AD) is a set of techniques which allows the numeric evaluation of derivatives of functions calculated by a computer program. |
Demonstration of Prospective Application of the Dual Number Automatic Differentiation for Uncertainty Propagation in Neutronic Calculations |
|
A Complex-variable Cohesive Finite Element Subroutine to Enable Efficient Determination of Interfacial Cohesive Material Parameters |
2021 |
D Ramirez-Tamayo, A Soulami, V Gupta, D Restrepo, A Montoya, H Millwater |
ZFEM, CTSE, Inverse Determination of Material Parameters, UEL, Automatic Differentiation |
A new complex variable version of a cohesive element is presented that provides highly accurate first order derivatives of the nodal displacements with respect to the cohesive fracture parameters. |
A Complex-variable Cohesive Finite Element Subroutine to Enable Efficient Determination of Interfacial Cohesive Material Parameters |
|
A block forward substitution method for solving the hypercomplex finite element system of equations |
2021 |
AM Aguirre-Mesa, MJ García, M Aristizabal, D Wagner, D Ramirez-Tamayo, A Montoya, H Millwater |
Hypercomplex Numbers, Dual Numbers, Hyperdual Numbers, Complex-variable FEM, CTSE |
The hypercomplex FEM, ZFEM, allows the analyst to compute highly-accurate arbitrary-order shape, material property, and loading derivatives by augmenting the traditional FEM with multiple imaginary degrees of freedom. |
A block forward substitution method for solving the hypercomplex finite element system of equations |
|
On the Numerical Performance of Derivative-Free Optimization Methods Based on Finite-Difference Approximations |
2021 |
H Shi, M Xuan, F Oztoprak, J Nocedal |
Derivative-free Optimization, Noisy Optimization, Zeroth-order Optimization, Nonlinear Optimization |
This paper reports the basic principles of the automatic differentiation method and some experiments on the sensitivity analysis of mechanical structures. |
On the Numerical Performance of Derivative-Free Optimization Methods Based on Finite-Difference Approximations |
|
Multidual Sensitivity Method in Ray-Tracing Transport Simulations |
2021 |
MR Balcer, H Millwater, JA Favorite |
Multidual Sensitivity, High-order Derivatives, Ray-tracing Transport Simulation |
The multidual differentiation method has been implemented in a ray-tracing transport simulation for the purpose of calculating arbitrary-order sensitivities of the uncollided particle leakage. |
Multidual Sensitivity Method in Ray-Tracing Transport Simulations |
|
Hyper-Dual Quaternions Representation of Rigid Bodies Kinematics |
2020 |
A Cohen, M Shoham |
Dual-quaternion, Hyper-dual Numbers, Hyper-dual Quaternion, Rigid Body Kinematics |
Hyper-Dual Quaternions, HDQ, are first introduced in this paper. |
Hyper-Dual Quaternions Representation of Rigid Bodies Kinematics |
|
Fundamental Theorem of Matrix Representations of Hyper-Dual Numbers for Computing Higher-Order Derivatives |
2020 |
Y Imoto, N Yamanaka, T Uramoto, M Tanaka, M Fujikawa, N Mitsume |
Hyper-dual Numbers, Matrix Representation, Higher-order Derivatives, HDN-M Differentation, Automatic Differentiation |
Hyper-dual numbers (HDN) are numbers defined by using nilpotent elements that differ from each other. |
Fundamental Theorem of Matrix Representations of Hyper-Dual Numbers for Computing Higher-Order Derivatives |
|
MultiZ: A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers |
2020 |
AM Aguirre-Mesa, MJ García, H Millwater |
Commutative Hypercomplex, Multicomplex, Multidual, Hyperdual, High-order Derivatives |
Multicomplex and multidual numbers are two generalizations of complex numbers with multiple imaginary axes, useful for numerical computation of derivatives with machine precision. |
MultiZ: A Library for Computation of High-order Derivatives Using Multicomplex or Multidual Numbers |
|
On Correctness of Automatic Differentiation for Non-Differentiable Functions |
2020 |
W Lee, H Yu, X Rival, H Yang |
Non-Differentiabilities, Automatic Differentiation, PAP, Intenstional Derivatives |
Differentiation lies at the core of many machine-learning algorithms, and is well-supported by popular autodiff systems, such as TensorFlow and PyTorch. |
On Correctness of Automatic Differentiation for Non-Differentiable Functions |
|
Order Truncated Imaginary Algebra for Computation of Multivariable High-order Derivatives in Finite Element Analysis |
2020 |
M Aristizabal |
FEM, High-order Derivatives, Hypercomplex Algebras, Numerical Differentiation, Order Truncated Imaginary Numbers |
Computation of sensitivities in engineering problems has become a necessity in many areas, including mechanical, chemical and biomedical engineering. |
Order Truncated Imaginary Algebra for Computation of Multivariable High-order Derivatives in Finite Element Analysis |
|
Mixed-Mode Stress Intensity Factors Computation in Functionally Graded Materials using a Hypercomplex Variable Finite Element Formulation |
2020 |
D Ramirez-Tamayo, M Balcer, A Montoya, H Millwater |
Hypercomplex Variable FEM, Complex Variable Sensitivity Method, J-integral |
The hypercomplex variable FEM, ZFEM, is extended to compute the mode I and mode II energy release rates (ERR) for functionally graded materials. |
Mixed-Mode Stress Intensity Factors Computation in Functionally Graded Materials using a Hypercomplex Variable Finite Element Formulation |
|
Algorithm 1008: Multicomplex Number Class for Matlab, with a Focus on the Accurate Calculation of Small Imaginary Terms for Multicomplex Step Sensitivity Calculations |
2020 |
J Casado, R Hewson |
Numerical Differentiation, Multicomplex Numbers, Multicomplex Step, MATLAB |
A MATLAB class for multicomplex numbers was developed with particular attention paid to the robust and accurate handling of small imaginary components. |
Algorithm 1008: Multicomplex Number Class for Matlab, with a Focus on the Accurate Calculation of Small Imaginary Terms for Multicomplex Step Sensitivity Calculations |
|
Higher-order Automatic Differentiation with Dual Numbers |
2020 |
L Szirmay-Kalos |
Dual Numbers, Higher-order Automatic Differentiation |
In engineering applications, we often need the derivatives of functions defined by a program. |
Higher-order Automatic Differentiation with Dual Numbers |
|
Dual Number-based Variational Data Assimilation: Constructing Exact Tangent Linear and Adjoint Code From Nonlinear Model Evaluations |
2019 |
J Mattern, C Edwards, C Hill |
Automatic Differentiation, Dual Numbers, Adjoint Models |
Dual numbers allow for automatic, exact evaluation of the numerical derivative of high-dimensional functions at an arbitrary point with minimal coding effort. |
Dual Number-based Variational Data Assimilation: Constructing Exact Tangent Linear and Adjoint Code From Nonlinear Model Evaluations |
|
Some Applications in Classical Mechanics of the Double and the Dual Numbers |
2019 |
G.F. Torres del Castillo |
Dual Numbers, Hamiltonian Mechanics, Symmetry Groups |
The complex numbers are employed in classical physics as a useful tool, usually taking advantage of their basic algebraic properties, but not as an essential ingredient; the objects of physical interest are real numbers or real-valued functions. |
Some Applications in Classical Mechanics of the Double and the Dual Numbers |
|
Quaternion and Octonion-Based Finite Element Analysis Methods for Computing Multiple First Order Derivatives |
2019 |
M Aristizabal, D Ramirez-Tamayo, M García, A Aguirre-Mesa, A Montoya, H Millwater |
Quaternions, Cayley-Dickson Numbers, Numerical Differentiation, First Order Derivatives, Complex Step |
The complex Taylor series expansion (CTSE) method for computing accurate first order derivatives is extended in this work to quaternion, octonion and any order Cayley-Dickson algebra. |
Quaternion and Octonion-Based Finite Element Analysis Methods for Computing Multiple First Order Derivatives |
|
A Stiffness Derivative Local Hypercomplex Variable Finite Element Method for Computing the Energy Release Rate |
2019 |
AM Aguirre-Mesa, D Ramirez-Tamayo, MJ García, A Montoya, H Millwater |
Hypercomplex, Complex Variable FEM, CTSE |
A “local” hypercomplex variable FEM, L-ZFEM, is proposed for the computation of the energy release rate (ERR) using the stiffness derivative equation. |
A Stiffness Derivative Local Hypercomplex Variable Finite Element Method for Computing the Energy Release Rate |
|
A Finite Element-based Adaptive Energy Response Function Method for 2D Curvilinear Progressive Fracture |
2019 |
D Wagner, MJ García, A Montoya, H Millwater |
Curvilinear Progressive Fracture, Hypercomplex FEM, ZFEM, Multicomplex, Multidual |
An adaptive arbitrary-order curvilinear progressive 2D crack growth algorithm is presented. The method is automated such that the full crack path from inception to failure is computed with multiple finite element analyses. |
A Finite Element-based Adaptive Energy Response Function Method for 2D Curvilinear Progressive Fracture |
|
FDOT: A Fast, Memory-Efficient and Automated Approach for Discrete Adjoint Sensitivity Analysis Using the Operator Overloading Technique |
2019 |
R Djeddi, K Ekici |
Automatic Differentiation, Operator Overloading, Discrete Adjoint, Sensitivity Analysis, Object-oriented Programming, Computational Fluid Dynamics |
A new toolbox based on operator overloading is introduced for automatic differentiation of scientific computing codes – and in particular legacy computational fluid dynamics solvers that are developed using Fortran. |
FDOT: A Fast, Memory-Efficient and Automated Approach for Discrete Adjoint Sensitivity Analysis Using the Operator Overloading Technique |
|
Automatic Differentiation for Error Analysis of Monte Carlo Data |
2019 |
A Ramos |
Automatic Differentiation, Taylor Series, Forward Accumulation, Hyper-dual Numbers |
Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. |
Automatic Differentiation for Error Analysis of Monte Carlo Data |
|
A Review of Automatic Differentiation and its Efficient Implementation |
2019 |
C Margossian |
Automatic Differentiation, Forward Mode, Reverse Mode, Operator Overloading, High-order Differentiation |
Derivatives play a critical role in computational statistics, examples being Bayesian inference using Hamiltonian Monte Carlo sampling and the training of neural networks. |
A Review of Automatic Differentiation and its Efficient Implementation |
|
Efficient Estimate of Residual Stress Variance using Complex Variable Finite Element Methods |
2019 |
R Fielder, H Millwater, A Montoya, P Golden |
Residual Stress, Variance Approximation, Complex FEM, Sensitivity Analysis, Autofrettage Process |
The incorporation of residual stress states into structural components is often used to improve fatigue performance. |
Efficient Estimate of Residual Stress Variance using Complex Variable Finite Element Methods |
|
Use of Dual Numbers in Kinematical Analysis of Spatial Mechanisms. Part I: Principle of the Method |
2019 |
S Alaci, R Pentiuc, I Doroftei, F Ciornei |
Kinematical Chain, Dual Numbers |
The general methodology of solving a problem of kinematical analysis of a spatial mechanism is presented. |
Use of Dual Numbers in Kinematical Analysis of Spatial Mechanisms. Part I: Principle of the Method |
|
Use of Dual Numbers in Kinematical Analysis of Spatial Mechanisms. Part II: Applying the Method for the Generalised Cardan Mechanism |
2019 |
S Alaci, R Pentiuc, I Doroftei, F Ciornei |
Dual Numbers, Dual Matrix, Cardan Joint |
The equations which permit obtaining the displacements from kinematical pairs were deduced in the first part of the paper. |
Use of Dual Numbers in Kinematical Analysis of Spatial Mechanisms. Part II: Applying the Method for the Generalised Cardan Mechanism |
|
Variational Updates for Thermomechanically Coupled Gradient-Enhanced Elastoplasticity - Implementation Based on Hyper-Dual Numbers |
2018 |
V Fohrmeister, A Bartels, J Mosler |
Hyper-dual Numbers, Numerical Differentiation, Variational Constitutive Updates, Thermomechanical Coupling, Gradient-enhanced Elastoplasticity |
This paper deals with the implementation of thermomechanically coupled gradient-enhanced elastoplasticity at finite strains. |
Variational Updates for Thermomechanically Coupled Gradient-Enhanced Elastoplasticity - Implementation Based on Hyper-Dual Numbers |
|
Complex Variable Finite Element Method for Mixed-Mode Fracture and Interface Cracks |
2018 |
D Ramirez-Tamayo, A Montoya, H Millwater |
Mixed-Mode Fracture, Virtual Crack Extension Method, ZFEM |
Mixed mode fracture is frequently encountered in laminated composites and dissimilar materials in which the crack is lying on the interface. |
Complex Variable Finite Element Method for Mixed-Mode Fracture and Interface Cracks |
|
A Complex Variable Virtual Crack Extension Finite Element Method for Elastic-Plastic Fracture Mechanics |
2018 |
A Montoya, D Ramirez-Tamayo, H Millwater, M Kirby |
Nonlinear Fracture, Energy Release Rate, Virtual Crack Extension Method, ZFEM, Elastic-Plastic Fracture |
The virtual crack extension (VCE) approach for computing the energy release rate using a complex variable finite element method (ZFEM) is extended to nonlinear materials undergoing plastic deformation. |
A Complex Variable Virtual Crack Extension Finite Element Method for Elastic-Plastic Fracture Mechanics |
|
A Virtual Crack Extension Method for Thermoelastic Fracture using a Complex Variable Finite Element Method |
2018 |
D Ramirez-Tamayo, A Montoya, H Millwater |
Strain Energy Release Rate, Thermoelastic Fracture, Virtual Crack Extension Method, J-integral, ZFEM, CTSE |
A virtual crack extension (VCE) technique using the complex variable finite element method (ZFEM) has been developed and demonstrated for thermoelastic fracture problems. |
A Virtual Crack Extension Method for Thermoelastic Fracture using a Complex Variable Finite Element Method |
|
Automatic Differentiation in Machine Learning: A Survey |
2018 |
A Baydin, B Pearlmutter, A Radul, J Siskind |
Backpropagation, Differentiable Programming, Automatic Differentiation |
Derivatives, mostly in the form of gradients and Hessians, are ubiquitous in machine learning. |
Automatic Differentiation in Machine Learning: A Survey |
|
Automatic Differentiation in ML: Where We Are and Where We Should Be Going |
2018 |
B van Merriënboer, O Breuleux, A Bergeron, P Lamblin |
Automatic Differentiation, Machine Learning, Operator Overloading, Source Transformation, Myia |
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). |
Automatic Differentiation in ML: Where We Are and Where We Should Be Going |
|
Tangent: Automatic Differentiation Using Source-Code Transformation for Dynamically Typed Array Programming |
2018 |
B van Merriënboer, D Moldovan, A Wiltschko |
Automatic Differentiation, Source-code Transformation, Higher-order Derivatives |
The need to efficiently calculate first- and higher-order derivatives of increasingly complex models expressed in Python has stressed or exceeded the capabilities of available tools. |
Tangent: Automatic Differentiation Using Source-Code Transformation for Dynamically Typed Array Programming |
|
The Simple Essence of Automatic Differentiation |
2018 |
C Elliott |
Automatic Differentiation, Program Calculation, Category Theory |
Automatic differentiation (AD) in reverse mode (RAD) is a central component of deep learning and other uses of large-scale optimization. |
The Simple Essence of Automatic Differentiation |
|
Validated Computation of the Local Truncation Error of Runge–Kutta Methods with Automatic Differentiation |
2018 |
O Mullier, A Chapoutot, J Alexandre dit Sandretto |
Automatic Differentiation, Interval Analysis, Affine Arithmetic, Validated Numerical Integration, Runge-Kutta Methods |
A novel approach to bound the local truncation error of explicit and implicit Runge–Kutta methods is presented. |
Validated Computation of the Local Truncation Error of Runge–Kutta Methods with Automatic Differentiation |
|
Fuzzy Dual Numbers |
2018 |
F Mora-Camino, C Cosenza |
Dual Numbers, Vectors, Matrices, Entropy |
The series “Studies in Fuzziness and Soft Computing” contains publications on various topics in the area of soft computing. |
Fuzzy Dual Numbers |
|
Principle of Transference - An Extension to Hyper-Dual Numbers |
2017 |
A Cohen, M Shoham |
Dual Numbers, Hyper-dual Numbers, Principle of Transference |
The algebra of hyper-dual numbers and hyper-dual vectors of order n, developed in this paper, follows the same rules as those of dual numbers and dual vectors. |
Principle of Transference - An Extension to Hyper-Dual Numbers |
|
Application of Hyper-Dual Numbers to Rigid Bodies Equations of Motion |
2017 |
A Cohen, M Shoham |
Dual Numbers, Hyper-dual Numbers, Rigid Body Dynamic, Robot Dynamics |
This paper formulates the equations of motion of a rigid body in a hyper-dual number form. |
Application of Hyper-Dual Numbers to Rigid Bodies Equations of Motion |
|
Computational Two-Mode Asymptotic Bifurcation Theory Combined with Hyper-Dual Numbers and Applied to Plate/Shell Buckling |
2017 |
F Fujii, M Tanaka, T Sasagawa, R Omote |
Asymptotic Theory, Hyper-dual Numbers, Singularity, Limit Point, Bifurcation |
For error-free computation of higher-order derivatives of a complex mathematical expression composed of elementary functions, hyper-dual numbers (HDNs) are receiving close attention in computational engineering and sciences. |
Computational Two-Mode Asymptotic Bifurcation Theory Combined with Hyper-Dual Numbers and Applied to Plate/Shell Buckling |
|
Sensitivity Analysis for Radiofrequency Induced Thermal Therapies Using the Complex Finite Element Method |
2017 |
JF Monsalvo, MJ García, H Millwater, Y Feng |
Sensitivity Analysis, CTSE, Bioheat Transfer Equation, Radio Frequency Ablation, Liver Cancer Treatment |
In radiofrequency induced thermal procedures for cancer treatment, the temperature of the cancerous tissue is raised over therapeutic values while maintaining the temperature of the surrounding tissue at normal levels. |
Sensitivity Analysis for Radiofrequency Induced Thermal Therapies Using the Complex Finite Element Method |
|
Sensitivity Analysis in Thermoelastic Problems using the Complex Finite Element Method |
2017 |
A Montoya, H Millwater |
CTSE, ZFEM, Thermal Stress, Thermoelasticity Sensitivity Analysis |
This article presents the complex finite element method (ZFEM) for the sensitivity analysis of thermoelastic systems. |
Sensitivity Analysis in Thermoelastic Problems using the Complex Finite Element Method |
|
Applications of Automatic Differentiation in Topology Optimization |
2017 |
S Nørgaard, M Sagebaum, N Gauger, B Lazarov |
Topology Optimization, Automatic Differentiation, Lattice Boltzmann |
The goal of this article is to demonstrate the applicability and to discuss the advantages and disadvantages of automatic differentiation in topology optimization. |
Applications of Automatic Differentiation in Topology Optimization |
|
Residual Stress Sensitivity Analysis using a Complex Variable Finite Element Method |
2017 |
R Fielder, A Montoya, H Millwater, P Golden |
Residual Stress, Autofrettage, Sensitivity Analysis, Complex Variable FEM, CTSE Method |
The application of the complex variable FEM, ZFEM, to structures containing residual stresses is detailed. |
Residual Stress Sensitivity Analysis using a Complex Variable Finite Element Method |
|
Parameterized Reduced Order Models From a Single Mesh Using Hyper-Dual Numbers |
2016 |
M Brake, J Fike, S Topping |
Model Reduction Theory, Parametrization, Uncertainty, Parameterized Reduction Order Model |
In order to assess the predicted performance of a manufactured system, analysts must consider random variations (both geometric and material) in the development of a model. |
Parameterized Reduced Order Models From a Single Mesh Using Hyper-Dual Numbers |
|
Implementation of Incremental Variational Formulations Based on the Numerical Calculation of Derivatives Using Hyper-Dual Numbers |
2016 |
M Tanaka, D Balzani, J Schroder |
Inelasticity, Incremental Variational Formulas, Finite Strains, Higher-order Hyper-dual Numbers |
In this paper, novel implementation schemes for the automatic calculation of internal variables, stresses, and consistent tangent moduli for incremental variational formulations (IVFs) describing inelastic material behavior are proposed. |
Implementation of Incremental Variational Formulations Based on the Numerical Calculation of Derivatives Using Hyper-Dual Numbers |
|
Higher-order Probabilistic Sensitivity Calculations using the Multicomplex Score Function Method |
2016 |
J Garza, H Millwater |
Score Function, Sensitivity Analysis, CTSE, Multicomplex Step Differentiation Method |
The score function method used to compute first order probabilistic sensitivities is extended in this work to arbitrary-order derivatives included mixed partial derivatives through the use of multicomplex mathematics. |
Higher-order Probabilistic Sensitivity Calculations using the Multicomplex Score Function Method |
|
A Source Transformation via Operator Overloading Method for the Automatic Differentiation of Mathematical Functions in MATLAB |
2016 |
M Weinstein, A Rao, |
Automatic Differentiation, Numerical Methods, MATLAB, Scientific Computation, Applied Mathematics |
A source transformation via operator overloading method is presented for computing derivatives of mathematical functions defined by MATLAB computer programs. |
A Source Transformation via Operator Overloading Method for the Automatic Differentiation of Mathematical Functions in MATLAB |
|
Application of Hyper-Dual Numbers to Multibody Kinematics |
2016 |
A Cohen, M Shoham |
Hyperdual Numbers, Rigid Body Kinematic, Automatic Differentiation |
Hyper-dual numbers (HDNs) are applied in this paper to multibody kinematics. First, the hyper-dual angle that encompasses a body’s position, orientation, as well as its velocity, is defined as an element of the hyper-dual transformation matrix. |
Application of Hyper-Dual Numbers to Multibody Kinematics |
|
TMB : Automatic Differentiation and Laplace Approximation |
2016 |
K Kristensen, A Nielson, C Berg, H Skaug, B Bell |
Automatic Differentiation, Random Effects, Latent Variables, C++ |
TMB is an open source R package that enables quick implementation of complex nonlinear random effects (latent variable) models in a manner similar to the established AD Model Builder package. |
TMB : Automatic Differentiation and Laplace Approximation |
|
Automatic Differentiation Variational Inference |
2016 |
A Kucukelbir, D Tran, R Ranganath, A Gelman, D Blei |
Algorithms, Automatic Differentiation, Probability, Gradients |
Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. |
Automatic Differentiation Variational Inference |
|
A Virtual Crack Extension Method to Compute Energy Release Rates using a Complex Variable Finite Element Method |
2016 |
H Millwater, D Wagner, A Baines, A Montoya |
Strain Energy Release Rate, Virtual Crack Extension Method, Stiffness Derivative Method, ZFEM, CTSE Method |
The complex valued finite element method, ZFEM, is proposed as a new virtual crack extension method to compute the energy release rate. |
A Virtual Crack Extension Method to Compute Energy Release Rates using a Complex Variable Finite Element Method |
|
A Hitchhiker’s Guide to Automatic Differentiation |
2016 |
P Hoffmann |
Automatic Differentiation, Forward Mode, Reverse Mode, Dual Numbers |
This article provides an overview of some of the mathematical principles of Automatic Differentiation (AD). |
A Hitchhiker’s Guide to Automatic Differentiation |
|
DrMAD: Distilling Reverse-Mode Automatic Differentiation for Optimizing Hyperparameters of Deep Neural Networks |
2016 |
J Fu, H Luo, J Feng, K Low, T Chua |
Automatic Differentiation, Reverse Mode |
The performance of deep neural networks is well-known to be sensitive to the setting of their hyperparameters. Recent advances in reverse-mode automatic differentiation allow for optimizing hyperparameters with gradients. |
DrMAD: Distilling Reverse-Mode Automatic Differentiation for Optimizing Hyperparameters of Deep Neural Networks |
|
Weight Functions and Stress Intensity Factors for Pin-Loaded Single-Edge Notch Bend Specimen |
2015 |
X Zhao, X Wu, D Tong |
Weight Function Method, Taylor Series Expansion Method, Pin-loaded Singe-notch Bend Specimen |
A recently developed pin-loaded single-edge notch bend specimen provides an alternative to the single-edge notch tension specimen commonly used for small-crack growth testing. |
Weight Functions and Stress Intensity Factors for Pin-Loaded Single-Edge Notch Bend Specimen |
|
Fully Parameterized Reduced Order Models Using Hyper-Dual Numbers and Component Mode Synthesis |
2015 |
M Bonney, D Kammer, M Brake |
Parameterized Reduction Order Model, Sensitivity, Hyper-dual Numbers |
The uncertainty of a system is usually quantified with the use of sampling methods such as Monte Carlo or Latin hypercube sampling. These sampling methods require many computations of the model and may include re-meshing. |
Fully Parameterized Reduced Order Models Using Hyper-Dual Numbers and Component Mode Synthesis |
|
A Highly Accurate 1st- and 2nd-Order Differentiation Scheme for Hyperelastic Material Models Based on Hyper-Dual Numbers |
2015 |
M Tanaka, T Sasagawa, R Omote, M Fujikawa, D Balzani, J Schroder |
Nonlinear Finite Element Method, Finite Deformations, Numerical Derivative, Strain Energy Function, Hyper-dual Numbers |
In this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models, which are derived in terms of the first and second derivatives of a strain energy function. |
A Highly Accurate 1st- and 2nd-Order Differentiation Scheme for Hyperelastic Material Models Based on Hyper-Dual Numbers |
|
Complex Finite Element Sensitivity Method for Creep Analysis |
2015 |
A Gomez-Farias, A Montoya, H Millwater |
Creep, Sensitivity Analysis, Pressure Vessel, Skeletal Point, Complex Step Method |
The complex finite element method (ZFEM) has been extended to perform sensitivity analysis for mechanical and structural systems undergoing creep deformation. |
Complex Finite Element Sensitivity Method for Creep Analysis |
|
Multidual Numbers and Their Multidual Functions |
2015 |
F Messelmi |
Multidual Numbers, Multidual Functions, Hyperholomorphicity, Generator Polynomial |
The purpose of this paper is to develop a general theory of multidual numbers. We start by defining the notion of multidual numbers and their algebraic properties. |
Multidual Numbers and Their Multidual Functions |
|
Automatic Implementation of Elasto-plastic Incremental Formulations at Finite Strains using Hyper-Dual Numbers |
2015 |
M Tanaka, D Balzani, J Schröder |
Optimization, Hyperdual Numbers |
Numerical simulation of standard dissipative materials undergoing finite strains remains an important and challenging topic in computational mechanics. |
Automatic Implementation of Elasto-plastic Incremental Formulations at Finite Strains using Hyper-Dual Numbers |
|
Automatic Implementation of Finite Strain Anisotropic Hyperelastic Models Using Hyper-dual Numbers |
2015 |
R Kiran, K Khandelwal |
Hyperdual Numbers, Finite Strain Anisotropic Hyperelastic Model |
The main aim of this paper is to automate the implementation of finite strain anisotropic hyperelastic models into a general finite element framework. |
Automatic Implementation of Finite Strain Anisotropic Hyperelastic Models Using Hyper-dual Numbers |
|
Multicomplex Newmark-Beta Time Integration Method for Sensitivity Analysis in Structural Dynamics |
2015 |
J Garza, H Millwater |
Sensitivity Analysis, Newmark-beta Algorithm, Multicomplex, High-order Derivatives |
This paper describes an extension of the standard Newmark-beta algorithm to the multicomplex mathematical domain such that time-dependent, high-order, high-accuracy derivatives of dynamic systems can be obtained along with the traditional response. |
Multicomplex Newmark-Beta Time Integration Method for Sensitivity Analysis in Structural Dynamics |
|
The Stan Math Library: Reverse-Mode Automatic Differentiation in C++ |
2015 |
B Carpenter, M Hoffman, M Brubaker, D Lee, P Li, M Betancourt |
Automatic Differentiation, C++, Reverse Mode, Jacobian, Gradients |
As computational challenges in optimization and statistical inference grow ever harder, algorithms that utilize derivatives are becoming increasingly more important. |
The Stan Math Library: Reverse-Mode Automatic Differentiation in C++ |
|
A comparison of different methods for calculating tangent-stiffness matrices in a massively parallel computational peridynamics code |
2014 |
MD Brothers, JT Foster, HR Millwater |
Newton's Method, Newton-Raphson, Numerical Differentiation, Complex-step, Automatic Differentiation, Tangent-stiffness |
In order to maintain the quadratic convergence properties of Newton’s method in quasi-static nonlinear analysis of solid structures it is crucial to obtain accurate, algorithmically consistent tangent-stiffness matrices. |
A comparison of different methods for calculating tangent-stiffness matrices in a massively parallel computational peridynamics code |
|
Finite Element Sensitivity for Plasticity using Complex Variable Methods |
2014 |
A Montoya, R Fielder, A Gomez-Farias, H Millwater |
Complex Variable FEM, Complex Variables, Nonlinear Finite Element Equations |
The complex variable FEM (ZFEM) has been enhanced in this research to compute derivatives with respect to shape, material properties, and loads for a nonlinear solid mechanics model undergoing plastic deformation. |
Finite Element Sensitivity for Plasticity using Complex Variable Methods |
|
On Automatic Differentiation and Algorithmic Linearization |
2014 |
A Griewank |
Jacobians, Taylor Expansion, Piecewise Linearization |
We review the methods and applications of automatic differentiation, a research and development activity, which has evolved in various computational fields since the mid 1950's. |
On Automatic Differentiation and Algorithmic Linearization |
|
Automatic Differentiation of Algorithms for Machine Learning |
2014 |
A Baydin, B Pearlmutter |
Automatic Differentiation, Machine Learning, Optimization |
Usability is achieved through a simple direct interface and a cleanly abstracted functional interface. |
Automatic Differentiation of Algorithms for Machine Learning |
|
Multicomplex Taylor Series Expansion for Computing High-order Derivatives |
2014 |
H Millwater, S Shirinkam |
CTSE, High-order Derivatives, Multicomplex Numbers |
Multicomplex Taylor series expansion (MCTSE) is a numerical method for calculating higher-order partial derivatives of a multivariable realvalued and complex-valued analytic function based on TSE without subtraction cancelation errors. |
Multicomplex Taylor Series Expansion for Computing High-order Derivatives |
|
DNAD, A Simple Tool for automatic differentiation of Fortran Codes Using Dual Numbers |
2013 |
W Yu, M Blair |
Automatic Differentiation, Sensitivity, Fortran 90/95/2003, Dual Numbers, DNAD |
DNAD (dual number automatic differentiation) is a simple, general-purpose tool to automatically differentiate Fortran codes written in modern Fortran (F90/95/2003) or legacy codes written in previous versions of the Fortran language. |
DNAD, A Simple Tool for automatic differentiation of Fortran Codes Using Dual Numbers |
|
Improved WCTSE Method for the Generation of 2D Weight Functions Through Implementation into a Commercial Finite Element Code |
2013 |
H Millwater, D Wagner, A Baines, K Lovelady |
Weight Functions, Stress Intensity Factors, FEM |
The WCTSE (Weight function Complex Taylor Series Expansion) method was recently proposed as a new method for generating 2D weight functions. |
Improved WCTSE Method for the Generation of 2D Weight Functions Through Implementation into a Commercial Finite Element Code |
|
The Tapenade Automatic Differentiation Tool: Principles, Model, and Specification |
2013 |
L Hascoet, V Pascual |
Quadrature, Numerical Differentiation, Algorithms, Performance, Source Transformation, Adjoint Compiler |
Tapenade is an Automatic Differentiation (AD) tool which, given a Fortran or C code that computes a function, creates a new code that computes its tangent or adjoint derivatives. |
The Tapenade Automatic Differentiation Tool: Principles, Model, and Specification |
|
Multi-objective Optimization Using Hyper-dual Numbers |
2013 |
J Fike |
Automatic Differentiation, Hyper-dual Numbers, Finite Difference, Optimization |
High-fidelity analysis tools can provide accurate predictions of the performance of a design. However, these tools often have a higher computational cost when compared to lower-fidelity tools. |
Multi-objective Optimization Using Hyper-dual Numbers |
|
Imbedded Dual-Number Automatic Differentiation for Computational Fluid Dynamics Sensitivity Analysis |
2013 |
R Spall, W Yu |
Sensitiviity Analysis, Fortran, Dual Numbers, Automatic Differentiation |
Dual number automatic differentiation was applied to two computational fluid dynamics codes, one written specifically for this purpose and one “legacy” fortran code. |
Imbedded Dual-Number Automatic Differentiation for Computational Fluid Dynamics Sensitivity Analysis |
|
Implications of Using Dual Number Derivatives With a Numerical Solution |
2013 |
S Pakalapati, H Sezer, I Celik |
Dual Numbers, Automatic Differentiation, Computation Fluid Dynamics, Numerical Modeling |
Dual number arithmetic is a well-known strategy for automatic differentiation of computer codes which gives exact derivatives, to the machine accuracy, of the computed quantities with respect to any of the involved variables. |
Implications of Using Dual Number Derivatives With a Numerical Solution |
|
Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models |
2013 |
J Martins, J Hwang |
Derivatives, Chain Rule, Finite Differences, Symbolic Differentiation |
This paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework. |
Review and Unification of Methods for Computing Derivatives of Multidisciplinary Computational Models |
|
Using Multicomplex Variables for Automatic Computation of High-Order Derivatives |
2012 |
G Lantoine, R Russell, T Dargent |
Algorithms, Performance, Sensitivity Analysis, High-order Derivatives, Multicomplex Numbers, Overloading, Multicomplex Step Differentiation, Automatic Differentiation |
The computations of the high-order partial derivatives in a given problem are often cumbersome or not accurate. To combat such shortcomings, a new method for calculating exact high-order sensitivities using multicomplex numbers is presented. |
Using Multicomplex Variables for Automatic Computation of High-Order Derivatives |
|
Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex-variables |
2012 |
W Anderson, J Newman, D Whitfield, E Nielsen |
Sensitivity Analysis, Navier-Stokes Equation, Complex Variables |
The use of complex variables for determining sensitivity derivatives for turbulent flows is examined. |
Sensitivity Analysis for Navier-Stokes Equations on Unstructured Meshes Using Complex-variables |
|
Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives |
2012 |
J Fike, J Alonso |
Automatic Differentiation, Hyper-dual Numbers |
Automatic Differentiation techniques are typically derived based on the chain rule of differentiation. |
Automatic Differentiation Through the Use of Hyper-Dual Numbers for Second Derivatives |
|
Imbedded Dual-Number Automatic Differentiation to Support CFD Analysis |
2012 |
R Spall, W Yu |
Computational Fluid Dynamics, Fortran, Turbulence, Computer Software, Dimenions, Fluids, Laminar Flow, Modeling |
Dual number automatic differentiation was applied to two different computational fluid dynamics codes, one written specifically for this purpose and one larger ''legacy'' Fortran code. |
Imbedded Dual-Number Automatic Differentiation to Support CFD Analysis |
|
2D Weight Function Development using a Complex Taylor Series Expansion Method |
2012 |
D Wagner, H Millwater |
CTSE, Weight Functions, FEM |
Weight functions are a critical component of a damage tolerance fracture control plan in that they allow the stress intensity factor to be computed quickly from the stress along the uncracked crack line. |
2D Weight Function Development using a Complex Taylor Series Expansion Method |
|
AD Model Builder: Using Automatic Differentiation for Statistical Inference of Highly Parameterized Complex Nonlinear Models |
2012 |
D Fournier, H Skaug, J Ancheta, J Ianelli, A Magnusson, M Maunder, A Nielson, J Sibert |
Automatic Differentiation Model Builder, Parameter Estimation, Optimization, Laplace Approximation, Seperability |
Many criteria for statistical parameter estimation, such as maximum likelihood, are formulated as a nonlinear optimization problem. |
AD Model Builder: Using Automatic Differentiation for Statistical Inference of Highly Parameterized Complex Nonlinear Models |
|
Fatigue Sensitivity Analysis Using Complex Variable Methods |
2012 |
A Voorhees, H Millwater, R Bagley, P Golden |
Fatigue Life Prediction, Sensitivity Methods, CTSE |
The sensitivity of the computed cycles-to-failure and other lifing estimates to the various input parameters is a valuable, yet largely unexploited, aspect of a fatigue lifing analysis. |
Fatigue Sensitivity Analysis Using Complex Variable Methods |
|
Recent Advances in Algorithmic Differentiation |
2012 |
S Forth, P Hovland, E Phipps, J Utke, A Walther |
Leibniz, Automatic Differentiation, Jacobian, Nonlinear Models, Matrix Functions |
Notwithstanding the superiority of the Leibniz notation for differential calculus, the dot-and-bar notation predominantly used by the Automatic Differentiation community is resolutely Newtonian. |
Recent Advances in Algorithmic Differentiation |
|
On Higher-order Differentiation in Nonlinear Mechanics |
2012 |
I Charpentier |
Nonlinear Problems, Bifurcation Analysis, Taylor Series, Asymptotic Methods, Automatic Differentiation, Diamant |
To overcome the problem, we have studied a new approach for higher order sensitivity analysis of the finite element method using automatic differentiation techniques. |
On Higher-order Differentiation in Nonlinear Mechanics |
|
Achieving Logarithmic Growth of Temporal and Spatial Complexity in Reverse Automatic Differentiation |
2011 |
A Griewank |
Gradient, Adjoint, Complexity, Checkpointing, Recursion, Automatic Differentiation |
In its basic form, the reverse mode of automatic differentiation yields gradient vectors at a small multiple of the computational work needed to evaluate the underlying scalar function. |
Achieving Logarithmic Growth of Temporal and Spatial Complexity in Reverse Automatic Differentiation |
|
Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers |
2011 |
J Fike, S Jongsma, J Alonso, E Weide |
Hyper-dual Numbers, Hessian, Jacobian, Optimization |
Hyper-dual numbers can be used to compute exact first and second derivatives in order to form gradients and Hessians for optimization methods. There is, however, an increased computational cost associated with the mathematics of hyper-dual numbers. |
Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers |
|
Complex-variable Methods For Shape Sensitivity of Finite Element Models |
2011 |
A Voorhees, H Millwater, R Bagley |
Design Sensitivities, Optimization, Sensitivity Methods, CTSE, Finite Differencing, Fourier Differentiation |
Shape sensitivity analysis of finite element models is useful for structural optimization and design modifications. Complex variable methods for shape sensitivity analysis have some potential advantages over other methods. |
Complex-variable Methods For Shape Sensitivity of Finite Element Models |
|
The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations |
2011 |
J Fike, J Alonso |
Complex-step Method, Derivatives, Hyperdual Numbers, Operator Overloading |
The complex-step approximation for the calculation of derivative information has two significant advantages: the formulation does not suffer from subtractive cancellation errors and it can provide exact derivatives. |
The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations |
|
AUTO_DERIV: Tool for Automatic Differentiation of a Fortran Code |
2010 |
S Stamatiadis, S Farantos |
Fortran 95, Automatic Differentiation |
AUTO_DERIV is a module comprised of a set of fortran 95 procedures which can be used to calculate the first and second partial derivatives (mixed or not) of any continuous function with many independent variables. |
AUTO_DERIV: Tool for Automatic Differentiation of a Fortran Code |
|
TaylUR 3, a Multivariate Arbitrary-order Automatic Differentiation Package for Fortran 95 |
2010 |
G.M. von Hippel |
Automatic Differentiation, Higher-order Derivatives, Fortran 95 |
This new version of TaylUR is based on a completely new core, which now is able to compute the numerical values of all of a complex-valued function's partial derivatives up to an arbitrary order, including mixed partial derivatives. |
TaylUR 3, a Multivariate Arbitrary-order Automatic Differentiation Package for Fortran 95 |
|
Linear Dual Algebra Algorithms and Their Application to Kinematics |
2009 |
E Pennestri, P Valentini |
Dual Algebra, Line Vector, Finite Rigid Body Motion Analysis, Infinitesimal Rigid Body Motion Analysis |
Mathematical and mechanical entities such as line vectors, screws, and wrenches can be conveniently represented within the framework of dual algebra. |
Linear Dual Algebra Algorithms and Their Application to Kinematics |
|
On the Implementation of Automatic Differentiation Tools |
2008 |
C Bischof, P Hovland, B Norris |
Semantic Transformation, Automatic Differentiation |
Automatic differentiation is a semantic transformation that applies the rules of differential calculus to source code. |
On the Implementation of Automatic Differentiation Tools |
|
Advanced Concepts for Automatic Differentiation based on Operator Overloading |
2008 |
A Kowarz |
Automatic Differentiation, Operator Overloading, Activity Analysis, Parallelism, Nested Taping |
Using the technique of Automatic Differentiation (AD), derivative information can be computed efficiently for any function that is given as source code in a supported programming languages. |
Advanced Concepts for Automatic Differentiation based on Operator Overloading |
|
Computing Sparse Hessians with Automatic Differentiation |
2008 |
A Walther |
Algorithms, Theory, Performance, Automatic Differentiation, Second Order Derivatives, Sparsity Pattern |
A new approach for computing a sparsity pattern for a Hessian is presented: nonlinearity information is propagated through the function evaluation yielding the nonzero structure. |
Computing Sparse Hessians with Automatic Differentiation |
|
OpenAD/F: A Modular Open-Source Tool for Automatic Differentiation of Fortran Codes |
2008 |
J Utke, U Naumann, M Fagan, N Tallent, M Strout, P Heimbach, C Hill, C Wunsch |
Algorithms, Performance, Automatic Differentiation, Source Transformation, Adjoint Compiler |
The Open/ADF tool allows the evaluation of derivatives of functions defined by a Fortran program. |
OpenAD/F: A Modular Open-Source Tool for Automatic Differentiation of Fortran Codes |
|
Linear Algebra and Numerical Algorithms Using Dual Numbers |
2007 |
E Pennestrì, R Stefanelli |
Dual Numbers, Kinematics and Dynamics of Spatial Linkages |
Dual number algebra is a powerful mathematical tool for the kinematic and dynamic analysis of spatial mechanisms. |
Linear Algebra and Numerical Algorithms Using Dual Numbers |
|
Complex Variable Method for Eigensolution Sensitivity Analysis |
2006 |
B Wang, A Apte |
Complex Variables, Sensitivity Analysis, Eigenvalue, Eigenvector, Automatic Differentiation |
The application of the complex variable method for eigenvalue and eigenvector sensitivity analysis is presented. |
Complex Variable Method for Eigensolution Sensitivity Analysis |
|
Nonlinear Robust Performance Analysis Using Complex-step Gradient Approximation |
2006 |
J Kim, D Bates, I Postlethwaite |
Robust Analysis, Optimisation, Nonlinear Systems |
In this paper, the complex-step method is applied in the context of numerical optimization problems involving dynamical systems modeled as nonlinear differential equations. |
Nonlinear Robust Performance Analysis Using Complex-step Gradient Approximation |
|
Automatic Differentiation |
2006 |
H Berland |
Automatic Differentiation, Divided Difference, Dual Numbers, Forward Mode, Source Transformation, Operator Overloading |
Automatic differentiation is introduced to an audience with basic mathematical prerequisites. |
Automatic Differentiation |
|
Automatic Differentiation of C++ Codes for Large-Scale Scientific Computing |
2006 |
R Bartlett, D Gay, E Phipps |
C++, Automatic Differentiation, Element Derivatives |
We discuss computing first derivatives for models based on elements, such as large scale finite-element PDE discretizations, implemented in the C++ programming language. |
Automatic Differentiation of C++ Codes for Large-Scale Scientific Computing |
|
A New Inverse Analysis Approach for Multi-region Heat Conduction BEM Using Complex-variable Differentiation Method |
2005 |
X Gao, M He |
Inverse Analysis, Multi-region BEM,Sensitivity Coefficients, Heat Conduction, Shape Optimization, Damage Identification |
This paper presents a new inverse analysis approach for identifying material properties and unknown geometries for multi-region problems using the Boundary Element Method (BEM). |
A New Inverse Analysis Approach for Multi-region Heat Conduction BEM Using Complex-variable Differentiation Method |
|
Dual Numbers and Supersymmetric Mechanics |
2005 |
A Frydryszak |
Dual Numbers, Rigid Body Movements, Field Theory |
Apart from the known practical applications to the description of rigid body movements in three-dimensional space and their natural presence in abstract differential algebra, play a role in field theory and are related to supersymmetry as well. |
Dual Numbers and Supersymmetric Mechanics |
|
ADF95: Tool for Automatic Differentiation of a Fortran Code Designed for Large Numbers of Independent Variables |
2005 |
C Straka |
Automatic Differentiation, Derivatives, Fortran 95, Implicit Solvers |
ADF95is a tool to automatically calculate numerical first derivatives for any mathematical expression as a function of user defined independent variables. Accuracy of derivatives is achieved within machine precision. |
ADF95: Tool for Automatic Differentiation of a Fortran Code Designed for Large Numbers of Independent Variables |
|
The Complex-step Derivative Approximation |
2003 |
J Martins, P Sturdza, J Alonso |
Algorithms, Performance, Automatic Differentiation, Forward Mode, Complex-step Derivative Approximation, Overloading, Gradients, Sensitivities |
The complex-step derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to alg. differentiation. |
The Complex-step Derivative Approximation |
|
The Connection Between the Complex-step Derivative Approximation and Algorithmic Differentiation |
2001 |
J Martins, P Sturdza, J Alonso |
Complex-step Derivative Approximation, Algorithmic Differentiation Theory, Fortran, C, C++ |
This paper presents improvements to the complexstep derivative approximation method which increase its accuracy and robustness. These improvements unveil the connection to algorithmic differentiation theory. |
The Connection Between the Complex-step Derivative Approximation and Algorithmic Differentiation |
|
Remark On Algorithm 746: New Features of PCOMP, a Fortran Code for Automatic Differentiation |
2000 |
M Liepelt, K Schittkowski, |
Automatic Differentiation, Forward Accumulation, Reverse Accumulation, Algorithms |
The software system PCOMP uses automatic differentiation to calculate derivatives of functions that are defined by the user in a modeling language similar to Fortran. |
Remark On Algorithm 746: New Features of PCOMP, a Fortran Code for Automatic Differentiation |
|
An Automated Method for Sensitivity Analysis Using Complex Variables |
2000 |
J Martins, I Kroo, J Alonso |
Complex-step Method, Numerical Algorithms, Fluid Dynamics, Automatic Differentiation |
The complex-step method for calculating sensitivities and its use in numerical algorithms is presented. A general procedure for the implementation of this method is described in detail and a script is developed that automates its implementation. |
An Automated Method for Sensitivity Analysis Using Complex Variables |
|
Automatic Differentiation of Algorithms |
2000 |
M Bartholomew-Biggs, S Brown, B Christianson, L Dixon |
Algorithms, Automatic Differentiation, Adjoint Programming, Checkpoints, Error Analysis, Function Approximation, Implicit Equations, Interval Analysis, Nonlinear Optimization, Optimal Control, Parallelism, Penalty Functions, Program Transformation, Variab |
We introduce the basic notions of automatic differentiation, describe some extensions which are of interest in the context of nonlinear optimization and give some illustrative examples. |
Automatic Differentiation of Algorithms |
|
ADIC: An Extensible Automatic Differentiation Tool for ANSI-C |
1999 |
C Bischof, L Roh, A Mauer-Oats |
Automatic Differentiation, Derivatives, Semantic Augmentation, Source Transformation |
In scientific computing, we often require the derivatives df/dx of a function f expressed as a program with respect to some input parameter(s) x, say. |
ADIC: An Extensible Automatic Differentiation Tool for ANSI-C |
|
Automatic Differentiation of Numerical Integration Algorithms |
1999 |
P Eberhard, C Bischof |
Automatic Differentiaton, Numerical Integration Algorithms, Variable-order ODE Integrators |
Automatic differentiation (AD) is a technique for automatically augmenting computer programs with statements for the computation of derivatives. |
Automatic Differentiation of Numerical Integration Algorithms |
|
Dual Numbers Representation of Rigid Body Dynamics |
1999 |
V Brodsky, M Shoham |
Dual Numbers, Vector Algebra, Spherical Kinematics, Spatial Kinematics,Rigid Body Motion |
A three-dimensional representation of rigid body dynamic equations becomes possible by introducing the dual inertia operator. |
Dual Numbers Representation of Rigid Body Dynamics |
|
Using Complex Variables to Estimate Derivatives of Real Functions |
1998 |
W Squire, G Trapp |
Divided Difference, Subtractive Cancellation, Complex Variables |
A method to approximate derivatives of real functions using complex variables, which avoids the subtractive cancellation errors inherent in classical derivative approximations, is described. |
Using Complex Variables to Estimate Derivatives of Real Functions |
|
Adifor 2.0: Automatic Differentiation of Fortran 77 Programs |
1996 |
C Bischof, P Khademi, A Mauer, A Carle |
Automatic Differentiation, Fortran 77 |
Numerical codes that calculate not only a result but also the derivatives of the variables with respect to each other facilitate sensitivity analysis, inverse problem solving, and optimization. |
Adifor 2.0: Automatic Differentiation of Fortran 77 Programs |
|
Algorithm 755: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++ |
1996 |
A Griewank, D Juedes, J Utke |
C++, C, Fortran, Automatic Differentiation, Chain Rule, Forward Mode, Gradients, Hessians, Overloading, Reverse Mode, Taylor Coefficients |
The C++ package ADOL-C described here facilitates the evaluation of first and higher derivatives of vector functions defined by computer programs written in C or C++. |
Algorithm 755: ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++ |
|
Algorithm 746: PCOMP—a Fortran Code for Automatic Differentiation |
1995 |
M Dobmann, M Liepelt, K Schittkowski |
Algorithms, Languages, Automatic Differentiation, Forward Accumulation, Reverse Accumulation |
Automatic differentiation is an interesting and important tool for all numerical algorithms that require derivatives, e.g., in nonlinear programming, optimal control, parameter estimation, and differential equations. |
Algorithm 746: PCOMP—a Fortran Code for Automatic Differentiation |
|
Higher-order Sensitivity Analysis of Finite Element Method by Automatic Differentiation |
1995 |
I Ozaki, F Kimura, M Berz |
Automatic Differentiation, Sensitivity Analysis, Finite Element Method, Fortran |
To design optimal mechanical structures, design sensitivity analysis technique using higher order derivatives are important. |
Higher-order Sensitivity Analysis of Finite Element Method by Automatic Differentiation |
|
The Application of Automatic Differentiation to Problems in Engineering Analysis |
1994 |
S Chinchalkar |
Automatic Differentiation, Forward Mode, Reverse Mode |
Automatic differentiation is a technique of computing the derivative of a function or a subroutine written in a higher level language such as FORTRAN or C. Significant progress has been made in this field in the last few years. |
The Application of Automatic Differentiation to Problems in Engineering Analysis |
|
Programming With Dual Numbers and its Applications in Mechanisms Design |
1994 |
H Cheng |
Dual Numbers, Mechanism Design |
Dual numbers are expressed in the form x+Ey where E^2. Dual metanumbers are defined in this paper as DualZero, DualInf and DualNaN. |
Programming With Dual Numbers and its Applications in Mechanisms Design |
|
ADIFOR-Generating Derivative Codes from Fortran Programs |
1992 |
C Bischof, A Carle, G Corliss, A Greiwank, P Hovland |
Numerical Methods, Automatic Differentiaton, Fortran 77 |
The numerical methods used in solving many scientific computing problems require computing derivatives of a function f: R^n to R^m. |
ADIFOR-Generating Derivative Codes from Fortran Programs |
|
On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, Spatial Kinematics |
1976 |
G Veldkamp |
Dual Numbers, Dual Vectors, Instantaneous Kinematics, Spatial Kinematics |
This paper gives a self-contained account of the algebra of dual quantities, the differential-geometry of dual curves, and their application to theoretical space kinematics. |
On the Use of Dual Numbers, Vectors and Matrices in Instantaneous, Spatial Kinematics |
|
Numerical Algorithms Based on the Theory of Complex Variable |
1967 |
J Lyness |
Complex Variables, Complex Algebra, Wave Mechanics |
Since its introduction in the early part of the nineteenth century, the theory of complex variables has played a steadily increasing role in mathematics and scientific research. |
Numerical Algorithms Based on the Theory of Complex Variable |
|
On Hypercomplex Numbers |
1908 |
J Wedderburn |
Hypercomplex Numbers, Difference Algebra, Matrix Algebra |
The object of this paper is, in the first place, to set the theory of hypercomplex numbers on a rational basis. |
On Hypercomplex Numbers |
|
Preliminary Sketch of Biquaternions |
1871 |
C M.A. |
Vectors, Hamiltonian, Quaternions |
The vectors of Hamilton are quantities having magnitude and direction but no particular position; the vector AB is regarded as identical with the vector CD when AB is equal and parallel to CD and in the same sense. |
Preliminary Sketch of Biquaternions |
