@misc{Griewank, author = {Griewank, Andreas}, title = {Sequential Evaluation of Adjoints and Higher Derivative Vectors by Overloading and Reverse Accumulation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-539}, number = {SC-91-03}, abstract = {Most nonlinear computations require the evaluation of first and higher derivatives of vector functions defined by computer programs. It is shown here how vectors of such partial derivatives can be obtained automatically and efficiently if the computer language allows overloading (as is or will be the case for C++, PASCAL-XSC, FORTRAN90, and other modern languages). Here, overloading facilitates the extension of arithmetic operations and univariate functions from real or complex arguments to truncated Taylor-series (or other user- defined types), and it generates instructions for the subsequent evaluation of adjoints. Similar effects can be achieved by precompilation of FORTRAN77 programs. The proposed differentiation algorithm yields gradients and higher derivatives at a small multiple of the run-time and RAM requirement of the original function evaluation program. {\bf Keywords:} Automatic Differentiation, Chain Rule, Overloading, Taylor Coefficients, Gradients, Hessians, Reverse Accumulation, Adjoint Equations. {\bf Abbreviated title:} Automatic Differentiation by Overloading.}, language = {en} } @misc{GriewankStreubelTischendorf, author = {Griewank, Andreas and Streubel, Tom and Tischendorf, Caren}, title = {On the abs-polynomial expansion of piecewise smooth functions}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-78448}, abstract = {Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order \$\bar d-1\$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function \$|...|\$ apart from arithmetic operations and \$\bar d\$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of \$\mathcal O({\bar d}^2)\$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in \$\bar d\$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.}, language = {en} } @article{GriewankHasenfelderRadonsetal.2018, author = {Griewank, Andreas and Hasenfelder, Richard and Radons, Manuel and Lehmann, Lutz and Streubel, Tom}, title = {Integrating Lipschitzian dynamical systems using piecewise algorithmic differentiation}, series = {Optimization Methods and Software}, volume = {33}, journal = {Optimization Methods and Software}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1378653}, pages = {1089 -- 1107}, year = {2018}, abstract = {In this article we analyse a generalized trapezoidal rule for initial value problems with piecewise smooth right-hand side F : IR^n -> IR^n based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of F. The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third-order interpolation polynomial for the numerical trajectory. In the smooth case, the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.}, language = {en} } @article{GriewankStreubelLehmannetal.2017, author = {Griewank, Andreas and Streubel, Tom and Lehmann, Lutz and Radons, Manuel and Hasenfelder, Richard}, title = {Piecewise linear secant approximation via algorithmic piecewise differentiation}, series = {Optimization Methods and Software}, volume = {33}, journal = {Optimization Methods and Software}, number = {4-6}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2017.1387256}, pages = {1108 -- 1126}, year = {2017}, abstract = {It is shown how piecewise differentiable functions F : IR^n -> IR^m that are defined by evaluation programmes can be approximated locally by a piecewise linear model based on a pair of sample points \check x and \hat x. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x - \check x||*||x - \hat x||). As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equalling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.}, language = {en} } @misc{HasenfelderLehmannRadonsetal., author = {Hasenfelder, Richard and Lehmann, Lutz and Radons, Manuel and Streubel, Tom and Strohm, Christian and Griewank, Andreas}, title = {Computational aspects of the Generalized Trapezoidal Rule}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68615}, abstract = {In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side F:IR^n -> IR^n. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a non-differentiability of F. In such a situation the investigated generalized trapezoidal rule achieves a higher convergence order than the classical method. While the asymptotic behavior of the generalized method was investigated in a previous work, in the present article we develop the algorithmic structure for efficient implementation strategies and estimate the actual computational cost of the latter. Moreover, energy preservation of the generalized trapezoidal rule is proved for Hamiltonian systems with piecewise linear right hand side.}, language = {en} } @misc{RadonsLehmannStreubeletal., author = {Radons, Manuel and Lehmann, Lutz and Streubel, Tom and Griewank, Andreas}, title = {An Open Newton Method for Piecewise Smooth Systems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-70418}, abstract = {Recent research has shown that piecewise smooth (PS) functions can be approximated by piecewise linear functions with second order error in the distance to a given reference point. A semismooth Newton type algorithm based on successive application of these piecewise linearizations was subsequently developed for the solution of PS equation systems. For local bijectivity of the linearization at a root, a radius of quadratic convergence was explicitly calculated in terms of local Lipschitz constants of the underlying PS function. In the present work we relax the criterium of local bijectivity of the linearization to local openness. For this purpose a weak implicit function theorem is proved via local mapping degree theory. It is shown that there exist PS functions f:IR^2 --> IR^2 satisfying the weaker criterium where every neighborhood of the root of f contains a point x such that all elements of the Clarke Jacobian at x are singular. In such neighborhoods the steps of classical semismooth Newton are not defined, which establishes the new method as an independent algorithm. To further clarify the relation between a PS function and its piecewise linearization, several statements about structure correspondences between the two are proved. Moreover, the influence of the specific representation of the local piecewise linear models on the robustness of our method is studied. An example application from cardiovascular mathematics is given.}, language = {en} } @misc{GriewankStreubelLehmannetal., author = {Griewank, Andreas and Streubel, Tom and Lehmann, Lutz and Hasenfelder, Richard and Radons, Manuel}, title = {Piecewise linear secant approximation via Algorithmic Piecewise Differentiation}, issn = {1438-0064}, doi = {10.1080/10556788.2017.1387256}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-61642}, abstract = {It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x - x̌|| ||x - x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton's method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.}, language = {en} } @misc{StreubelTischendorfGriewank, author = {Streubel, Tom and Tischendorf, Caren and Griewank, Andreas}, title = {Piecewise Polynomial Taylor Expansions - The Generalization of Fa{\`a} di Bruno's Formula}, issn = {1438-0064}, doi = {10.1007/978-3-030-55240-4_3}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-68859}, abstract = {We present an extension of Taylor's theorem towards nonsmooth evalua- tion procedures incorporating absolute value operaions. Evaluations procedures are computer programs of mathematical functions in closed form expression and al- low a different treatment of smooth operations and calls to the absolute value value function. The well known classical Theorem of Taylor defines polynomial approx- imation of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential or differential algebraic equations, for the construction of solvers for the continuous nonlinear op- timization of finite dimensional objective functions and for root solving of nonlinear systems of equations. The herein provided proof is construtive and allow efficiently designed algorithms for the execution and computation of generalized piecewise polynomial expansions. As a demonstration we will derive a k-step method on the basis of polynomial interpolation and the proposed generalized expansions.}, language = {en} } @misc{GriewankHasenfelderRadonsetal., author = {Griewank, Andreas and Hasenfelder, Richard and Radons, Manuel and Lehmann, Lutz and Streubel, Tom}, title = {Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation}, issn = {1438-0064}, doi = {10.1080/10556788.2017.1378653}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-64639}, abstract = {In this article we analyze a generalized trapezoidal rule for initial value problems with piecewise smooth right hand side \(F:R^n \to R^n\) based on a generalization of algorithmic differentiation. When applied to such a problem, the classical trapezoidal rule suffers from a loss of accuracy if the solution trajectory intersects a nondifferentiability of \(F\). The advantage of the proposed generalized trapezoidal rule is threefold: Firstly, we can achieve a higher convergence order than with the classical method. Moreover, the method is energy preserving for piecewise linear Hamiltonian systems. Finally, in analogy to the classical case we derive a third order interpolation polynomial for the numerical trajectory. In the smooth case the generalized rule reduces to the classical one. Hence, it is a proper extension of the classical theory. An error estimator is given and numerical results are presented.}, language = {en} } @article{GriewankStreubelTischendorf2020, author = {Griewank, Andreas and Streubel, Tom and Tischendorf, Caren}, title = {On the abs-polynomial expansion of piecewise smooth functions}, series = {Optimization Methods and Software}, journal = {Optimization Methods and Software}, publisher = {Taylor \& Francis}, doi = {10.1080/10556788.2020.1817448}, year = {2020}, abstract = {Tom Streubel has observed that for functions in abs-normal form, generalized Taylor expansions of arbitrary order \$\bar d-1\$ can be generated by algorithmic piecewise differentiation. Abs-normal form means that the real or vector valued function is defined by an evaluation procedure that involves the absolute value function \$|...|\$ apart from arithmetic operations and \$\bar d\$ times continuously differentiable univariate intrinsic functions. The additive terms in Streubel's expansion are abs-polynomial, i.e. involve neither divisions nor intrinsics. When and where no absolute values occur, Moore's recurrences can be used to propagate univariate Taylor polynomials through the evaluation procedure with a computational effort of \$\mathcal O({\bar d}^2)\$, provided all univariate intrinsics are defined as solutions of linear ODEs. This regularity assumption holds for all standard intrinsics, but for irregular elementaries one has to resort to Faa di Bruno's formula, which has exponential complexity in \$\bar d\$. As already conjectured we show that the Moore recurrences can be adapted for regular intrinsics to the abs-normal case. Finally, we observe that where the intrinsics are real analytic the expansions can be extended to infinite series that converge absolutely on spherical domains.}, language = {en} } @article{StreubelTischendorfGriewank2020, author = {Streubel, Tom and Tischendorf, Caren and Griewank, Andreas}, title = {Piecewise Polynomial Taylor Expansions - The Generalization of Fa{\`a} di Bruno's Formula}, series = {Modeling, Simulation and Optimization of Complex Processes HPSC 2018}, journal = {Modeling, Simulation and Optimization of Complex Processes HPSC 2018}, number = {Modeling, Simulation and Optimization of Complex Processes HPSC 2018}, publisher = {Springer International Publishing}, doi = {10.1007/978-3-030-55240-4_3}, pages = {63 -- 82}, year = {2020}, abstract = {We present an extension of Taylor's Theorem for the piecewise polynomial expansion of non-smooth evaluation procedures involving absolute value operations. Evaluation procedures are computer programs of mathematical functions in closed form expression and allow a different treatment of smooth operations or calls to the absolute value function. The well known classical Theorem of Taylor defines polynomial approximations of sufficiently smooth functions and is widely used for the derivation and analysis of numerical integrators for systems of ordinary differential- or differential-algebraic equations, for the construction of solvers for continuous non-linear optimization of finite dimensional objective functions and for root solving of non-linear systems of equations. The long term goal is the stabilization and acceleration of already known methods and the derivation of new methods by incorporating piecewise polynomial Taylor expansions. The herein provided proof of the higher order approximation quality of the new generalized expansions is constructive and allows efficiently designed algorithms for the execution and computation of the piecewise polynomial expansions. As a demonstration towards the ultimate goal we will derive a prototype of a {\\$}{\\$}k{\\$}{\\$}k-step method on the basis of polynomial interpolation and the proposed generalized expansions.}, language = {en} }