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  • 11
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    On the abs-polynomial expansion of piecewise smooth functions (2020)
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    Publication Date: 2023-03-31
    Description: 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: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 12
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    Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas Networks (2022)
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    Publication Date: 2023-03-31
    Language: English
    Type: doctoralthesis , doc-type:doctoralThesis
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  • 13
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    Integrating Lipschitzian Dynamical Systems using Piecewise Algorithmic Differentiation (2017)
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    Publication Date: 2023-03-31
    Description: 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: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
  • 14
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    Piecewise linear secant approximation via Algorithmic Piecewise Differentiation (2016)
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    Publication Date: 2023-03-31
    Description: 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: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 15
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    Piecewise Polynomial Taylor Expansions – The Generalization of Faà di Bruno’s Formula (2018)
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    Publication Date: 2023-03-31
    Description: 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: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 16
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    On the Numerical Treatment of Interlaced Target Values - Modeling, Optimization and Simulation of Regulating Valves in Gas Networks (2021)
    Details
    Publication Date: 2023-03-31
    Description: Due to the current and foreseeable shifts in energy production, the trading and transport operations of gas will become more dynamic, volatile, and hence also less predictable. Therefore, computer-aided support in terms of rapid simulation and control optimization will further broaden its importance for gas network dispatching. In this paper, we aim to contribute and openly publish two new mathematical models for regulators, also referred to as control valves, which together with compressors make up the most complex and involved types of active elements in gas network infrastructures. They provide full direct control over gas networks but are in turn controlled via target values, also known as set-point values, themselves. Our models incorporate up to six dynamical target values to define desired transient states for the elements' local vicinity within the network. That is, each pair of every two target values defines a bounding box for the inlet pressure, outlet pressure as well as the passing mass flow of gas. In the proposed models, those target values are prioritized differently and are constantly in competition with each other, which can only be resolved dynamically at run-time of either a simulation or optimization process. Besides careful derivation, we compare simulation and optimization results with predictions of the commercial simulation tool SIMONE.
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/pdf
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  • 17
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    On the Numerical Treatment of Interlaced Target Values - Modeling, Optimization and Simulation of Regulating Valves in Gas Networks (2023)
    Details
    Publication Date: 2023-08-17
    Description: Due to the current and foreseeable shifts towards carbon dioxide neutral energy production, which will likely result in balancing fluctuating renewable energy generation by transforming power-to-gas-to-power as well as building a large-scale hydrogen transport infrastructure, the trading and transport operations of gas will become more dynamic, volatile, and hence also less predictable. Therefore, computer-aided support in terms of rapid simulation and control optimization will further broaden its importance for gas network dispatching. In this paper, we aim to contribute and openly publish two new mathematical models for regulators, also referred to as control valves, which together with compressors make up the most complex and involved types of active elements in gas network infrastructures. They provide direct control over gas networks but are in turn controlled via target values, also known as set-point values, themselves. Our models incorporate up to six dynamical target values to define desired transient states for the elements' local vicinity within the network. That is, each pair of every two target values defines a bounding box for the inlet pressure, outlet pressure as well as the passing mass flow of gas. In the proposed models, those target values are prioritized differently and are constantly in competition with each other, which can only be resolved dynamically at run-time of either a simulation or optimization process. Besides careful derivation, we compare simulation and optimization results with predictions of the widely adopted commercial simulation tool SIMONE, serving as our substitute for actual real-world transport operations.
    Language: English
    Type: article , doc-type:article
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    OPUS
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