@misc{ArnoldBertholdHeinzetal., author = {Arnold, Thomas and Berthold, Timo and Heinz, Stefan and Vigerske, Stefan and Henrion, Ren{\´e} and Gr{\"o}tschel, Martin and Koch, Thorsten and Tischendorf, Caren and R{\"o}misch, Werner}, title = {A Jack of all Trades? Solving stochastic mixed-integer nonlinear constraint programs}, series = {MATHEON - Mathematics for Key Technologies}, volume = {1}, journal = {MATHEON - Mathematics for Key Technologies}, editor = {Deuflhard, Peter and Gr{\"o}tschel, Martin and H{\"o}mberg, Dietmar and Horst, Ulrich and Kramer, J{\"u}rg and Mehrmann, Volker and Polthier, Konrad and Schmidt, Frank and Sch{\"u}tte, Christof and Skutella, Martin and Sprekels, J{\"u}rgen}, publisher = {European Mathematical Society}, doi = {10.4171/137}, pages = {135 -- 146}, abstract = {Natural gas is one of the most important energy sources in Germany and Europe. In recent years, political regulations have led to a strict separation of gas trading and gas transport, thereby assigning a central role in energy politics to the transportation and distribution of gas. These newly imposed political requirements influenced the technical processes of gas transport in such a way that the complex task of planning and operating gas networks has become even more intricate. Mathematically, the combination of discrete decisions on the configuration of a gas transport network, the nonlinear equations describing the physics of gas, and the uncertainty in demand and supply yield large-scale and highly complex stochastic mixed-integer nonlinear optimization problems. The Matheon project "Optimization of Gas Transport" takes the key role of making available the necessary core technology to solve the mathematical optimization problems which model the topology planning and the operation of gas networks. An important aspect of the academic impact is the free availability of our framework. As a result of several years of research and development, it is now possible to download a complete state-of-the-art framework for mixed-integer linear and nonlinear programming in source code at http://scip.zib.de}, language = {en} } @incollection{BennerGrundelHimpeetal.2019, author = {Benner, Peter and Grundel, Sara and Himpe, Christian and Huck, Christoph and Streubel, Tom and Tischendorf, Caren}, title = {Gas Network Benchmark Models}, series = {Applications of Differential-Algebraic Equations: Examples and Benchmarks}, booktitle = {Applications of Differential-Algebraic Equations: Examples and Benchmarks}, publisher = {Springer International Publishing}, isbn = {978-3-030-03718-5}, doi = {10.1007/11221_2018_5}, pages = {171 -- 197}, year = {2019}, abstract = {The simulation of gas transportation networks becomes increasingly more important as its use-cases broaden to more complex applications. Classically, the purpose of the gas network was the transportation of predominantly natural gas from a supplier to the consumer for long-term scheduled volumes. With the rise of renewable energy sources, gas-fired power plants are often chosen to compensate for the fluctuating nature of the renewables, due to their on-demand power generation capability. Such an only short-term plannable supply and demand setting requires sophisticated simulations of the gas network prior to the dispatch to ensure the supply of all customers for a range of possible scenarios and to prevent damages to the gas network. In this work we describe the modeling of gas networks and present benchmark systems to test implementations and compare new or extended models.}, 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} } @inproceedings{StreubelStrohmTrunschkeetal.2018, author = {Streubel, Tom and Strohm, Christian and Trunschke, Philipp and Tischendorf, Caren}, title = {Generic Construction and Efficient Evaluation of Network DAEs and Their Derivatives in the Context of Gas Networks}, series = {Operations Research Proceedings 2017}, booktitle = {Operations Research Proceedings 2017}, publisher = {Springer International Publishing}, isbn = {978-3-319-89920-6}, doi = {10.1007/978-3-319-89920-6_83}, pages = {627 -- 632}, year = {2018}, abstract = {We present a concept that provides an efficient description of differential-algebraic equations (DAEs) describing flow networks which provides the DAE function f and their Jacobians in an automatized way such that the sparsity pattern of the Jacobians is determined before their evaluation and previously determined values of f can be exploited. The user only has to provide the network topology and local function descriptions for each network element. The approach uses automatic differentiation (AD) and is adapted to switching element functions via the abs-normal-form (ANF).}, 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} }