Numerik großer Differentiell‐Algebraischer Gleichungen – Simulation und Optimierung

Bock, Hans Georg; Schlder, J. P.; Schulz, Volker

doi:10.1002/9783527624867.ch2

Cited by 4 publications

(7 citation statements)

References 28 publications

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“…, y N for specified adaptive components h n and k n using automatic differentiation (AD) techniques either in forward or in adjoint mode. Adjoint IND was first described in [8] for Runge-Kutta integration schemes and later on in [10] for BDF methods. Applying adjoint IND to the BDF scheme (7) we obtain the discrete adjoint scheme…”

Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

“…In this section, we focus on the asymptotic behavior of the solutions of the discrete adjoint scheme (10). Therefore, we consider a nominal BDF method of constant order k with constant stepsizes h using a self-starting procedure for y 1 , .…”

Section: Convergence Analysis Of Classical Adjoints and Weak Adjointsmentioning

confidence: 99%

“…The discrete adjoints given by (10) are the exact derivatives of the nominal integration scheme (7) (beside round-off errors). Furthermore, for a BDF scheme with constant order k, the discrete adjoint λ 1 converges with the same order k to the value λ(t s ) of the adjoint solution of (2), cf.…”

Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

“…Due to the inconsistency of the discrete adjoint scheme (10) with the adjoint differential equation ( 2) discretization and optimization of (5) do not commute in the commonly used Hilbert space setting. This gives rise to the question for a new functional-analytic setting that is suitable for multistep methods.…”

Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

“…are equivalent to the BDF scheme (7) with prescribed stepsizes and orders together with its discrete adjoint scheme (10).…”

Section: Finite-dimensional Optimality Conditionsmentioning

confidence: 99%

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Approximation of weak adjoints by reverse automatic differentiation of BDF methods

et al. 2013

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With this contribution, we shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions. We devise a finite element Petrov-Galerkin interpretation of the BDF method together with its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the finite element approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its asymptotic convergence in the space of normalized functions of bounded variation. We also obtain asymptotic convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over the existing theory on global error estimation techniques from finite element methods to BDF methods.

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Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

Section: Convergence Analysis Of Classical Adjoints and Weak Adjointsmentioning

confidence: 99%

Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

Section: Adjoint Differentiation Of Bdf Integration Schemesmentioning

confidence: 99%

“…are equivalent to the BDF scheme (7) with prescribed stepsizes and orders together with its discrete adjoint scheme (10).…”

Section: Finite-dimensional Optimality Conditionsmentioning

confidence: 99%

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Approximation of weak adjoints by reverse automatic differentiation of BDF methods

et al. 2013

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Analyse und Anwendung dynamischer Optimierungsalgorithmen für Batch‐Prozesse

Wozny

1996

Chemie Ingenieur Technik

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Immer haufiger finden in der chemischen Industrie Batch-Prozesse Anwendungen. Der Vorteil von Batch-Prozessen liegt in der Flexibilitat, den erzielbaren Reinheiten, sowie in der Moglichkeit zu haufigem Produktwechsel. Zur wirtschaftlichen Fuhrung von Batch-Prozessen ist es notwendig, optimale Fuhrungsstrategien zu entwickeln. Hierzu sind geeignete Optimierungsalgorithmen erforderlich. Aus diesem Grund werden in dieser Arbeit zwei unterschiedliche Optimierungsalgorithmen untersucht, namlich die differentiell dynamische Programmierung und die simultane Optimierung mit dem orthogonalen Kollokationsverfahren. Es werden die unterschiedlichen Merkmale herausgearbeitet und die Leistungsfahigkeit analysiert. Simulationsergebnisse von Batch-Reaktoren und -Kolonnen zeigen die Anwendbarkeit der Algorithmen.

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Mathematical Aspects of CFD-based Optimization

Bock

Schulz

Optimization and Computational Fluid Dynamics

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Numerik großer Differentiell‐Algebraischer Gleichungen – Simulation und Optimierung

Cited by 4 publications

References 28 publications

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

Analyse und Anwendung dynamischer Optimierungsalgorithmen für Batch‐Prozesse

Mathematical Aspects of CFD-based Optimization

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