Approximation of weak adjoints by reverse automatic differentiation of BDF methods
| Autor: | Mario S. Mommer, Leonard Wirsching, Dörte Beigel, Hans Georg Bock |
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| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Backward differentiation formula
Automatic differentiation Differential equation 65L06 65L60 49K40 65L20 Applied Mathematics Numerical analysis Mathematical analysis Numerical Analysis (math.NA) Mathematics::Spectral Theory Mathematics::Numerical Analysis Computational Mathematics Adjoint equation Ordinary differential equation Bounded variation FOS: Mathematics Numerical differentiation Mathematics - Numerical Analysis Mathematics |
| Zdroj: | Numerische Mathematik |
| ISSN: | 0029-599X |
| DOI: | 10.1007/s00211-013-0570-4 |
| Popis: | 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. 29 pages |
| Databáze: | OpenAIRE |
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