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Volumn 12, Issue , 2003, Pages 321-398

A mathematical view of automatic differentiation

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EID: 24644437042     PISSN: 09624929     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0962492902000132     Document Type: Review
Times cited : (93)

References (99)
  • 2
    • 0001061872 scopus 로고
    • Numerical differentiation proceedings for non-exact data
    • R. Anderssen and P. Bloomfield (1974), 'Numerical differentiation proceedings for non-exact data', Numer. Math. 22, 157-182.
    • (1974) Numer. Math. , vol.22 , pp. 157-182
    • Anderssen, R.1    Bloomfield, P.2
  • 4
    • 85022001969 scopus 로고    scopus 로고
    • An optimal control approach to error control and mesh adaptation in finite element methods
    • Cambridge University Press
    • R. Becker and R. Rannacher (2001), An optimal control approach to error control and mesh adaptation in finite element methods, in Acta Numerica, Vol. 10, Cambridge University Press, pp. 1-102.
    • (2001) Acta Numerica , vol.10 , pp. 1-102
    • Becker, R.1    Rannacher, R.2
  • 5
    • 0015680909 scopus 로고
    • Logical reversibility of computation
    • C. Bennett (1973), 'Logical reversibility of computation', IBM J. Research Development 17, 525-532.
    • (1973) IBM J. Research Development , vol.17 , pp. 525-532
    • Bennett, C.1
  • 7
    • 0027903541 scopus 로고
    • Structured second- and higher-order derivatives through univariate Taylor series
    • C. Bischof, G. Corliss and A. Griewank (1993), 'Structured second- and higher-order derivatives through univariate Taylor series', Optim. Methods Software 2, 211-232.
    • (1993) Optim. Methods Software , vol.2 , pp. 211-232
    • Bischof, C.1    Corliss, G.2    Griewank, A.3
  • 8
    • 0030245387 scopus 로고    scopus 로고
    • The ADIFOR 2.0 system for the automatic differentiation of Fortran 77 programs
    • C. Bischof, A. Carle, P. Khademi and A. Mauer (1996), 'The ADIFOR 2.0 system for the automatic differentiation of Fortran 77 programs', IEEE Comput. Sci. Engr. 3, 18-32.
    • (1996) IEEE Comput. Sci. Engr. , vol.3 , pp. 18-32
    • Bischof, C.1    Carle, A.2    Khademi, P.3    Mauer, A.4
  • 10
    • 0019032178 scopus 로고
    • Sensitivity theory for general systems of nonlinear equations
    • D. Cacuci, C. Weber, E. Oblow and J. Marable (1980), 'Sensitivity theory for general systems of nonlinear equations', Nucl. Sci. Engr. 88, 88-110.
    • (1980) Nucl. Sci. Engr. , vol.88 , pp. 88-110
    • Cacuci, D.1    Weber, C.2    Oblow, E.3    Marable, J.4
  • 13
    • 0028436529 scopus 로고
    • Utilization of automatic differentiation in control algorithms
    • S. Campbell, E. Moore and Y. Zhong (1994), 'Utilization of automatic differentiation in control algorithms', IEEE Trans. Automatic Control 39, 1047-1052.
    • (1994) IEEE Trans. Automatic Control , vol.39 , pp. 1047-1052
    • Campbell, S.1    Moore, E.2    Zhong, Y.3
  • 16
    • 0000187532 scopus 로고
    • Automatic Hessians by reverse accumulation
    • B. Christianson (1992), 'Automatic Hessians by reverse accumulation', IMA J. Numer. Anal 12, 135-150.
    • (1992) IMA J. Numer. Anal , vol.12 , pp. 135-150
    • Christianson, B.1
  • 17
    • 0028698107 scopus 로고
    • Reverse accumulation and attractive fixed points
    • B. Christianson (1994), 'Reverse accumulation and attractive fixed points', Optim. Methods Software 3, 311-326.
    • (1994) Optim. Methods Software , vol.3 , pp. 311-326
    • Christianson, B.1
  • 18
    • 0031639751 scopus 로고    scopus 로고
    • Reverse accumulation and implicit functions
    • B. Christianson (1998), 'Reverse accumulation and implicit functions', Optim. Methods Software 9, 307-322.
    • (1998) Optim. Methods Software , vol.9 , pp. 307-322
    • Christianson, B.1
  • 19
    • 0032655686 scopus 로고    scopus 로고
    • Cheap Newton steps for optimal control problems: Automatic differentiation and Pantoja's algorithm
    • B. Christianson (1999), 'Cheap Newton steps for optimal control problems: Automatic differentiation and Pantoja's algorithm', Optim. Methods Software 10, 729-743.
    • (1999) Optim. Methods Software , vol.10 , pp. 729-743
    • Christianson, B.1
  • 20
    • 0030711856 scopus 로고    scopus 로고
    • Automatic differentiation of computer programs in a parallel computing environment
    • (H. Power and J. C. Long, eds), Computational Mechanics Publications, Southampton
    • B. Christianson, L. Dixon and S. Brown (1997), Automatic differentiation of computer programs in a parallel computing environment, in Applications of High Performance Computing in Engineering V (H. Power and J. C. Long, eds), Computational Mechanics Publications, Southampton, pp. 169-178.
    • (1997) Applications of High Performance Computing in Engineering V , pp. 169-178
    • Christianson, B.1    Dixon, L.2    Brown, S.3
  • 21
    • 0000421410 scopus 로고
    • Estimation of sparse Jacobian matrices and graph coloring problems
    • T. Coleman and J. Moré (1984), 'Estimation of sparse Jacobian matrices and graph coloring problems', SIAM J. Numer. Anal. 20, 187-209.
    • (1984) SIAM J. Numer. Anal. , vol.20 , pp. 187-209
    • Coleman, T.1    Moré, J.2
  • 23
    • 0003709683 scopus 로고
    • LANCELOT, a fortran package for large-0 scale nonlinear optimization (release A)
    • Springer, Berlin
    • A. Conn, N. Gould and P. Toint (1992), LANCELOT, a Fortran Package for Large-0 Scale Nonlinear Optimization (Release A), Vol. 17 of Computational Mathematics, Springer, Berlin.
    • (1992) Computational Mathematics , vol.17
    • Conn, A.1    Gould, N.2    Toint, P.3
  • 26
    • 24644447435 scopus 로고    scopus 로고
    • Reverse automatic differentiation for optimum design: From adjoint state assembly to gradient computation
    • INRIA Sophia-Antipolis
    • F. Courty, A. Dervieux, B. Koobus and L. Hascoët (2002), Reverse automatic differentiation for optimum design: From adjoint state assembly to gradient computation, Technical report, INRIA Sophia-Antipolis.
    • (2002) Technical Report
    • Courty, F.1    Dervieux, A.2    Koobus, B.3    Hascoët, L.4
  • 27
    • 84952419741 scopus 로고
    • On the estimation of sparse Jacobian matrices
    • A. Curtis, M. Powell and J. Reid (1974), 'On the estimation of sparse Jacobian matrices', J. Inst. Math. Appl. 13, 117-119.
    • (1974) J. Inst. Math. Appl. , vol.13 , pp. 117-119
    • Curtis, A.1    Powell, M.2    Reid, J.3
  • 29
    • 0002473211 scopus 로고
    • Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems
    • J. Dunn and D. Bertsekas (1989), 'Efficient dynamic programming implementations of Newton's method for unconstrained optimal control problems', J. Optim. Theory Appl. 63, 23-38.
    • (1989) J. Optim. Theory Appl. , vol.63 , pp. 23-38
    • Dunn, J.1    Bertsekas, D.2
  • 30
    • 0001666357 scopus 로고
    • Note sur une nouvelle formule de calcule differentiel
    • Faa di Bruno (1856), 'Note sur une nouvelle formule de calcule differentiel', Quar. J. Math. 1, 359-360.
    • (1856) Quar. J. Math. , vol.1 , pp. 359-360
    • Bruno, F.D.1
  • 31
  • 34
  • 35
    • 24644501323 scopus 로고    scopus 로고
    • On the performance of derivative code generated by TAMC
    • Manuscript, FastOpt, Hamburg, Germany. Submitted to
    • R. Giering and T. Kaminski (2000), On the performance of derivative code generated by TAMC. Manuscript, FastOpt, Hamburg, Germany. Submitted to Optim. Methods Software See www.FastOpt.de/papers/perftamc.ps.gz.
    • (2000) Optim. Methods Software
    • Giering, R.1    Kaminski, T.2
  • 38
    • 0026855130 scopus 로고
    • Automatic differentiation and iterative processes
    • J. C. Gilbert (1992), 'Automatic differentiation and iterative processes', Optim. Methods Software 1, 13-21.
    • (1992) Optim. Methods Software , vol.1 , pp. 13-21
    • Gilbert, J.C.1
  • 40
    • 0034433978 scopus 로고    scopus 로고
    • An introduction to the adjoint approach to design
    • M. B. Giles and N. A. Pierce (2001), 'An introduction to the adjoint approach to design', Flow, Turbulence and Combustion 65, 393-415.
    • (2001) Flow, Turbulence and Combustion , vol.65 , pp. 393-415
    • Giles, M.B.1    Pierce, N.A.2
  • 41
    • 85021895182 scopus 로고    scopus 로고
    • Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality
    • Cambridge University Press
    • M. B Giles and E. Süli (2002), Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, in Acta Numerica, Vol. 11, Cambridge University Press, pp. 145-236.
    • (2002) Acta Numerica , vol.11 , pp. 145-236
    • Giles, M.B.1    Süli, E.2
  • 42
    • 0012614704 scopus 로고    scopus 로고
    • C++ classes for linking optimization with complex simulations
    • M. Gockenbach, M. J. Petro and W. Symes (1999), 'C++ classes for linking optimization with complex simulations', ACM Trans. Math. Software 25, 191-212.
    • (1999) ACM Trans. Math. Software , vol.25 , pp. 191-212
    • Gockenbach, M.1    Petro, M.J.2    Symes, W.3
  • 43
    • 0003319841 scopus 로고    scopus 로고
    • Evaluating derivatives, principles and techniques of algorithmic differentiation
    • SIAM, Philadelphia
    • A. Griewank (2000), Evaluating Derivatives, Principles and Techniques of Algorithmic Differentiation, Vol. 19 of Frontiers in Applied Mathematics, SIAM, Philadelphia.
    • (2000) Frontiers in Applied Mathematics , vol.19
    • Griewank, A.1
  • 45
    • 0141738320 scopus 로고    scopus 로고
    • Reduced functions, gradients and Hessians from fixed point iteration for state equations
    • A. Griewank and C. Faure (2002), 'Reduced functions, gradients and Hessians from fixed point iteration for state equations', Numer. Alg. 30, 113-139.
    • (2002) Numer. Alg. , vol.30 , pp. 113-139
    • Griewank, A.1    Faure, C.2
  • 46
    • 10644238699 scopus 로고    scopus 로고
    • Detecting Jacobian sparsity patterns by Bayesian probing
    • A. Griewank and C. Mitev (2002), 'Detecting Jacobian sparsity patterns by Bayesian probing', Math. Prog. 93, 1-25.
    • (2002) Math. Prog. , vol.93 , pp. 1-25
    • Griewank, A.1    Mitev, C.2
  • 47
    • 33751118192 scopus 로고    scopus 로고
    • Accumulating Jacobians by vertex, edge or face elimination
    • (L. Andriamampianina, B. Philippe, E. Kamgnia and M. Tchuente, eds), Imprimerie Saint Paul, Camerun and INRIA, France
    • A. Griewank and U. Naumann (2003a), Accumulating Jacobians by vertex, edge or face elimination, in Proceedings of the 6th African Conference on Research in Computer Science (L. Andriamampianina, B. Philippe, E. Kamgnia and M. Tchuente, eds), Imprimerie Saint Paul, Camerun and INRIA, France, pp. 375-383.
    • (2003) Proceedings of the 6th African Conference on Research in Computer Science , pp. 375-383
    • Griewank, A.1    Naumann, U.2
  • 48
    • 21044440826 scopus 로고    scopus 로고
    • Accumulating Jacobians as chained sparse matrix products
    • A. Griewank and U. Naumann (2003b), 'Accumulating Jacobians as chained sparse matrix products', Math. Prog. 95, 555-571.
    • (2003) Math. Prog. , vol.95 , pp. 555-571
    • Griewank, A.1    Naumann, U.2
  • 50
    • 0010666451 scopus 로고    scopus 로고
    • Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation
    • A. Griewank and A. Walther (2000), 'Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation', Trans. Math. Software 26, 19-45.
    • (2000) Trans. Math. Software , vol.26 , pp. 19-45
    • Griewank, A.1    Walther, A.2
  • 52
    • 0034402962 scopus 로고    scopus 로고
    • Evaluating higher derivative tensors by forward propagation of univariate Taylor series
    • A. Griewank, J. Utke and A. Walther (2000), 'Evaluating higher derivative tensors by forward propagation of univariate Taylor series', Math. Comp. 69, 1117-1130.
    • (2000) Math. Comp. , vol.69 , pp. 1117-1130
    • Griewank, A.1    Utke, J.2    Walther, A.3
  • 55
    • 0242616934 scopus 로고    scopus 로고
    • The data-dependence graph of adjoint programs
    • INRIA, Sophia-Antipolis, France
    • L. Hascoët (2001), The data-dependence graph of adjoint programs, Research Report 4167, INRIA, Sophia-Antipolis, France. See www.inria.fr/rrrt/ rr-4167.html.
    • (2001) Research Report , vol.4167
    • Hascoët, L.1
  • 60
    • 0032292043 scopus 로고    scopus 로고
    • Computing a sparse Jacobian matrix by rows and columns
    • A. S. Hossain and T. Steihaug (1998), 'Computing a sparse Jacobian matrix by rows and columns', Optim. Methods Software 10, 33-48.
    • (1998) Optim. Methods Software , vol.10 , pp. 33-48
    • Hossain, A.S.1    Steihaug, T.2
  • 61
    • 24644509369 scopus 로고    scopus 로고
    • Sparsity issues in the computation of Jacobian matrices
    • Department of Computer Science, University of Bergen, Norway
    • S. Hossain and T. Steihaug (2002), Sparsity issues in the computation of Jacobian matrices, Technical Report 223, Department of Computer Science, University of Bergen, Norway.
    • (2002) Technical Report , vol.223
    • Hossain, S.1    Steihaug, T.2
  • 62
    • 38249027744 scopus 로고
    • Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations
    • M. Iri, T. Tsuchiya and M. Hoshi (1988), 'Automatic computation of partial derivatives and rounding error estimates with applications to large-scale systems of nonlinear equations', Comput. Appl. Math. 24, 365-392.
    • (1988) Comput. Appl. Math. , vol.24 , pp. 365-392
    • Iri, M.1    Tsuchiya, T.2    Hoshi, M.3
  • 63
    • 0000618224 scopus 로고
    • Original Japanese version appeared in J. Information Processing 26 (1985), 1411-1420.
    • (1985) J. Information Processing , vol.26 , pp. 1411-1420
  • 64
    • 50149099813 scopus 로고
    • Ob odnoj matematicheskoj simvolike, udobnoj pri provedenii vychislenij na mashinakh
    • L. Kantorovich (1957), 'Ob odnoj matematicheskoj simvolike, udobnoj pri provedenii vychislenij na mashinakh', Doklady Akademii Nauk. SSSR 113, 738-741.
    • (1957) Doklady Akademii Nauk. SSSR , vol.113 , pp. 738-741
    • Kantorovich, L.1
  • 65
    • 0019029712 scopus 로고
    • Automatic differentiation of computer programs
    • G. Kedem (1980), 'Automatic differentiation of computer programs', ACM Trans. Math. Software 6, 150-165.
    • (1980) ACM Trans. Math. Software , vol.6 , pp. 150-165
    • Kedem, G.1
  • 69
    • 0021004359 scopus 로고
    • Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms
    • S. Linnainmaa (1983), 'Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms', BIT 23, 346-359.
    • (1983) BIT , vol.23 , pp. 346-359
    • Linnainmaa, S.1
  • 70
    • 24644490242 scopus 로고
    • with examples, Habilitationsschrift, Institute for Applied Mathematics, University of Karlsruhe
    • R. Lohner (1992), Verified computing and programs in PASCAL-XSC, with examples, Habilitationsschrift, Institute for Applied Mathematics, University of Karlsruhe.
    • (1992) Verified Computing and Programs in PASCAL-XSC
    • Lohner, R.1
  • 73
    • 1542373167 scopus 로고    scopus 로고
    • Sensitivity analysis and real-time control of parametric optimal control problems using boundary value methods
    • (M. Grötschel, S. Krumke and J. Rambau, eds), Springer, Berlin/Heidelberg/New York, Chapter I
    • H. Maurer and D. Augustin (2001), Sensitivity analysis and real-time control of parametric optimal control problems using boundary value methods, in Online Optimization of Large Scale Systems (M. Grötschel, S. Krumke and J. Rambau, eds), Springer, Berlin/Heidelberg/New York, Chapter I, pp. 17-55.
    • (2001) Online Optimization of Large Scale Systems , pp. 17-55
    • Maurer, H.1    Augustin, D.2
  • 78
    • 24644435068 scopus 로고    scopus 로고
    • Directions for computing multivariate Taylor series
    • (200x), Technical report, Davidson College, Davidson. To appear in
    • R. Neidinger (200x), Directions for computing multivariate Taylor series. Technical report, Davidson College, Davidson. To appear in Math. Comp.
    • Math. Comp.
    • Neidinger, R.1
  • 79
    • 0345341465 scopus 로고
    • Observations on computational methodologies for use in large-scale, gradient-based, multi-disciplinary design incorporating advanced CFD codes
    • Technical Memorandum 104206, NASA Langley Research Center.
    • P. Newman, G.-W. Hou, H. Jones, A. Taylor and V. Korivi (1992), Observations on computational methodologies for use in large-scale, gradient-based, multi-disciplinary design incorporating advanced CFD codes, Technical Memorandum 104206, NASA Langley Research Center. AVSCOM Technical Report 92-B-007.
    • (1992) AVSCOM Technical Report , vol.92 B-007
    • Newman, P.1    Hou, G.-W.2    Jones, H.3    Taylor, A.4    Korivi, V.5
  • 82
    • 0000353268 scopus 로고
    • The consistent initialization of differential-algebraic systems
    • C. Pantelides (1988), 'The consistent initialization of differential-algebraic systems', SIAM J. Sci. Statist. Comput. 9, 213-231.
    • (1988) SIAM J. Sci. Statist. Comput. , vol.9 , pp. 213-231
    • Pantelides, C.1
  • 83
    • 0024014522 scopus 로고
    • Differential dynamic programming and Newton's method
    • J. D. O. Pantoja (1988), 'Differential dynamic programming and Newton's method', Internat. J. Control 47, 1539-1553.
    • (1988) Internat. J. Control , vol.47 , pp. 1539-1553
    • Pantoja, J.D.O.1
  • 84
    • 0032262788 scopus 로고    scopus 로고
    • Solving high-index DAEs by Taylor series
    • J. Pryce (1998), 'Solving high-index DAEs by Taylor series', Numer. Alg. 19, 195-211.
    • (1998) Numer. Alg. , vol.19 , pp. 195-211
    • Pryce, J.1
  • 85
    • 0038292580 scopus 로고
    • Differentiation and generation of Taylor coefficients in Pascal-SC
    • (U. Kulisch and W. L. Miranker, eds), Academic Press, New York
    • L. Rail (1983), Differentiation and generation of Taylor coefficients in Pascal-SC, in A New Approach to Scientific Computation (U. Kulisch and W. L. Miranker, eds), Academic Press, New York, pp. 291-309.
    • (1983) A New Approach to Scientific Computation , pp. 291-309
    • Rail, L.1
  • 86
    • 0001126902 scopus 로고    scopus 로고
    • Circumventing storage limitations in variational data assimilation studies
    • J. Restrepo, G. Leaf and A. Griewank (1998), 'Circumventing storage limitations in variational data assimilation studies', SIAM J. Sci. Comput. 19, 1586-1605.
    • (1998) SIAM J. Sci. Comput. , vol.19 , pp. 1586-1605
    • Restrepo, J.1    Leaf, G.2    Griewank, A.3
  • 87
    • 33747698416 scopus 로고    scopus 로고
    • Graph-theoretic characterization of structural controllability for singular DAE
    • K. Röbenack and K. Reinschke (2000), 'Graph-theoretic characterization of structural controllability for singular DAE', Z. Angew. Math. Mech. 8 Suppl. 1, 849-850.
    • (2000) Z. Angew. Math. Mech. , vol.8 , Issue.1 SUPPL. , pp. 849-850
    • Röbenack, K.1    Reinschke, K.2
  • 89
    • 0001435054 scopus 로고
    • Algorithmic aspects of vertex elimination on directed graphs
    • D. Rose and R. Tarjan (1978), 'Algorithmic aspects of vertex elimination on directed graphs', SIAM J. Appl. Math. 34, 177-197.
    • (1978) SIAM J. Appl. Math. , vol.34 , pp. 177-197
    • Rose, D.1    Tarjan, R.2
  • 91
    • 4244177441 scopus 로고
    • Optimal error estimates for Gaussian elimination in floating-point arithmetic
    • F. Stummel (1981), 'Optimal error estimates for Gaussian elimination in floating-point arithmetic', GAMM, Numerical Analysis 62, 355-357.
    • (1981) GAMM, Numerical Analysis , vol.62 , pp. 355-357
    • Stummel, F.1
  • 95
    • 0000238006 scopus 로고    scopus 로고
    • Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one-dimensional flow
    • D. Venditti and D. Darmofal (2000), 'Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow', J. Comput. Phys. 104, 204-227.
    • (2000) J. Comput. Phys. , vol.104 , pp. 204-227
    • Venditti, D.1    Darmofal, D.2
  • 96
    • 0022161487 scopus 로고
    • Automatic computation of derivatives with the use of the multilevel differentiating technique, I: Algorithmic basis
    • Y. Volin and G. Ostrovskii (1985), 'Automatic computation of derivatives with the use of the multilevel differentiating technique, I: Algorithmic basis', Comput. Math. Appl. 11, 1099-1114.
    • (1985) Comput. Math. Appl. , vol.11 , pp. 1099-1114
    • Volin, Y.1    Ostrovskii, G.2
  • 99
    • 2442601132 scopus 로고    scopus 로고
    • Introduction to adjoint techniques and the MM5 adjoint modeling system
    • Mesocale and Microscale Meteorology Division, National Center for Atmospheric Research, Boulder, Colorado
    • X. Zou, F. Vandenberghe, M. Pondeca and Y.-H. Kuo (1997), Introduction to adjoint techniques and the MM5 adjoint modeling system, NCAR Technical Note NCAR/TN-435-STR, Mesocale and Microscale Meteorology Division, National Center for Atmospheric Research, Boulder, Colorado.
    • (1997) NCAR Technical Note , vol.NCAR-TN-435-STR
    • Zou, X.1    Vandenberghe, F.2    Pondeca, M.3    Kuo, Y.-H.4


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