Using AD to solve BVPs in MATLAB

ABSTRACT

The MATLAB program bvp4c solves two--point boundary value problems (BVPs) of considerable generality. The numerical method requires partial derivatives of several kinds. To make solving BVPs as easy as possible, the default in bvp4c is to approximate these derivatives with finite differences. The solver is more robust and efficient if analytical derivatives are supplied. In this article we investigate how to use automatic differentiation (AD) to obtain the advantages of analytical derivatives without giving up the convenience of finite differences. In bvp4cAD we have approached this ideal by a careful use of the MAD AD tool and some modification of bvp4c.

REFERENCES

• 1: Ascher, U., Mattheij, R., and Russell, R. 1995. Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia, PA, USA.
• 2: Bischof, C., Lang, B., and Vehreschild, A. 2003. Automatic differentiation for MATLAB programs. Proc. Appl. Math. Mech 2, 1, 50--53.
• 3: Bischof, C. H., Carle, A., Hovland, P. D., Khademi, P., and Mauer, A. 1998. ADIFOR 2.0 user's guide (Revision D). Tech. rep., Mathematics and Computer Science Division Technical Memorandum no. 192 and Center for Research on Parallel Computation Tech. Rep. CRPC-95516-S. See www.mcs.anl.gov/adifor.
• 4: C. H. Bischof , L. Roh , A. J. Mauer-Oats, ADIC: an extensible automatic differentiation tool for ANSI-C, Software—Practice & Experience, v.27 n.12, p.1427-1456, Dec. 1997 [doi>10.1002/(SICI)1097-024X(199712)27:12<1427::AID-SPE138>3.3.CO;2-H]
• 5: Thomas F. Coleman , Fadil Santosa , Arun Verma, Efficient calculation of Jacobian and adjoint vector products in the wave propagational inverse problem using automatic differentiation, Journal of Computational Physics, v.157 n.1, p.234-255, Jan. 1, 2000 [doi>10.1006/jcph.1999.6373]
• 6: Thomas F. Coleman , Arun Verma, ADMIT-1 : Automatic Differentiation and MATLAB Interface Toolbox, Cornell University, Ithaca, NY, 1998
• 7: Thomas F. Coleman , Arun Verma, The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation, SIAM Journal on Scientific Computing, v.19 n.4, p.1210-1233, July 1998 [doi>10.1137/S1064827595295349]
• 8: Thomas F. Coleman , Arun Verma, ADMIT-1: automatic differentiation and MATLAB interface toolbox, ACM Transactions on Mathematical Software (TOMS), v.26 n.1, p.150-175, March 2000 [doi>10.1145/347837.347879]
• 9: FastOpt 2003. Transformation of Algorithms in Fortran, Manual, Draft Version, TAF Version 1.6. FastOpt. See http://www.FastOpt.com/taf.
• 10: Forth, S. A. 2001. User guide for MAD - a Matlab automatic differentiation toolbox. Applied Mathematics and Operational Research Report AMOR 2001/5, Cranfield University (RMCS Shrivenham), Swindon, SN6 8LA, UK. June.
• 11: Andreas Griewank, Evaluating derivatives: principles and techniques of algorithmic differentiation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000
• 12: Andreas Griewank , David Juedes , Jean Utke, Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++, ACM Transactions on Mathematical Software (TOMS), v.22 n.2, p.131-167, June 1996 [doi>10.1145/229473.229474]
• 13: Jacek Kierzenka , Lawrence F. Shampine, A BVP solver based on residual control and the Maltab PSE, ACM Transactions on Mathematical Software (TOMS), v.27 n.3, p.299-316, September 2001 [doi>10.1145/502800.502801]
• 14: MathWorks. 2002. Matlab 6.5 release notes.
• 15: Matlab 2000. Matlab 6. The MathWorks, Inc., 3 Apple Hill Dr., Natick, MA 01760.
• 16: Pryce, J. D. and Reid, J. K. 1998. ADO1, a Fortran 90 code for automatic differentiation. Tech. Rep. RAL-TR-1998-057, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 OQX, England. See ftp://matisa.cc.rl.ac.uk/pub/reports/prRAL98057.ps.gz.
• 17: Lawrence C. Rich , David R. Hill, Automatic differentiation in MATLAB, Applied Numerical Mathematics, v.9 n.1, p.33-43, Jan. 1992 [doi>10.1016/0168-9274(92)90065-L]
• 18: Ian Gladwell , Larry Shampine , Skip Thompson, Solving ODEs with MATLAB, Cambridge University Press, New York, NY, 2003
• 19: Lawrence F. Shampine , Mark W. Reichelt, The MATLAB ODE Suite; , SIAM Journal on Scientific Computing, v.18 n.1, p.1-22, January. 1997 [doi>10.1137/S1064827594276424]
• 20: Shampine, L., Kierzenka, J., and Reichelt, M. n.d. Solving boundary value problems for ordinary differential equations in Matlab with bvp4c. See ftp://ftp.mathworks.com/pub/doc/papers/bvp/.
• 21: Tapenade. 2003. The TAPENADE tutorial http://www-sop.inria.fr/tropics/tapenade/tutorial.html. Web Site.
• 22: Vehreschild, A. 2001. Semantic augmentation of MATLAB programs to compute derivatives. M.S. thesis, Institute for Scientific Computing, Aachen University, Germany.
• 23: Arun Verma , Thomas F. Coleman, Structured automatic differentiation, 1998

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.4 Quadrature and Numerical Differentiation
          Subjects: Automatic differentiation

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.7 Ordinary Differential Equations
          Subjects: Boundary value problems
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Reliability and robustness; Efficiency

Keywords:
AD, BVP, Matlab