Solving partial differential equations via sparse SDP relaxations

Pacific Journal of Optimization

Volumn 4, Issue 2, 2008, Pages 213-241

  Mevissen, Martin ^a   Kojima, Masakazu ^a   Nie, Jiawang ^b   Takayama, Nobuki ^c  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c KOBE UNIVERSITY   (Japan)

Author keywords

Differential algebraic equation; Optimal control; Ordinary differential equation; Partial differential equation; Polynomial optimization; Semidefinite programming relaxation; Sparsity

Indexed keywords

EID: 77950444358     PISSN: 13489151     EISSN: 13498169     Source Type: Journal    
DOI: None     Document Type: Article

References (22)

1. J.R.S. Blair and B. Peyton, An introduction to chordal graphs and clique trees, inGraph Theory and Sparse Matrix Computation, A. George, J.R. Gilbert and J.W. Liu(eds.), Springer-Verlag, New York, 1993, pp. 1-29.
2. W.L. Briggs, V.E. Henson and S.F. McCormick, A Multigrid Tutorial, SIAM, Philadelphia,2000.
3. R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol 1 (1931), Chapter4, The method of variation.
4. A. George and J.W. Liu, Computer Solution of Large Sparse Positive Definite Systems,Prentice-Hall, Englewood Cliffs, N.J., 1981.
5. J.L. Guermond, A finite element technique for solving first-order PDEs in LP, SIAMJ. Numerical Anal. 42 (2004) 714-737.
6. J.L. Guermond, and B. Popov, Linear advection with ill-posed boundary conditions viaL1-minimization, Int. Numer. Anal. Modeling 4 (2007) 39-47
7. T. Gunji, S. Kim, M. Kojima, A. Takeda, K. Fujisawa and T. Mizutani, PHo M - a Polyhedral Homotopy Continuation Method for Polynomial Systems, Research Reportson Mathematical and Computing Sciences, Dept. of Math. and Comp. Sciences, Tokyo Inst. of Tech., B-386 (2003).
8. S. Kim, M. Kojima and H. Waki, Generalized Lagrangian duals and sums of squaresrelaxations of sparse polynomial optimization problem, SIAM J. Optim. 15 (2005) 697-719.
9. M. Kojima, S. Kim and H. Waki, Sparsity in sums of squares of polynomials, Math.Program. 103 (2005) 45-62.
10. J.B. Lasserre, Global optimization with polynomials and the problem of moments,SIAM J. Optim. 11 (2001) 796-817.
11. J.B. Lasserre, Convergent SDP-relaxations in polynomial optimization with sparsity,SIAM J. Optim. 17 (2006) 822-843.
12. J. Macki and A. Strauss, Introduction to Optimal Control Theory, Springer-Verlag, New York, 1982.
13. M. Mimura, Asymptotic behaviors of a parabolic system related to a planktonic preyand predator model, SIAM J. Applied Mathematics 37 (1979) 499-512.
14. J. Nocedal and S.J. Wright, Numerical Optimization, Series in Operations Research,Springer, New York, 2006.
15. J. Rauch and J. Smoller, Qualitative theory of the Fitz Hugh-Nagumo equations, Advancesin Mathematics 27 (1978) 12-44.
16. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York,1983.
17. Sparse POP Home Page, http://www.is.titech.ac.jp/~kojima/SparsePOP.
18. M. Tabata, A finite difference approach to the number of peaks of solutions for semilinearparabolic problems, J. Math. Soc. Japan 32 (1980) 171-192.
19. U. Trottenberg, C.W. Oostelee and A. Schüller, Multigrid, Academic Press, London2001.
20. H. Waki, S. Kim, M. Kojima and M. Muramatsu, Sums of squares and semidefiniteprogram relaxations for polynomial optimization problems with structured sparsity,SIAM J. Optim. 17 (2006) 218-242.
21. H. Waki, S. Kim, M. Kojima and M. Muramatsu, Sparse POP: a Sparse Semidefinite Programming Relaxation of Polynomial Optimization Problems, Research Reports on Mathematical and Computing Sciences, Dept. of Math. and Comp. Sciences, Tokyo Inst. of Tech., B-414 (2005).
22. H. Yokota, Introduction to Numerical Analysis II, web book (in Japanese),http://next1.cc.it-hiroshima.ac.jp/MULTIMEDIA/numeanal2/node24.html.