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A quadratically constrained minimization problem arising from PDE of Monge–Ampère type

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Abstract

This note develops theory and a solution technique for a quadratically constrained eigenvalue minimization problem. This class of problems arises in the numerical solution of fully-nonlinear boundary value problems of Monge–Ampère type. Though it is most important in the three dimensional case, the solution method is directly applicable to systems of arbitrary dimension. The focus here is on solving the minimization subproblem which is part of a method to numerically solve a Monge–Ampère type equation. These subproblems must be evaluated many times in this numerical solution technique and thus efficiency is of utmost importance. A novelty of this minimization algorithm is that it is finite, of complexity \(\mathcal{O}(n^3)\), with the exception of solving a very simple rational function of one variable. This function is essentially the same for any dimension. This result is quite surprising given the nature of the constrained minimization problem.

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References

  1. Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. Am. Math. Soc., Providence, RI (1995)

    MATH  Google Scholar 

  2. Chang, A., Guan, P., Yang, P.: Monge–Ampère type equations and applications. BIRS workshop report. http://www.birs.ca/workshops/2003/03w5067/report03w5067.pdf (2003). Accessed 27 May 08

  3. Dean, E.J., Glowinski, R.: Numerical solution of the two-dimensional Monge–Ampère equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 339(12), 887–892 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Dean, E.J., Glowinski, R.: On the numerical solution of a two-dimensional Pucci’s equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I 341, 375–380 (2005)

    MATH  MathSciNet  Google Scholar 

  5. Fairlie, D.B., Leznov, A.N.: General solutions of the Monge–Ampère equation in n-dimensional space. J. Geom. Phys. 16, 385–390 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fairlie, D.B., Leznov, A.N.: General solutions of the complex Monge–Ampère equation in two dimensional space. http://arxiv.org/abs/solv-int/9909014/ Preprint August (1999). Accessed 27 May 08

  7. Glowinski, R., Dean, E.J., Guidoboni, G., Juarez, L.H., Pan, T.W.: Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two- dimensional Monge–Ampère equation. Jpn. J. Ind. Appl. Math. 25, 1–63 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gutierrez, C.: The Monge–Ampère Equation. Birkhauser, Boston, MA (2001)

    MATH  Google Scholar 

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Correspondence to Danny C. Sorensen.

Additional information

This work was supported in part by the NSF through Grants DMS-0412267, DMS-9972591, CCF-0634902 and ACI-0325081.

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Sorensen, D.C., Glowinski, R. A quadratically constrained minimization problem arising from PDE of Monge–Ampère type. Numer Algor 53, 53–66 (2010). https://doi.org/10.1007/s11075-009-9300-5

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  • DOI: https://doi.org/10.1007/s11075-009-9300-5

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