THE MONGE-AMP` ERE EQUATION: NOVEL FORMS AND NUMERICAL SOLUTION - PowerPoint PPT Presentation

THE MONGE-AMP` ERE EQUATION: NOVEL FORMS AND NUMERICAL SOLUTION V.A. Zheligovsky (vlad@mitp.ru) , O.M. Podvigina International Institute of Earthquake Prediction Theory and Mathematical Geophysics, 84/32 Profsoyuznaya St., Moscow, Russian Federation ; U. Frisch Observatoire de la Cˆ ote d’Azur, U.M.R. 6529, BP 4229, 06304 Nice Cedex 4, France Zheligovsky V., Podvigina O., Frisch U. The Monge–Amp` ere equation: various forms and numerical methods. J. Computational Physics , 229 , 2010, 5043-5061 [http://arxiv.org/abs/0910.1301]

Monge-Amp` ere equation: det � u x i ,x j � = f ( x ) • The “2nd order divergence” form • The Fourier integral form • The convolution integral form • A test problem with a cosmological flavour • Numerical solution: – Fixed point algorithm for the regular part of the MAE – Algorithm with continuation in a parameter and discrepancy minimisation – Algorithm with improvement of convexity and discrepancy minimisation • Results of solution of the test problem

Existence and regularity of solutions to the MAE: • A.V. Pogorelov. The Minkowski multidimensional problem. Nauka, Moscow, 1975. (Geometric approach, convexity.) • I.J. Bakelman. Convex analysis and nonlinear geometric elliptic equations. Springer-Verlag, 1994. • L.A. Caffarelli, X. Cabr´ e. Fully nonlinear elliptic equations. American Mathematical Society colloquium publications, vol. 43. Amer. Math. Soc., Providence, Rhode Island, 1995. (The PDE approach, viscous solutions.) Methods for discrete optimal transportation problem: ask Andrei Sobolevsky. Numerical algorithms linked to the geometric interpretation of the MAE:   2 • V.I. Oliker, L.D. Prussner. On the numerical solution of the equation ∂ 2 z ∂ 2 z  ∂ 2 z ∂y 2 −  = f ∂x 2 ∂x∂y and its discretizations, I. Numerische Mathematik, 54 (1988) 271–293. (A mesh comprised of 25 points.) • D. Michaelis, S. Kudaev, R. Steinkopf, A. Gebhardt, P. Schreiber, A. Br¨ auer. Incoherent beam shaping with freeform mirror. Nonimaging optics and efficient illumination systems V. Eds. R. Winston, R.J. Koshel. Proc. of SPIE, vol. 7059 (2008) 705905. (A 55 × 55 mesh, 15 min. of a Pentium 4.)

Application of algorithms for saddle-point optimisation to the two-dimensional MAE: • J.-D. Benamou, Y. Brenier. A computational fluid mechanics solution to the Monge– Kantorovich mass transfer problem. Numerische Mathematik, 84 (2000) 375–393. • E.J. Dean, R. Glowinski. Numerical solution of the two-dimensional elliptic Monge–Amp` ere equation with Dirichlet boundary conditions: an augmented Lagrangian approach. C. R. Acad. Sci. Paris, Ser. I, 336 (2003) 779–784. Various numerical approaches: • E.J. Dean, R. Glowinski. Numerical solution of the two-dimensional elliptic Monge–Amp` ere equation with Dirichlet boundary conditions: a least-squares approach. C. R. Acad. Sci. Paris, Ser. I, 339 (2004) 887–892. • E.J. Dean, R. Glowinski. Numerical methods for fully nonlinear elliptic equations of the Monge–Amp` ere type. Comput. Methods Appl. Mech. Engrg. 195 (2006) 1344–1386. • X. Feng, M. Neilan. Galerkin methods for the fully nonlinear Monge–Amp` ere equation (arXiv:0712.1240) . • X. Feng, M. Neilan. Mixed finite element methods for the fully nonlinear Monge–Amp` ere equation based on the vanishing moment method (arXiv:0712.1241) . • G. Loeper, F. Rapetti. Numerical solution of the Monge–Amp` ere equation by a Newton’s algorithm. C. R. Acad. Sci. Paris, Ser. I, 340 (2005) 319–324. (A pseudospectral Newton’s algorithm.)

• J.-D. Benamou, B.D. Froese, A.M. Oberman. Two numerical methods for the elliptic Mon- ge–Amp` ere equation. Preprint, 2009 [www.divbyzero.ca/froese/w/images/4/40/MA.pdf] . � u = ∇ − 2 u 2 x 1 x 1 + u 2 x 2 x 2 + 2 u 2 x 1 x 2 + 2 f An iterative Newton–Krylov solver with preconditioning (finite differences, modest accuracy required, discrepancies the order of 10 − 3 − 10 − 4 acceptable): • G.L. Delzanno, L. Chac´ on, J.M. Finn, Y. Chung, G. Lapenta. An optimal robust equidis- tribution method for two-dimensional grid adaptation based on Monge–Kantorovich opti- mization. J. Comput. Physics, 227 (2008) 9841–9864. ( 256 × 256 grid, contrast ratio = 8886 , discrepancy= 7 . 78 × 10 − 5 , 70 s. of a 2.4 GHz Intel Xeon processor.) • J.M. Finn, G.L. Delzanno, L. Chacon. Grid generation and adaptation by Monge–Kantoro- vich optimization in two and three dimensions. Proceedings of the 17th International Meshing Roundtable (2008) 551–568.

The “2nd order divergence” form � u ( ω ) e i ω · x d ω . A Fourier integral solution in R N : u = R N � det � a ij � ≡ 1 N � � ε i 1 ...i N ε j 1 ...j N a i n j n ( ε p 1 ...p N is the unit antisymmetric tensor of rank N ) N ! i 1 ,...,iN , n =1 j 1 ,...,jN det � u x i x j � = ( − 1) N � N � � u ( ω ) ω i n ω j n e i ω · x d ω ⇒ ε i 1 ...i N ε j 1 ...j N R N � N ! n =1 i 1 ,...,iN , j 1 ,...,jN     = ( − 1) N � � N N � � � � ω n ω n R N ...  ε i 1 ...i N   ε j 1 ...j N  i n j n N ! R N i 1 ,...,i N n =1 j 1 ,...,j N n =1     N N � � ω n · x u ( ω n )  exp  d ω 1 ... d ω N ×   i � n =1 n =1 � � � � = ( − 1) N � � N − 1 � � � R N det 2 � ω 1 , ..., ω N − 1 , ω − ω n R N ... � � � � N ! � n =1     N − 1 N − 1 � �  e i ω · x d ω 1 ... d ω N − 1 d ω u ( ω n ) ω n ×   � u  ω − � n =1 n =1 � � ω 1 , ..., ω N � � � � is the N × N matrix comprised of columnar vectors ω 1 , ..., ω N ) (

    = ( − 1) N � � N − 1 N − 1 R N det 2 � � � � � � � ω 1 , ..., ω N − 1 u ( ω n ) ω n  e i ω · x d ω 1 ... d ω N − 1 d ω . R N ... , ω   � u  ω − � � N ! n =1 n =1 “Reverse engineering” yields the “second-order divergence” form of the MAE in R N : � � 1 � ε i 1 ... i N ε j 1 ... j N u x i 1 x j 1 ... u x iN − 1 x jN − 1 u = f. N ! x iN x jN i 1 ,...,i N ,j 1 ,...,j N If u is space-periodic, and φ is a smooth function with a finite support, 1 � � � ε i 1 ... i N ε j 1 ... j N R N u x i 1 x j 1 ... u x iN − 1 x jN − 1 u φ x iN x jN d x = R 3 fφ d x . N ! i 1 ,...,iN , j 1 ,...,jN ∀ u ∈ W 2 N − 1 ( T N ) the integrals are ::::::::::::::::: well-defined :::: (by the Sobolev embedding theorem, ∇ u ∈ L 2( N − 1) ( T N ) ⇒ u ∈ L ∞ ( T N ) ). By contrast, integrals in the similar identity obtained by one integration by parts well-defined for u ∈ W 2 N − 1 ( T N ). are ::::: not :::: :::::::::::::::::

The Fourier integral form of the MAE � For a space-periodic solution, 0 = T 3 f d x . To accommodate f > 0 (of interest in cosmology), let u = c 2 | x | 2 + u � , � u � � = 0 . � 1 �·� denotes the average: � f � = lim [ − L,L ] 3 f ( x ) d x . (2 L ) 3 L →∞ c N = � f � ; f/c N = f ( ω ) e i ω · x d ω . � � R N ∇ 2 u � = � ϕ ( ω ) e i ω · x d ω , u � = � u � ( ω ) e i ω · x d ω � R N � R N u � ( ω ) = − � ϕ ( ω ) / | ω | 2 . ⇒ �

� � � � x i x j � = 1 � � R N det 2 det � u � R N ... � � i ω 1 , ..., i ω N − 1 , i ω − � N − 1 � � n =1 ω n N !     N − 1 N − 1 � �  e i ω · x d ω 1 ... d ω N − 1 d ω ϕ ( ω n ) ω n ×   � ϕ  ω − � n =1 n =1 ( i a is a unit vector in the direction of a ). Consider the term of order m < N in u � :   ( − 1) m � � � � � u � ( ω ) ω i n ω j n e i ω · x d ω ε i 1 ...i N ε j 1 ...j N   δ i n j n R N � N ! i 1 ,...,i N ,j 1 ,...,j N | σ | = m n : i n ,j n ∈ σ n : i n or j n ∈ /σ � (the sum | σ | = m is over all subsets σ ⊂ { 1 , ..., N } of cardinality m ; δ i n j n is the Kronecker symbol)   2 = ( − 1) m � � m � � � ω n R N ...  ε j 1 ...j m p 1 ...p N − m  j n m ! R N 1 ≤ p 1 <...<p N − m ≤ N j 1 ,...,j m n =1     m m � � ω n · x  exp  d ω 1 ... d ω m u � ( ω n ) ×  i  � n =1 n =1 � � � � = R N ... R N A m i ω 1 , ..., i ω m − 1 , i ω − � m − 1 n =1 ω n     m − 1 m − 1 � �  e i ω · x d ω 1 ... d ω m − 1 d ω , ϕ ( ω n ) ω n ×   � ϕ  ω − � n =1 n =1

A m ( i 1 , ..., i m ) ≡ 1 � M 2 p 1 ...p N − m ( i 1 , ..., i m ) where m ! 1 ≤ p 1 <...<p N − m ≤ N is the sum of squares of all minors of rank m , � M p 1 ...p N − m ( i 1 , ..., i m ) ≡ ε j 1 ...j m p 1 ...p N − m ( i 1 ) j 1 ... ( i m ) j m , j 1 ,...,j m obtained by crossing out rows of numbers p 1 < ... < p N − m from the N × m matrix � i 1 , ..., i m � � � � M m ≡ � , comprised of m columnar vectors i 1 , ..., i m . The Fourier integral form of the MAE: � � � � N � ϕ ( ω ) + R N ... R N A m i ω 1 , ..., i ω m − 1 , i ω − � m − 1 � n =1 ω n m =2     m − 1 m − 1 � �  d ω 1 ... d ω m − 1 = � ϕ ( ω n ) ω n ×   � ϕ  ω − f ( ω ) , ∀ ω � = 0 . � n =1 n =1