Discrete complex analysis Convergence results M. Skopenkov 123 - PowerPoint PPT Presentation

Discrete complex analysis Convergence results M. Skopenkov 123 joint work with A. Bobenko 1 National Research University Higher School of Economics 2 Institute for Information Transmission Problems RAS 3 King Abdullah University of Science and Technology Embedded graphs, St. Petersburg, 27–31.10.2014 M. Skopenkov Discrete complex analysis

Discretizations of complex analysis Discrete complex analysis ւ ↓ ց z 2 Q z 2 z 3 z 1 z 1 z 3 z 4 . . . f ( z 1 ) − f ( z 3 ) = f ( z 2 ) − f ( z 4 ) . . . f ( z 1 ) + f ( z 2 ) + f ( z 3 ) = 0 z 1 − z 3 z 2 − z 4 Dynnikov–Novikov Isaacs, Ferrand, . . . Thurston ↓ ↓ ↓ integrable systems numerical analysis conformal network theory geometry statistical physics M. Skopenkov Discrete complex analysis

Overview 1 Discrete analytic functions in a planar domain 2 Discrete analytic functions in a Riemann surface 3 Convergence via energy estimates M. Skopenkov Discrete complex analysis

1 Discrete analytic functions in a planar domain M. Skopenkov Discrete complex analysis

Main definitions z 2 A graph Q ⊂ C is a quadrilateral lattice ⇔ each bounded face is a quadrilateral Q A function f : Q → C is discrete analytic ⇔ z 3 f ( z 1 ) − f ( z 3 ) = f ( z 2 ) − f ( z 4 ) z 1 z 1 − z 3 z 2 − z 4 for each face z 1 z 2 z 3 z 4 with the vertices listed z 4 clockwise. Re f is called discrete harmonic. Q Q Q square lattice rhombic lattice orthogonal lattice Isaacs,Ferrand (1940s) Duffin (1960s) Mercat (2000s) M. Skopenkov Discrete complex analysis

The Dirichlet boundary value problem Problem. Prove convergence of discrete harmonic functions to their continuous counterparts as h → 0 . Square lattices , C 0 : Lusternik, 1926. Square lattices , C ∞ : Courant–Friedrichs–Lewy, 1928. Rhombic lattices , C 0 : Ciarlet–Raviart, 1973 (implicitly). Rhombic lattices , C 1 : Chelkak–Smirnov, 2008. ∂ Ω Ω The Dirichlet problem in a domain Ω is to find a continuous function u Ω , g : Cl Ω → R having given boundary values g : ∂ Ω → R ∂ Q and such that ∆ u Ω , g = 0 in Ω . Q The Dirichlet problem on Q is to find a discrete harmonic function u Q , g : Q → R having given boundary values g : ∂ Q → R . M. Skopenkov Discrete complex analysis

Existence and Uniqueness Theorem Existence and Uniqueness Theorem (S. 2011). The Dirichlet problem on any finite quadrilateral lattice has a unique solution . Example (Tikhomirov, 2011): no maximum principle! 0 0 1 0 M 0 0 1 0 √ √ ± cot π 2 M (cot π 2 M (cot π z 0 ± i ± 8 + i ) ± 8 − i ) 8 f ( z ) M (1 + i ) 1 0 0 2 Mi Re f ( z ) M 1 0 0 0 Both f ( z ) and the shape of Q depends on a prameter M . M. Skopenkov Discrete complex analysis

Convergence Theorem for the Dirichlet Problem A sequence { Q n } is nondegenerate uniform ⇔ ∃ const > 0 : the angle between the diagonals and the ratio of the diagonals in each quadrilateral face are > const , the number of vertices in each disk of radius Size ( Q n ) is < const − 1 , where Size ( Q n ) := maximal edge length. Convergence Theorem for BVP (S. 2013). Let Ω ⊂ C be a bounded simply-connected domain. Let g : C → R be a smooth function. Take a nondegenerate uniform sequence of finite orthogonal lattices { Q n } such that Size ( Q n ) , Dist ( ∂ Q n , ∂ Ω) → 0 . Then the solution u Q n , g : Q n → R of the Dirichlet problem on Q n uniformly converges to the solution u Ω , g : Ω → R of the Dirichlet problem in Ω . M. Skopenkov Discrete complex analysis

2 Discrete analytic functions in Riemann surfaces M. Skopenkov Discrete complex analysis

Riemann surfaces Riemann surface Analytic functions planar domain functions u ( x , y ) + iv ( x , y ) s.t. ∂ u ∂ x = ∂ v ∂ y , ∂ u ∂ y = − ∂ v ∂ x quotient C by a lattice doubly periodic analytic functions complex algebraic curve analytic functions in both w and z a nm z n w m + · · · + a 00 = 0 polyhedral surface continuous functions which are analytic on each face α d β β β α β α d α M. Skopenkov Discrete complex analysis

Discrete Riemann surfaces R a polyhedral surface T its triangulation α e T T 0 the set of vertices � T 1 the set of oriented edges l e T 2 the set faces e r e t e h e A discrete analytic function is a pair ( u : T 0 → R , v : T 2 → R ) such that ∀ e ∈ � T 1 β e v ( l e ) − v ( r e ) = cot α e + cot β e ( u ( h e ) − u ( t e )) . 2 (Duffin, Pinkall–Polthier, Desbrun–Meyer–Schr¨ oder, Mercat) Remark. T is a Delauney triangulation of R 2 ⇒ u ⊔ iv is discrete analytic on Q (in the sense of Part 1 of the slides). M. Skopenkov Discrete complex analysis

Discrete Abelian integrals of the 1st kind p : � R → R the universal covering { α, β } the basis of π 1 ( R ) 0 1 2 { d α , d β } the automorphisms of p 3 / 2 3 / 2 d β 3 / 2 3 / 2 0 1 2 α 1 / 2 1 / 2 β β α 1 / 2 1 / 2 2 0 1 α β d α p S 1 × S 1 � T → T ≈ A discrete Abelian integral of the 1st kind with periods A , B ∈ C is a discrete analytic function T 0 → R , Im f : � T 2 → R ) such that ∀ z ∈ � ( Re f : � T 0 , ∀ w ∈ � T 2 [ Re f ]( d α z ) − [ Re f ]( z ) = Re A ; [ Re f ]( d β z ) − [ Re f ]( z ) = Re B ; [ Im f ]( d α w ) − [ Im f ]( w ) = Im A ; [ Im f ]( d β w ) − [ Im f ]( w ) = Im B . M. Skopenkov Discrete complex analysis

Discrete Abelian integrals of the 1st kind p : � R → R the universal covering { α k , β k } g the basis of π 1 ( R ) k =1 1 { d α k , d β k } g 0 2 the automorphisms of p k =1 3 / 2 3 / 2 d β 1 3 / 2 3 / 2 1 2 0 α 1 1 / 2 1 / 2 β 1 β 1 α 1 1 / 2 1 / 2 0 1 2 α 1 β 1 d α 1 S 1 × S 1 p � T → T ≈ A discrete Abelian integral of the 1st kind with periods A 1 , . . . , A g , B 1 , . . . , B g ∈ C is a discrete analytic function T 0 → R , Im f : � T 2 → R ) such that ∀ z ∈ � ( Re f : � T 0 , ∀ w ∈ � T 2 Re f ( d α k z ) − Re f ( z ) = Re A k ; Re f ( d β k z ) − Re f ( z ) = Re B k ; Im f ( d α k w ) − Im f ( w ) = Im A k ; Im f ( d β k w ) − Im f ( w ) = Im B k . M. Skopenkov Discrete complex analysis

Period matrix Existence & Uniqueness Theorem (Bobenko–S. 2012) ∀ A ∈ C there is a discrete Abelian integral of the 1st kind with the A-period A . It is unique up to constant. The discrete period matrix Π T ( period matrix Π T ) is the B-period of the discrete Abelian integral (Abelian integral) of the 1st kind with the A-period 1 . 1 0 2 It is a 1 × 1 matrix for a surface of genus 1 . 3 / 2 3 / 2 d β 3 / 2 3 / 2 0 1 2 Notation. 1 / 2 1 / 2 γ z := 2 π (the sum of angles meeting at z ) − 1 1 / 2 1 / 2 2 0 1 γ z > 1 ⇔ “curvature” > 0 d α γ R := min z ∈T 0 { 1 , γ z } Π T = i = Π R M. Skopenkov Discrete complex analysis

Existence and Uniqueness Theorem Existence & Uniqueness Theorem (Bobenko–S. 2012) For any numbers A 1 , . . . , A g ∈ C there exist a discrete Abelian integral of the 1st kind with A-periods A 1 , . . . , A g . It is unique up to constant. T 0 → R , Im φ l T 2 → R ) be the unique T : � T : � Let φ l T = ( Re φ l (up to constant) discrete Abelian integral of the 1st kind with A-periods A k = δ kl . The discrete period matrix Π T is the g × g matrix whose T , . . . , φ g columns are the B-periods of φ 1 T . 2 0 1 3/2 3/2 Example. For R = C / ( Z + η Z ) : 0 2 3/2 1 3/2 Re φ 1 T ( z ) = Re z , 1/2 1/2 Im φ 1 T ( w ) = Im w ∗ , 2 0 1 1/2 1/2 where w ∗ is the circumcenter of a face w . M. Skopenkov Discrete complex analysis

The complex structure on polyhedral surfaces Polyhedral metric ❀ complex structure T 2 with a triangle in C by an Identify each face w ∈ � orientation-preserving isometry. A function f : � R → C is analytic , if it is continuous and its restriction to the interior of each face is analytic. R : � Let φ l R → C be the unique (up to constant) Abelian integral of the 1st kind with A-periods A k = δ kl . The period matrix Π R is the g × g matrix whose columns are R , . . . , φ g the B-periods of φ 1 R . γ z := 2 π (the sum of angles meeting at z ) − 1 γ z > 1 ⇔ “curvature” > 0 γ R := min z ∈T 0 { 1 , γ z } M. Skopenkov Discrete complex analysis

Convergence Theorem for Period Matrices Convergence Theorem for Period Matrices (Bobenko–S. 2013) ∀ δ > 0 ∃ Const δ, R , const δ, R > 0 such that for any triangulation T of R with the maximal edge length h < const δ, R and with the minimal face angle > δ we have   h , if γ R > 1 / 2;  � Π T − Π R � ≤ Const δ, R · h | log h | , if γ R = 1 / 2;   h 2 γ R , if γ R < 1 / 2 . Corollary. The discrete period matrices of a sequence of triangulations of the surface with the maximal edge length tending to zero and with face angles bounded from zero converge to the period matrix of the surface. M. Skopenkov Discrete complex analysis

Numerical computation Model surface: R β 1 T n α 1 n β − 1 2 n α 2 Computations using a software by S. Tikhomirov: � Π T n − Π R � · h − 2 γ R � Π T n − Π R � n 8 0.611 1.22 16 0.363 1.15 32 0.220 1.11 64 0.136 1.08 128 0.084 1.07 256 0.053 1.06 M. Skopenkov Discrete complex analysis