Capacitance Matrix Method

Transcription

1 Capacitance Matrix Method Marcus Sarkis New England Numerical Analysis Day at WPI, 2019 Thanks to Maksymilian Dryja, University of Warsaw 1

2 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 2

3 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 3

4 Timeline s In 1949, Sherman-Morrison-Woodbury identity (A + UDV T ) 1 = A 1 [A 1 U(D 1 + V T A 1 U) 1 V T A 1 ] In 1952, conjugated gradient method introduced by Hestenes-Stiefel In the 1960th years, there existed fast solvers for FD discretizations of some elliptic problems on rectangular regions based on FFT In 1968, the description of the CMM is credited to Oscar Buneman (see R. W. Hockney Formation and stability of virtual electrodes in a cylinder)

5 Timeline s Early 70 s: Buzbee, Dorr, George, Golub, Hockney, others In 1976, Proskurowski-Widlund, On the numerical solution of Helmholtz s equation by CMM Domain Imbedding: Shieh 1978 and 1979, O Leary-Widlund 1979, Proskurowski-Widlund 1980 (FEM), others Pierre-Louis Lions, Variational Alternating Schwarz Ficticius Component Method: Astrakhantsev 1978 and 1985, Matsokin and Skripko 1983, Matsokin and Nepomnyaschikh 1985 Domain Decomposition: Bjørstad-Widlund 1981 and 1986, Dryja 1982 and 1984, Dihn-Glowinski-Périaux 1985, others In 1982, Dryja, A capacitance matrix method for Dirichlet problem on polygon region

6 Timeline s In 1985, W. Hackbusch, Multi-Grid Methods and Applications In 1986, H. Yserentant, On the multilevel splitting of finite element spaces. In 1986, Bramble-Basciak-Schatz, The construction of precond. for elliptic problems by substructures I In January 1987, Paris, First International Symposium of Domain Decomposition Methods.

7 Capacitance Matrix Method-CMM 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 7

8 CMM: B is a Fast Solver Let A R n n invertible matrix. Find the solution x of Ax = b Let B R n n invertible matrix which has the same rows as the matrix A, except the last p rows (in general p n) [ ] [ ] [ ] A11 A A = 12 b1 A11 A b = B = 12 A 21 A 22 b 2 B 21 B 22 See that Bx = [ b1 ] 8

9 CMM: Capacitance Matrix We look for solution of the form ˆb = [ b1 ˆb2 ] x = B 1ˆb + B 1 I np w p ˆb2 arbitrary where [ 0 I np = After some algebra, w p satisfies Cw p = g p where I pp ] C = I pp I T np(b A)B 1 I np and g p = b 2 ˆb 2 +I T np(b A)B 1ˆb or C = I T npab 1 I np and g p = b 2 I T npab 1ˆb C R p p is invertible and called the Capacitance Matrix 9

10 Implementations To find x, solve Cw p = g p and x = B 1ˆb + B 1 I np w p To solve Cw p = g p use a iterative method Each vector multiplication to C requires a B 1 fast solver Can build C by multiplying Ce i, 1 i p To build C: cost O(p n log 2 (n) Factorization of C: cost O(p 3 ) With an optimal preconditioner κ(p 1 C) = O(1) Cost O(n) log 2 (n) 10

11 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 11

12 Applications Capacitance Matrix Method: Domain Imbedding-DI: After B FFT Domain Decomposition-Iterative Substructuring: After B localization

13 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 13

14 Domain Imbedding-Dirichlet Problem: u = f on Ω and u = 0 on Ω Rectangular region R Ω and Ω a Ω Ω b = R\Ω a Γ = Ω a Ω b Finite Difference ( h x a = b a in Ω h a) ) and (x 2 = 0 on Γ h ) B aa 0 B a2 x a b a Ax = 0 B bb B b2 x b = 0 = b 0 0 I 22 x 2 0 FD ( h x = ˆb in R h ) and (zero Dirichlet on R) B aa 0 B a2 Ω h a B = 0 B bb B b2 Ω h b B 2a B 2b B 22 Γ h

15 Domain Imbedding-Dirichlet The capacitance matrix C = S 1 = (B 22 i={a,b} B 2i B 1 ii B i2 ) 1 C = I pp Inp(B T A)B 1 I np = InpAB T 1 I np Preconditioners κ(( p ) 1 2 C) = O(1). Dryja 1982 κ(h 1 S 1 H T ) = O(1 + log 2 h). Yserentant 1986

16 Domain Imbedding-Neumann u + cu = f in Ω and n u = 0 on Ω B aa 0 B a2 x a Ax = 0 B bb B b2 x b = B 2a 0 B (1) 22 x 2 Well conditioned Capacitance Matrix b a 0 0 = b C = S a (S) 1 = (B (a) 22 B 2a B 1 aa B a2 )(S) 1

17 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 17

18 Proskurowski-Widlund, 1976 Modification of a rectangular domain R. W. Hocknew Five-points discretization of Laplace equation with a number of electrodes nodes are introduced in the interior of a rectangular region or on straight line segment to which one of several mesh points are assigned In Proskurowski-Widlund 76, the goal is to solve the Poisson problem in complex geometry (Dirichlet or Neumann problems) using FD. In Proskurowski-Widlund 76, the CMM is interpreted using classical potential theory using single-layer dipole (ansatz). And in this case C is SPD with condition number O(h) and CG is used

19 Proskurowski-Widlund 76 x = B 1ˆb + B 1 I np w p This equation is interpreted using the 76 paper notation u = Gf + GDµ G Discrete Green s Function Here µ is the dipole density (jump of normal derivatives) Cµ = (I p + Z T GD)µ = Z T Gf = g G is translated invariant (FFT techniques). To build C O(nlog 2 n + p 2 )

20 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 20

21 L-Shaped Domain Poisson Equation on L-shaped region Ω = Ω a Γ Ω b Ω a = (0, x 1 ) (0, y 2 ) Γ = {x 1 } (0, y 1 ) Ω b = (x 1, x 2 ) (0, y 1 ) FDM with Homogeneous Dirichlet BC. Solve Ax = b [ ] [ A11 A A = 12 Ω h a Ω h b b1 b = A 21 A 22 Γ h b 2 ] We choose B of the form [ A11 A B = 12 0 I 22 ] C = A 22 A 21 A 1 11 A 12 = S, the Schur complement of A with respect to A 22

22 Remarks κ(c) = O(1/h) κ(( p ) 1 2 C) = O(1). Dryja 1982 If we choose C = I 22 B = [ A11 A 12 0 S ] 22

23 Sort of BDDC with Vertex Constraint Several substructures Ω i and interface Γ Ω = N i=1ω i Γ = Ω\( N i=1ω i ) Let V be the set of vertices of the substructures [ ] [ A11 A A = 12 (Ω h \Γ h ) V A11 A B = 12 A 21 A 22 Γ h \V 0 I 22 ] C = A 22 A 21 A 1 11 A 12 Preconditioner K = blockdiag{( ij ) 1 2 } Fij Γ 23

24 Neumann-Dirichlet Method Dryja, Proskurowski, Widlund, Checkerboard distribution local solvers. Or two subdomains Ω = Ω (D) Ω (N) = ( i ND Ω (D) i ) ( i NN Ω (N) i ) [ ] A11 A A = 12 A 21 A 22 Ω (D) h (interior) Ω (N) h B = v T 2 A 22 u 2 = a Ω (N)(u 2, v 2 ) + a Ω (D)(u 2, v 2 ) C = (A 22 A 21 A 1 11 A 12 )(A (N) 22 ) 1 κ(s(a (N) 22 ) 1 ) = O(1 + log H/h) 2 [ A11 A 12 0 A (N) 22 ] 24

25 Outline 1 Timeline 2 Capacitance Matrix Method-CMM 3 Applications 4 Domain Imbedding 5 Proskurowski-Widlund 76 6 Domain Decomposition-Iterative Substructuring 7 Conclusion 25

26 Conclusions THANK YOU 26