Preconditioners for ill conditioned (block) Toeplitz systems: facts and ideas Paris Vassalos Department of Informatics, Athens University of Economics and Business, Athens, Greece. Email:pvassal@aueb.gr, pvassal@uoi.gr Joint work with D. Noutsos Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Problem We are interested in the fast and efficient solution of nm × nm BTTB systems T n , m ( f ) x = b where f is nonnegative real-valued belonging to C 2 π, 2 π defined in the fundamental domain Q = ( − π, π ] 2 and is a priori known. The entries of the coefficient matrix are given by 1 � f ( x , y ) e − i ( jx + ky ) dxdy , t j , k = 4 π 2 Q for j = 0 , ± 1 , . . . , ± ( n − 1 ) and k = 0 , . . . , ± ( m − 1 ) . Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Connection between f and T nm ( f ) The main connection between T nm ( f ) and the generating function is described by the the following result: Theorem If f ∈ C [ − π, π ] 2 and c < C the extreme values of f ( x , y ) on Q. Then every eigenvalue λ of the block Toeplitz matrix T satisfies the strict inequalities c < λ < C Moreover as m , n → ∞ then λ min ( T nm ( f )) → c and λ max ( T nm ( f )) → C Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Crucial relationship The basis for the construction of effective preconditioners is described by the following theorem Theorem Let f , g ≥ 0 ∈ C [ − π, π ] 2 (f and g not identically zero). Then for every m , n the matrix T − 1 nm ( g ) T nm ( f ) has eigenvalues in the open interval ( r , R ) , where f f r = inf and R = sup g g Q Q Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Negative result for τ algebra preconditioners Theorem (Noutsos,Serra,Vassalos, TCS (2004)) Let f be equivalent to p k ( x , y ) = ( 2 − 2 cos ( x )) k + ( 2 − 2 cos ( y )) k with k ≥ 2 and let β be a fixed positive number independent of n. Then for every sequence { P n } with P n ∈ τ , n = ( n 1 , n 2 ) , and such that λ max ( P − 1 n T n ( f )) ≤ β (1) uniformly with respect to ˆ n, we have (a) the minimal eigenvalue of P − 1 n T n ( f ) tends to zero. (b) the number # { λ ( n ) ∈ σ ( P − 1 n T n ( f )) : λ ( n ) → N ( n ) →∞ 0 } tends to infinity as N (ˆ n ) tends to infinity. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Negative result for τ algebra preconditioners Theorem (Noutsos,Serra,Vassalos, TCS (2004)) Let f be equivalent to p k ( x , y ) = ( 2 − 2 cos ( x )) k + ( 2 − 2 cos ( y )) k with k ≥ 2 and let α be a fixed positive number independent of n = ( n 1 , n 2 ) . Then for every sequence { P n } with P n ∈ τ and such that λ min ( P − 1 n T n ( f )) ≥ α (2) uniformly with respect to n, we have (a) the maximal eigenvalue of P − 1 n T n ( f ) tends to ∞ . (b) the number # { λ ( n ) ∈ σ ( P − 1 n T n ( f )) : λ ( n ) → N ( n ) →∞ ∞} tends to infinity as N ( n ) tends to infinity. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
How to solve: 2D- ill conditioned problem Direct methods: Levinson type methods cost O ( n 2 m 3 ) ops while superfast methods O ( nm 3 log 2 n ) . Stability problems. Not optimal. PCG method with matrix algebra preconditioners. Cost O ( k ( ǫ ) nm log nm ) , where k ( ǫ ) is the required number of iterations and depends from the condition number of T nm . PCG method where the preconditioner is band Toeplitz matrix. Under some assumptions, the cost is optimal ( O ( nm log nm ) ). Multigrid methods: Promising but in early stages. Cost O ( nm log nm ) ops. G. Fiorentino and S. Serra-Capizzano (1996), T. Huckle and J. Staudacher (2002),H.W. Sun, X.Q. Jin and Q.S. Chang (2004), A. Arico, M. Donatelli and S. Serra-Capizzano (2004). Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
More on Band Preconditioners S. Serra-Capizzano (BIT, (1994)) and M. Ng (LAA, (1997)) proposed as preconditioner the band BTTB matrix generated by the minimum trigonometric polynomial g which has the same roots with f . Let f = g · h with h > 0. Then D. Noutsos, S. Serra Capizzano and P . Vassalos (Numer. Math. (2006)) proposed as preconditioners the band BTTB matrix generated by g · ˆ h where ˆ h : is the trigonometric polynomial arises from the Fourier 1 approximation on h . arises from the Lagrange interpolation of h at 2 D 2 Chebyshev points, or from the interpolation of h using the 2 D Fejer kernel. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Preliminaries We assume that the nonnegative function f has isolated zeros ( x 1 , y 1 ) , ( x 2 , y 2 ) ,. . . , ( x k , y k ) , on Q each one of multiplicities ( 2 µ 1 , 2 ν 1 ) , ( 2 µ 2 , 2 ν 2 ) , . . . , ( 2 µ k , 2 ν k ) . Then, f can be written as f = g · w where k � [( 2 − 2 cos ( x − x i )) µ i + ( 2 − 2 cos ( y − y i )) ν i ] g = i = 1 and w , is strictly positive on Q . Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
New Proposal For the system T nm ( f ) x = b we define and propose as a preconditioner the product of matrices √ √ K nm ( f ) = A nm ( w ) T nm ( g ) A nm ( w ) = A nm ( h ) T nm ( g ) A nm ( h ) with h = √ w , A nm ∈ { τ, C , H} , where { τ, C , H} is the set of matrices belonging to Block τ , Block Circulant and Block Hartley algebra, respectively. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Construction of 2 D algebras v [ n ] A Q n � n � � v [ n ] sin ( jv [ n ] π i 2 τ = ) n + 1 n + 1 i i i , j = 1 e i jv [ n ] v [ n ] � � = 2 π i 1 C F n = √ n i i n v [ n ] = 2 π i H Re ( F n ) + Im ( F n ) i n The matrices C ( h ) , τ ( h ) , H ( h ) can be written as � � f ( v [ nm ] ) · Q H A nm ( h ) = Q nm · Diag nm where v [ nm ] = v [ n ] × v [ m ] � and Q nm = Q n Q m Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Obviously K nm ( f ) has all the properties that a preconditioner must satisfied, i.e, is symmetric, is positive definite, The cost for the solution of the arbitrary system K nm ( f ) x = b is of order O ( nm log nm ) . O ( nm log nm ) for the “inversion” of A nm ( h ) by 2D FFT, and O ( nm ) for the “inversion” of block band Toeplitz matrix T n , m ( g ) which can be done by multigrid methods. So, the only condition that must be fulfill, in order to be a competitive preconditioner, is the spectrum of � − 1 T nm ( f ) being bounded from above and below. K A � nm ( f ) Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Useful Definition Definition We say that a function h is a (( k 1 , k 2 ) , ( x 0 , y 0 )) -smooth function if ∂ l 1 + l 2 ∂ x l 1 ∂ y l 2 h ( x 0 , y 0 ) = 0 , l 1 < k 1 , l 2 < k 2 , and l 1 + l 2 < max { k 1 , k 2 } and ∂ l 1 + l 2 ∂ x l 1 ∂ y l 2 h ( x 0 , y 0 ) is bounded for l 1 = k 1 , l 2 = 0 and l 2 = k 2 , l 1 = 0 , and l 1 + l 2 = max { k 1 , k 2 } , l 1 < k 1 , l 2 < k 2 . Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
τ Case: Clustering Theorem Let f belongs to the Wiener class. Then for every ǫ the spectrum of nm ( f )] − 1 T nm ( f ) [ K τ lies in [ 1 − ǫ, 1 + ǫ ] , for n , m sufficient large, except of O ( m + n ) outliers. Thus, we have weak clustering around unity of the spectrum of the preconditioned matrix. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
τ Case: Bounds Theorem Let f ∈ C 2 π, 2 π even function on Q with roots ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x k , y k ) , each one of multiplicities ( 2 µ 1 , 2 ν 1 ) , ( 2 µ 2 , 2 ν 2 ) , . . . , ( 2 µ k , 2 ν k ) , respectively. If g is the trigonometric polynomial of minimal degree that rises the roots of f and h is ( µ i − 1 , ν i − 1 ) smooth function at the roots ( x i , y i ) , i = 1 ( 1 ) k, then the nm ( f )] − 1 T nm ( f ) spectrum of the preconditioned matrix P τ = [ K τ is bounded from above as well as bellow: c < λ min ( P τ ) < λ max ( P τ ) < C with c , C constants independent of n , m Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Circulant Case: Clustering Theorem Let f belongs to the Wiener class on Q. Then for every ǫ the spectrum of � − 1 T nm ( f ) K C � nm ( f ) lies in [ 1 − ǫ, 1 + ǫ ] for n , m sufficient large except O ( m + n ) outliers. Thus, we have a weak clustering of the spectrum of the preconditioned matrix around unity. Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
Circulant Case: Bound of Spectrum Theorem Let f ∈ C 2 π, 2 π even function on Q with roots ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x k , y k ) , on Q each one of multiplicities ( 2 µ 1 , 2 ν 1 ) , ( 2 µ 2 , 2 ν 2 ) , . . . , ( 2 µ k , 2 ν k ) respectively. If g is the trigonometric polynomial of minimal degree that rises the roots of f and h is ( µ i , ν i ) smooth function at the roots ( x i , y i ) , i = 1 ( 1 ) k then the spectrum of the preconditioned matrix � − 1 T nm ( f ) is bounded above as well as bellow: P C = � K C nm ( f ) c < λ min ( P C ) < λ max ( P C ) < C with c , C constants independent of n , m Paris Vassalos Preconditioners for ill conditioned (block) Toeplitz systems: facts a
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