Infinite Dimensional Preconditioners V.B. Kiran Kumar Department of Mathematics Cochin University of Science And Technology International Workshop on Operator Theory and its Applications TU Chemnitz, Germany
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Contents Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems Further Problems
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Consider the linear system Ax = b , where A is an n × n matrix. There are several algorithms like Gauss elimination to solve this system. The computational complexity is a problem when n is large. Various iteration methods are used to obtain an approximate solution. There are several ways of iteration. For example, split the matrix A = S − T and consider the iteration by Sx k + 1 = Tx k + b , k = 1 , 2 , . . . .
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Consider the linear system Ax = b , where A is an n × n matrix. There are several algorithms like Gauss elimination to solve this system. The computational complexity is a problem when n is large. Various iteration methods are used to obtain an approximate solution. There are several ways of iteration. For example, split the matrix A = S − T and consider the iteration by Sx k + 1 = Tx k + b , k = 1 , 2 , . . . .
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F The choice of S must be in such a way that it must be invertible so that we can compute x k + 1 from x k . Also, the iteration must converge at a faster rate. That is the sequence { x k } must converge fast. We have to find an optimal choice of S to meet both requirements. Consider the following iteration Cx j + 1 = ( C − A ) x j + b . A preconditioner C can be viewed as an approximation to A that is efficiently invertible and can be used to obtain an approximate solution to the equation Ax = b .
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F The choice of S must be in such a way that it must be invertible so that we can compute x k + 1 from x k . Also, the iteration must converge at a faster rate. That is the sequence { x k } must converge fast. We have to find an optimal choice of S to meet both requirements. Consider the following iteration Cx j + 1 = ( C − A ) x j + b . A preconditioner C can be viewed as an approximation to A that is efficiently invertible and can be used to obtain an approximate solution to the equation Ax = b .
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F It was observed from numerical experiments that the convergence of this iteration process is much faster if the eigenvalues of C − 1 ( C − A ) are clustered around 0 . That means the eigenvalues of C − 1 A must be clustered around 1 . From simple computations, we get if � A − 1 ( C − A ) � < 1 , � A − 1 ( C − A ) � ρ ( C − 1 ( C − A )) ≤ 1 −� A − 1 ( C − A ) � , where ρ ( C − 1 ( C − A )) is the spectral radius.
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Remark Hence we have to choose C such that � ( C − A ) � is small. Definition (Frobenius Norm) For A , B ∈ M n ( C ) the Frobenius norm is defined by n � 2 � A � 2 � � � F = � A j , k j , k = 1 induced by the classical Frobenius scalar product, � A , B � = trace ( B ∗ A ) For a given matrix A ∈ M n ( C ) , our aim is to obtain a preconditioner C such that the Frobenius norm � A − C � F is minimum.
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Remark Hence we have to choose C such that � ( C − A ) � is small. Definition (Frobenius Norm) For A , B ∈ M n ( C ) the Frobenius norm is defined by n � 2 � A � 2 � � � F = � A j , k j , k = 1 induced by the classical Frobenius scalar product, � A , B � = trace ( B ∗ A ) For a given matrix A ∈ M n ( C ) , our aim is to obtain a preconditioner C such that the Frobenius norm � A − C � F is minimum.
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Circulant Preconditioners Here we discuss the circulant preconditioners for Toeplitz sys- tem obtained by Tony Chan in [9]. Consider the Toeplitz matrix with symbol f . a 0 a − 1 · · · · · a − ( n − 1 ) a 1 a 0 a − 1 · · · · · · · · a 1 a 0 a − 1 · · A n ( f ) := (1) · · a 1 a 0 a − 1 · · · · · · · a ( n − 1 ) · · · · · · · · a 0 where a j is the j th Fourier coefficient of f .
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F The circulant matrices are special types of Toeplitz matrices with the following form. · · · · · c 0 c 1 c ( n − 1 ) c ( n − 1 ) c 0 c 1 · · · · c ( n − 2 ) · · c 0 c 1 · · C n := (2) · · · c 0 c 1 · · · · · · · c 1 c 2 · · · · · · · c 0 We are looking for a circulant matrix C with the Frobenius norm � ( C − A ) � F is small.
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F The circulant matrices are special types of Toeplitz matrices with the following form. · · · · · c 0 c 1 c ( n − 1 ) c ( n − 1 ) c 0 c 1 · · · · c ( n − 2 ) · · c 0 c 1 · · C n := (2) · · · c 0 c 1 · · · · · · · c 1 c 2 · · · · · · · c 0 We are looking for a circulant matrix C with the Frobenius norm � ( C − A ) � F is small.
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Tony Chan 1988 In [9], Tony Chan obtained explicitly the optimal circulant preconditioner. It is given by the circulant matrix C n where c i = ( n − i ) a i + ia − ( n − i ) ; i = 0 , 1 , . . . n − 1 . n In fact this is obtained by simply taking average of n elements traveling along diagonal and parallel lines. Remark The above optimal preconditioners are more efficient, compared to the then existing preconditioners (due to Strang [8]) as observed in [9].
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F Tony Chan 1988 In [9], Tony Chan obtained explicitly the optimal circulant preconditioner. It is given by the circulant matrix C n where c i = ( n − i ) a i + ia − ( n − i ) ; i = 0 , 1 , . . . n − 1 . n In fact this is obtained by simply taking average of n elements traveling along diagonal and parallel lines. Remark The above optimal preconditioners are more efficient, compared to the then existing preconditioners (due to Strang [8]) as observed in [9].
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F After the circulant preconditioner obtained by Tony Chan [9], in the same spirit, several researchers have considered the problem to get efficient preconditioners such as Hartly, ǫ − circulant etc. [1, 2]. The philosophy behind these developments is to identify the preconditioners as members in some matrix algebra. The key observation is that circulant matrices are precisely elements in the commutative algebra M U n of matrices defined as follows. M U n = { A ∈ M n ( C ) ; U n ∗ AU n complex diagonal } , � 2 π ijk � 1 where U n = √ n e , j , k = 0 , 1 , . . . n − 1 . n
Introduction Circulant Preconditioners Matrix Algebras Infinite Dimensional Case Spectral Approximation Problems. F After the circulant preconditioner obtained by Tony Chan [9], in the same spirit, several researchers have considered the problem to get efficient preconditioners such as Hartly, ǫ − circulant etc. [1, 2]. The philosophy behind these developments is to identify the preconditioners as members in some matrix algebra. The key observation is that circulant matrices are precisely elements in the commutative algebra M U n of matrices defined as follows. M U n = { A ∈ M n ( C ) ; U n ∗ AU n complex diagonal } , � 2 π ijk � 1 where U n = √ n e , j , k = 0 , 1 , . . . n − 1 . n
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