Basic Operator Theory with Karhunen Loéve Expansion Wafa Abu Zarqa, Maryam Abu Shamala, Samiha Al Aga, Khould Al Qitieti, Reem Al Rashedi Department of Mathematical Sciences College of Science United Arab Emirates University April 2, 2009
1 Introduction 2 Basic Operator Theory on L 2 [a; b] 3 Expansion Theorems 4 Integral Operator 5 Karhunen Loéve Expansion 6 Summary
Introduction 1 Basic Operator Theory on L 2 [a; b]
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem Eigenvalue Problem
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem Eigenvalue Problem 4 Karhunen Loéve (KL) Expansion
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem Eigenvalue Problem 4 Karhunen Loéve (KL) Expansion Series Representation of Random Variable
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem Eigenvalue Problem 4 Karhunen Loéve (KL) Expansion Series Representation of Random Variable 5 Examples of KL Expansion
Introduction 1 Basic Operator Theory on L 2 [a; b] Linear, Bounded, Continuous, Symmetric, Compact Operator Resolvent and Spectrum 2 Expansion Theorems 3 Integral Operator Mercer s Theorem Eigenvalue Problem 4 Karhunen Loéve (KL) Expansion Series Representation of Random Variable 5 Examples of KL Expansion Brownian Motion, Centered Brownian Motion, Brownian Bridge
Setup Function Space L 2 = L 2 [a; b] with finite number a and b: Set of all real valued function x (t ) on the interval [a; b] such that Z b x 2 (t ) dt < 1 a Inner Product on L 2 : Norm of x : kx k = Z b (x ; y) = a q (x ; x ) = x (t )y(t ) dt Z b a x 2 (t ) dt! 1=2
Definitions I Operator A on L 2 assigns a function x 2 L 2 to a function A(x ) 2 L 2. A is linear, if A satisfies A(x + y) = A(x ) + A(y); where x ; y 2 L 2 and, are constants.
Definitions II A is bounded, if norm kak of A is finite: kak = sup kax k < 1: kx k=1 A is continuous, if x n! x (L 2 norm) =) Ax n! Ax (L 2 norm)
Definitions III A is symmetric, if for all x ; y 2 L 2, (Ax ; y) = (x ; Ay) A is compact, if for every bounded sequence of x n, the sequence of image A(x n ) has a convergent subsequence.
Resolvent and Spectrum Let A be compact linear and a real number. Resolvent (A): Set of all such that A I is bijective, where I is the identity operator. Spectrum (A) = R (A) Eigenvalue of A and Eigenvector x corresponding to, if they satisfy Ax = x
Orthogonal Set f n g is an orthonormal set on L 2 [a; b] if it satisfies Z b ( n ; m ) = a n (t ) m (t ) dt = n;m ; where n;m is the Kronecker delta. Examples on L 2 [ ; ]: f n g 1 n= 1 and f ng 1 n= 1 defined by n (t ) = 1 p sin(nt ); n (t ) = 1 p cos(nt )
Gram Schmidt Process Given a set ff n g of linearly independent elements, Gram Schmidt process generates an orthogonal set f n g: Step 1: 1 = f 1 Step 2: Step n: 2 = f 2 n = f n n X1 i=1 (f 2 ; 1 ) k 1 k 2 1 (f n ; i ) k i k 2 i
Orthogonality of Eigenvectors Let A be a symmetric linear operator on L 2 [a; b]. Then eigenvectors corresponding to distinct eigenvalues of A are orthogonal, i.e., Ax = x and Ay = y with 6= =) (x ; y) = 0
Expansion Theorem Let A be a compact and symmetric linear operator on L 2. Let fe n g 1 n=1 be an orthonormal set formed by the eigenfunctions corresponding to non zero eigenvalues of A. Then any element x 2 L 2 has the representation: x = h + 1X (x ; e n )e n ; n=1 where h is the projection of x onto E 0 = f y 2 L 2 j Ay = 0 g.
Example For a given function f (x ) = e x on L 2 [ ; ], we compare the error between f (x ) and the series approximation f app. f app consists of orthonormal functions generated by f1; x ; : : : ; x 20 g through Gram Schmidt: Largest : 0:639711; Smallest : 1:21104 10 10 f app consists of orthonormal functions fsin(nx )= p g 5 n=1: Largest : 1:86283; Smallest : 0:900526
Figure Black: Given Function, Red: Series Approximation f app, Blue: Error
Integral Operator We consider the integral operator A on L 2 [a; b] defined by Z b (Ax )(t ) = a R(t ; s)x (s) ds: A is bounded as well as continuous and compact. If R(t ; s) is nonnegative definite as well as continuous and symmetric, then all eigenvalues of A are nonnegative.
Mercer s Theorem Let fe n g 1 n=1 be an orthonormal basis for the space spanned by the eigenvectors corresponding to the nonzero (hence positive) eigenvalues of A. If the basis is taken so that e n is an eigenvector corresponding to the eigenvalue n, then 1X R(t ; s) = n=1 n e n (t )e n (s); t ; s 2 [a; b]:
Karhunen Loéve Expansion I Let X (t ), t 2 [a; b], be a random variable with mean 0 and have continuous covariance R(t ; s). Then, X (t ) = 1X k=1 Z k e k (t ); a t b: 1 e k s are eigenfunctions of the integral operator corresponding to R(t ; s).
Karhunen Loéve Expansion II 2 e k s forms an orthonormal basis for the space spanned by the eigenfunctions corresponding to the nonzero eigenvalues. 3 Z k s are given by Z b a x (t )e k (t ) dt. 4 Z k s are orthogonal random variables with zero mean and variance k (eigenvalue corresponding to e k ).
Example of KL Expansion Brownian Motion blah...
Example of KL Expansion Centered Brownian Motion blah
Example of KL Expansion I Brownian Bridge The Brownian bridge B b (t ) = B (t ) tb (1) has covariance Cov[B b (t ); B b (s)] = min ft ; sg st : Consider the integral operator A b defined by (A b x ) (t ) = Z 1 0 Cov[B b(t ); B b (s)]x (s) ds:
Example of KL Expansion II Brownian Bridge 1 Eigenvalue and Orthonormal Eigenfunction of A: n = (n) 2 ; e n (t ) = p 2 sin(nt ) 2 KL expansion of B b (t ): p 1X B b (t ) = 2 where Z k = nz k and Z 1 Z k = 0 B b(t )e k (t ) dt. Z k k=1 sin(nt ) n ;
Summary Thank you!