Exercise sheet 4 To be handed in on Thursday, 12.05.2016

Wissenschaftliches Rechnen II/Scientific Computing II

Sommersemester 2016 Prof. Dr. Jochen Garcke Dipl.-Math. Sebastian Mayer

Exercise sheet 4 To be handed in on Thursday, 12.05.2016

1 Group exercises

G 1. Let k : Ω

→ R be positive semi-definite. Provide details for the proof of Theorem 34. Concretely,

a) Recall H = span{k(x, ·) | x ∈ Ω} with inner product h·, ·i

given by hf, g,

H =

n

X

i=1 m

X

j=

a

b

k(x

, y

) for f = P

i=1

a

k(x

, ·), g = P

j=1

b

K(y

, ·). Let (h

)

, h

∈ H, be a Cauchy sequence. Show that the pointwise limit h(t) := lim

h

(t) exists.

b) Let N

be the set of pointwise limits of arbitrary Cauchy sequences in H. For g, f ∈ N

, let

hf, gi

:= lim

n→∞

hf

, g

i

.

Verify that hf, f i = 0 if and only if f = 0, that is, h·, ·i is indeed a scalar product.

c) Show that N

is complete with respect to the norm induced by h·, ·i

. G 2. (Interpolation and discrete Fourier transform)

For even n ∈ N , consider again the space of complex-valued trigonometric polynomials H

introduced in Sheet 2, Exercise G2. Further consider the sampling points

x

= 2πk

n ∈ [−π, π], k = −n/2, . . . , 0, . . . , n/2 − 1.

a) Consider the subspace H e

= {f ∈ H

: hf, e

i = 0}, which has the kernel D e

(x, y) = 2 P

k=1

cos(k(x − y)). Show that the matrix D e

:=

( D e

(x

, x

))

is the identity, that is,

D e

(2πj/n, 2πl/n) =

( 2n : j = l 0 : j 6= l.

Hint: Use the following fact about geometric series: for r 6= 1, we have P

k=0

r

=

1−rⁿ⁺¹ 1−r

.

b) Assume to be given observations (x

, y

)

. Determine the polynomial g

∈ H e

which solves the interpolation problem

g

(x

) = y

, i = −n/2, . . . , n/2 − 1.

Determine the Fourier coefficients of g

.

Remark: The Fourier coefficients of g

are the discrete Fourier transform of the

vector (y

)

.

2 Homework

H 1. (Sums of kernel spaces)

Let k

, k

: Ω

→ R be two positive semi-definite mappings. Consider k = k

+ k

given by

k(x, y) := k

(x, y) + k

(x, y).

Show the native space N

is given by

N

= {f

+ f

| f

∈ N

, f

∈ N

} and the norm fufills

kf k

= min{kf

k

1

+ kf

k

2

: f = f

+ f

, f

∈ N

}.

Hint: Consider the product space H = N

× N

with norm given by k(f

, f

)k

= kf

k

1

+ kf

k

2

and find a suitable closed subspace.

(10 Punkte) H 2. (Kernel smoothing spline)

For even n ∈ N , let H

be the space of trigonometric polynomials as introduced in Sheet 2, G2. Let x

, . . . , x

be the sampling points given in G2. Further, let f : [−π, π] → C be a 2π-periodic function. You are given noisy observations

y

= f (x

) +

, i = −n/2, . . . , n/2 − 1.

The goal is to solve the following kernel smoothing spline optimization problem explicitly:

fλ

min

1 2n

n/2

X

i=−n/2

y

− f

(x

)

+ λ 2π

Z

−π

(f

(x))

dx. (1) Proceed as follows:

a) Determine the trigonometric polynomial f

∈ H

which solves the interpolation problem

f

= y

, i = −n/2, . . . , n/2 − 1.

b) Use a) to reformulate the optimization problem (1) in terms of the Fourier coefficients of f

and f

. Determine the Fourier coefficients of the trigonometric polynomial f

∈ H

which minimizes (1).

c) Compare the Fourier coefficients of f

and f

and interpret the difference between f

and f

.

(10 Punkte)

2