Fachbereich Mathematik & Informatik
Freie Universität Berlin
Prof. Dr. Carsten Gräser, Tobias Kies
4th exercise for the lecture
Numerics IV
Winter Term 2016/2017
http://numerik.mi.fu-berlin.de/wiki/WS_2016/NumericsIV.php
Due: Tuesday, Nov 22nd, 2016
Exercise 1 (4 TP)
Let Ω ⊆ Rd and ψ, ψ ∈ L2 (Ω) such that ψ ≤ ψ almost everywhere and define
K := v ∈ L2 (Ω) | ψ ≤ v ≤ ψ a. e. .
For f ∈ L2 (Ω) we consider the projection problem
min ku − f k2L2 (Ω) .
(1)
u∈K
a) Show that (1) admits a unique solution.
b) Compute the solution of (1) explicitly.
Exercise 2 (2 TP)
Let A ∈ Rn×n symmetric
Qn positive definite and ain, bi ∈ R such that ai ≤ bi for i ∈
{1, . . . , n}. Define K := i=1 [ai , bi ], suppose f ∈ R , u ∈ K and assume the variational
inequality
∀v ∈ K : hAu, v − ui ≥ hf, v − ui
is fulfilled. Show that for all i ∈ {1, . . . , n} with ui ∈ (ai , bi ) the equality (Au)i = fi
holds.
Exercise 3 (2 TP)
Let X a Banach space, K ⊆ X convex and f : K → R ∪ {∞} convex. Show that the
subdifferential of f is monotone, i. e. prove
∀x, y ∈ K ∀u ∈ ∂f (x), v ∈ ∂f (y) : hu − v, x − yi ≥ 0.
Please turn over...
Exercise 4 (4 TP)
Let X a normed vector space, K ⊆ X nonempty, convex and closed and suppose an
elliptic bilinear form a : X × X → R. Define the operator induced by a by
!
X→R
0
A : X −→ X ,
x 7−→
y 7→ a(x, y)
and denote IK the characteristic function over K, i. e. we have for all x ∈ X
(
0
if x ∈ K
IK (x) =
∞ if x ∈
/ K.
Suppose b ∈ X 0 and that x ∈ X fulfills (A + ∂IK )x 3 b. Show that x is then the solution
to the minimization problem
1
min hA(y), yi − b(y).
y∈K 2
Exercise 5 (4 TP)
Let X a normed vector space and J : X → R ∪ {∞} convex. The polar function of J is
defined as
J ∗ : X 0 −→ R ∪ {∞},
x0 7−→ sup x0 (x) − J(x) .
x∈X
Verify that
J∗
is convex, prove
∂J(x) = x0 ∈ X 0 | J ∗ (x0 ) = x0 (x) − J(x)
and show that for all x ∈ X and x0 ∈ X 0 holds
x ∈ (∂J)−1 (x0 ) =⇒ Ex ∈ (∂J ∗ )(x0 )
where Ex ∈ X 00 is defined by (Ex)(y 0 ) = y 0 (x) for all y 0 ∈ X 0 .
Remark: For a Banach space X and f : X ⊇ K → R ∪ {∞} convex the subdifferential
of f in x ∈ K is given by
∂f (x) := v ∈ X 0 | ∀y ∈ K : f (x) + hv, y − xi ≤ f (y)
and its inverse in x0 ∈ X 0 by
(∂f )−1 (x0 ) := x ∈ X | x0 ∈ ∂f (x) .