Math 641-600 — Fall 2017
Current Assignment
Assignment 7 - Due Monday, October 30, 2017.
- Read sections 3.1, 3.2 and my notes
on X-ray
Tomography and on Bounded
Operators & Closed Subspaces.
- Do the following problems.
- Section 2.2: 25(a,b), 26(b), 27(a)
- Let $S^{1/n}(1,0)$ be the space of piecewise linear splines, with
knots at $x_j=j/n$, and let $N_2(x)$ be the linear B-spline ("tent
function", see Keener, p. 81 or my notes on splines.)
- Let $\phi_j(x):= N_2(nx +1 -j)$. Show that
$\{\phi_j(x)\}_{j=0}^n$ is a basis for $S^{1/n}(1,0)$.
- Let $S_0^{1/n}(1,0):=\{s\in S^{1/n}(1,0):s(0)=s(1)=0\}$. Show that
$S_0^{1/n}(1,0)$ is a subspace of $S^{1/n}(1,0)$ and that
$\{\phi_j(x)\}_{j=1}^{n-1}$ is a basis for it.
- Let $H_0$ be the set of all $f\in C^{(0)}[0,1]$ such that
$f(0)=f(1)=0$ and that $f'$ is piecewise continuous. Show that
$\langle f,g\rangle_{H_0} :=\int_0^1f'(x)g'(x)dx$ defines a real
inner product on $H_0$.
- We want to use a Galerkin method to numerically solve the
boundary value problem (BVP): −u" = f(x), u(0) = u(1) = 0,
f ∈ C[0,1]
- Weak form of the problem. Let H_{0} be as in the
previous problem. Suppose that $v\in H_0$. Multiply both sides of
the equation above by $v$ and use integration by parts to show that
$ \langle u,v\rangle_{H_0} = \langle f,v\rangle_{L^2[0,1]}$. This
is called the ``weak'' form of the BVP.
- Conversely, suppose that u ∈ H_{0} is also in
C^{(2)}[0,1] and that u satisfies
⟨u,v⟩_{H0} = ∫_{0}^{1} f(x)
v(x) dx for all v ∈ H_{0}.
Show that u satisfies the BVP.
- Note that $S_0:=S_0^{1/n}(1,0)$ is a subspace of $H_0$ and let
$s_n\in S_0$ satisfy $\|u-s_n\|_{H_0} = \min_{s\in S_0}\|u -
s\|_{H_0}$; thus, $s_n$ is the least-squares approximation to u from
∈ S_{0}. Expand $s_n$ in the basis from problem 2(b):
$s_n = \sum_{j=1}^{n-1}\alpha_j\phi_j$. Use the normal equations and
part (a) above to show that the $\alpha_j$'s satisfy $G\alpha =
\beta$, where $\beta_j= \langle f,\phi_j\rangle_{L^2[0,1]}$ and $G_{kj}
=\langle \phi_j,\phi_k\rangle_{H_0}$
- Show that
$
G=\begin{pmatrix} 2n& -n &0 &\cdots &0\\
-n & 2n& -n &0 &\cdots \\
0&-n& 2n& \ddots &\ddots \\
\vdots &\cdots &\ddots &\ddots &-n\\
0 &\cdots &0 &-n &2n
\end{pmatrix}
$
- Let V be a Banach space. Show that a linear operator L:V → V
is bounded if and only if L is continuous.
- Consider the Sobolev space $H^1[0,1]$, with the inner product
$\langle f, g\rangle_{H^1} := \int_0^1 \big(f(x)\overline {g(x)} +
f('x)\overline {g'(x)}\big)dx$. For $f\in H^1$, let $Df=f'$. Show that
$D:H^1[0,1]\to L^2[0,1]$ is bounded, and that $\|D\|_{H^1 \to L^2}=1$.
Updated 10/23/2017.