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{\bf Math 273}: {\bf Homework \#1} Assigned on October 9. 

Due to: Teaching Assistant Eric Radke. 

Due date: one week from the date of the assignment. Late homework is accepted. 

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\noindent{\bf [1]} Compute the gradient $\nabla f(x)$ and Hessian $\nabla^2 f(x)$ of the function 
$$f(x)=100(x_2-x_1^2)^2+(1-x_1)^2.$$ 
Show that $x^*=(1,1)^T$ is the only local minimizer of this function, and that the Hessian matrix at that point is positive definite. 

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\noindent{{\bf [2]} Let $a$ be a given $n$-vector, and $A$ be a given $n\times n$ symmetric matrix. Compute the gradient and Hessian of $f_1(x)=a^Tx$ and $f_2(x)=x^TAx$. 

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\noindent{{\bf [3]} Suppose that $f$ is a convex function. Show that the set of global minimizers of $f$ is a convex set. 

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\noindent{{\bf [4]} Suppose that $\hat f(z)=f(x)$, where $x=Sz+s$ for some $S\in R^{n\times n}$ and $s\in R^n$. Show that 
$$\nabla \hat f(z)=S^T\nabla f(x),\ \ \ \ \nabla ^2\hat f(z)=S^T\nabla^2 f(x)S.$$ 

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\noindent{{\bf [5]} Computation of the Euler-Lagrange equation in the continuous case. 

(a) Consider the minimization problem
$$\inf_{u}F(u)=\int_{x_0}^{x_1} L(x,u(x),u'(x))dx,$$
with $u(x_0)=u_0$, $u(x_1)=u_1$ given constants, and $L$ a sufficiently smooth
function. Obtain formally the Euler-Lagrange equation of the minimization problem that is satisfied by a smooth\
 optimal $u$. 

Hint: Consider test functions $v$, such that $v(x_0)=v(x_1)=0$. Since $u$ is a minimizer, we must 
have $F(u)\leq F(u+\varepsilon v)$ for all such sufficiently smooth functions $v$ and every 
real $\epsilon$. Apply integration by parts to obtain the desired result. You should obtain a 
second-order differential equation.

(b) Let now $u(x,t)$ be a smooth solution of the time-dependent partial differential equation (PDE)
$$\frac{\partial u}{\partial t}=\frac{\partial}{\partial x} L_{u'}(x,u,u')-L_u(x,u,u'),$$
with $u(x,0)=u_0(x)$ on $(x_0,x_1)$ and $u(x_0,t)=U_0$, $u(x_1,t)=U_1$ for $t\geq 0$. 
Show that the function $E(t)=F(u(\cdot,t))$ is decreasing in time, where
$F(u)=\int_{x_0}^{x_1}L(x,u,u')dx.$


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{\bf [Notes]} 

$\bullet$ Let $\Omega$ be an open and bounded subset of $R^d$, with Lipschitz-continuous (or sufficien\
tly smooth) boundary $\partial\Omega$. Let $\vec{n}=(n_1,n_2,...,n_d)$ be the exterior unit normal to \
$\partial\Omega$.
 Recall the following fundamental Green's formula, or integration by parts formula: given two function\
s $u,v$ (with $u$, $v$, and all their 1st order partial derivatives belonging to $L^2(\Omega)$, or $u,\
v\in H^1(\Omega)$), then
$$\int_{\Omega}u v_{x_i} dx=-\int_{\Omega} u_{x_i} v dx +\int_{\partial\Omega} u v n_i dS.$$


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