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Let ${𝑥}_{𝑘}$, $𝑘=0,1,\dots ,𝑁$ be a 1D discretized signal on $[0,1]$. We assume that the measured signal ${𝑦}_{𝑘}$ is noisy:

where ${𝑛}_{𝑘}$ is noise. In order to remove the noise, we start an iterative estimation process. Given the iterate ${𝑥}_{𝑘}^{(𝑛)}$ let us define the error estimator

for $𝑧\in ℝ$. The parameter $𝜆$ regulates how much we weigh the bias toward the initial measurement ${𝑦}_{𝑘}$ relative to the bias toward the median value that brings about smoothness and kills the noise. We update our estimator as follows:

(i) Show that when $𝜆=1$ we have

(ii) Define the total cost function as (here we denote $𝑧=({𝑧}_1,\dots ,{𝑧}_{𝑁})$ while before we denote by $𝑧$ a scalar)

and write the iteration process as

Show that if the above iteration process converges:

then the limit is the critical point of the cost function

Consider a random walk that does the following. A particle starts at a point $𝑥\in ℝ$ and has a clock attached to it. The clock rings at a random time $𝜏>0$, with $\mathbb{P}(𝜏>𝑡)=exp(-𝑡)$. At that time, the particle jumps from the position $𝑥$ to a new position ${𝑦}_1\in {ℝ}^{𝑛}$, with

for any open set $𝐴\subset ℝ$. Here, $𝐾(𝑥)$ is a fixed rapidly decaying smooth non-negative function such that

After the jump the clock is reset and the process starts anew: the particle stays at ${𝑦}_1$ until the clock rings, when it jumps to the position ${𝑦}_2$ such that

and so on.

(i) Fix a bounded continuous compactly supported function $𝑓(𝑥)$ and define

Here, $𝑋(𝑡)$ is the location of the random walk that started at time $𝑡=0$ at the location $𝑥$. Show that $𝑢(𝑡,𝑥)$ satisfies the integro-differential equation

with the initial condition $𝑢(𝑡=0,𝑥)=𝑓(𝑥)\ge 0$.

(ii) Consider a family of solutions ${𝑢}^{𝜀}(𝑡,𝑥)$ to this equation with a fixed smooth compactly supported initial condition ${𝑢}^{𝜀}(0,𝑥)=𝑓(𝑥)$ and the kernel ${𝐾}_{𝜀}(𝑥)$ of the form ${𝐾}_{𝜀}(𝑥)={𝜀}^{-𝑛}𝐾(𝑥/𝜀)$, with a fixed non-negative smooth and compactly supported function $𝐾(𝑥)$ that satisfies ${\int }_{ℝ}𝐾(𝑥)d𝑥=1$. Show that the functions ${𝑤}^{𝜀}(𝑡,𝑥)={𝑢}^{𝜀}(𝑡/𝜀,𝑥)$ converge, as $𝜀\downarrow 0$, locally uniformly to a limit and that the limit is the solution to

with some constant $𝐷>0$ and the initial condition . Identify the diffusivity $𝐷>0$ in terms of the kernel $𝐾(𝑥)$.

Let $𝐴$ be a symmetric positive definite $𝑚\times 𝑚$ matrix, ${𝐷}_{𝑗}$, $𝑗=1,\dots ,𝑛$, be symmetric matrices, and let $𝑣=({𝑣}_1,\dots ,{𝑣}_{𝑚})$ be the vector solution to the system

with the initial condition $𝑣(0,𝑥)={𝑣}_0(𝑥)$ that is smooth and rapidly decaying at infinity. Show that the energy defined as

is a conserved quantity: $𝐸(𝑡)=𝐸(0)$ for all $𝑡\ge 0$.

Consider the following two regularizations of the Burgers' equation:

and

where $𝜀>0$ is a fixed number (you should think of it as being small).

(i) Look for special solutions of the first equation of the form $𝑢(𝑡,𝑥)=𝑈(𝑥-𝑐𝑡)$ with $𝑈(𝑥)$ such that $𝑈(𝑥)\to 1$ as $𝑥\to -\infty$, $𝑈(𝑥)\to 0$ as $𝑥\to +\infty$, and appropriately chosen $𝑐$, and such that $0<𝑈(𝑥)<1$ for all $𝑥\in ℝ$. (Hint: get an ODE for $𝑈(𝑥)$ and solve it.) You may assume that all derivatives of $𝑈(𝑥)$ tend to zero at infinity. This is a viscous approximation of the inviscid shock.

(ii) Show that the second equation has no such solution for $𝑣(𝑡,𝑥)$.

Consider the PDE

with an initial condition $𝑢(0,𝑥)=𝑓(𝑥)$, ${𝑢}_{𝑡}(0,𝑥)=𝑔(𝑥)$. Here, $𝑓(𝑥)$ and $𝑔(𝑥)$ are smooth bounded functions, and $𝑐(𝑥)$ is a smooth function such that there exist ${𝑐}_1>0$, ${𝑐}_2>0$ so that ${𝑐}_1\le 𝑐(𝑥)\le {𝑐}_2$ for all $𝑥\in {ℝ}^{𝑛}$.

(i) Fix ${𝑥}_0\in {ℝ}^{𝑛}$ and ${𝑅}_0>0$ and for $0<𝑡<{𝑅}_0/{𝑐}_2$ define

Show that

(ii) Suppose that $𝑓(𝑥)=𝑔(𝑥)=0$ for all $𝑥\in {ℝ}^{𝑛}$ such that $|𝑥|\ge 𝑅$. Show that $𝑢(𝑡,𝑥)=0$ if $|𝑥|\ge 𝑅+{𝑐}_2𝑡$, so that waves propagate at most with the speed ${𝑐}_2$.

Let $𝑢(𝑡,𝑥)$ be the solution to the heat equation

with an initial condition ${𝑢}_0(𝑥)\in {𝐶}_{𝑐}(ℝ)$.

(i) Show that there exists a constant $𝑀>0$ that depends on ${𝑢}_0$ so that

and

(ii) Consider the function

Show that $𝑤(𝑡,𝑥)$ satisfies

and that there exists $𝐾>0$ that depends on ${𝑢}_0$ such that

(iii) Use part (i) to estimate how close $𝑢(𝑡,𝑥)$ is to the Gaussian

(i) Let $𝐻(𝑝)={|𝑝|}^{𝑟}/𝑟$, for some $𝑟\in (1,\infty )$. Show that its Legendre transform is $𝐿(𝑦)={|𝑦|}^{𝑠}/𝑠$ with $𝑠\in (1,\infty )$ satisfying $1/𝑟+1/𝑠=1$.

(ii) Let ${𝑎}_{𝑖𝑗}$ be a symmetric positive-definite matrix, and $𝑏\in {ℝ}^{𝑛}$, and set

Compute its Legendre transform.

Suppose that a function $ℎ(𝑥)$ satisfies $|ℎ(𝑥)|\le 𝑀(1+|𝑥|)$ with some constant $𝑀>0$, and $𝐻(𝑝)$ is a convex function that is super-linear at infinity:

Set

where $𝐿(𝑦)$ is the Legendre transform of $𝐻(𝑝)$. Show that actually

where $𝑅={sup}_{𝑥\in {ℝ}^{𝑛}}|\nabla 𝐻(\nabla ℎ(𝑥))|$. Use this to show that Hamilton-Jacobi equations have a finite speed of propagation.

Let $\mathrm{\Omega }\subset {ℝ}^{𝑛}$ be a bounded domain with a smooth boundary.

(i) Fix $𝜆>0$ and consider a sequence of functions defined as follows: ${𝑢}_0\equiv 0$ and, given ${𝑢}_{𝑛}$, the function ${𝑢}_{𝑛+1}$ is the solution of the Poisson equation

with the boundary condition ${𝑢}_{𝑛+1}(𝑥)=0$ for $𝑥$ on the boundary $𝜕\mathrm{\Omega }$. Show that ${𝑢}_1(𝑥)\ge {𝑢}_0(𝑥)$ for . Use induction to show that ${𝑢}_{𝑛+1}(𝑥)\ge {𝑢}_{𝑛}(𝑥)$ for all .

(ii) If the limit $𝑢(𝑥)$ of ${𝑢}_{𝑛}(𝑥)$ exists then it satisfies

with the boundary condition $𝑢(𝑥)=0$ for $𝑥$ on the boundary $𝜕\mathrm{\Omega }$ (you do not need to prove this). Show that there is so that this equation can not have a non-negative solution for .

Let $𝑓(𝑥)$, $𝑥\in {ℝ}^{𝑛}$ be a smooth and compactly supported function and $𝑘>0$ and consider the following PDE:

(i) Show that this equation has a unique bounded solution that decays at infinity.

(ii) Is this true if $𝑘=0$? Does the answer depend on dimension?