Lectures

Lecture 1 (1/5)

• Key questions: distributions, sampling, features of a system
• Random Sampling
• Ising Model
• The main idea is that sampling allows integration, scales better than grids, and solves impossible sums.

Lecture 2 (1/7)

Example 1:

Let $x(t)$ be a particle with the following ODE

$$\begin{array}{r}\frac{dx}{dt}=-\frac{du}{dx}\end{array}$$

Let $u(x)=\frac{1}{2}{x}^2$. Then

$$\begin{array}{rl}u(x)& =\frac{1}{2}{x}^2\\ \frac{du}{dx}& =x\end{array}$$

such that

$$\frac{dx}{dt}=-x$$

Upon solving this ODE, we get

$$x(t)=x(0)\cdot e^{-t}$$

So, as $t\to \infty ,x\to 0$.

Example 2:

We could have a double well

$$u(x)=\frac{1}{4}(x^2-1{)}^2$$

In the SDE case, given

$$dx=-u^{\prime }(x)dt+b(t)d{B}_t$$

In order, the terms are:

• $dx$ is the change $x$
• $u^{\prime }(x)$ is the deterministic force
• $b(t)d{B}_t$ is some “random” force. In particular, $d{B}_t$ is the Brownian increment, or “white noise”.

So when we calculate the position of the particle via $x(t)$, the $u^{\prime }(t)$ term tells us how we move via the force, with some randomness via $b(t)d{B}_t$. This leads to

• new frequency in a system

what does frequency mean?

• new types of resonance/transport effects.

Lecture 3 (1/9)

Reference for this section is Rubinstein Ch.2

• Inversion
• Transformation of Random Variables

Lecture 4 (1/12)

Reference: Rubinstein Ch. 2, 2.4, 2.3.4

• Box-Muller Method
• Acceptance-Rejection

Lecture 5 (1/14)

• Acceptance-Rejection
• [[Acceptance-Rejection#theorem-uniformity-on-region-a|Theorem (Uniformity on Region $A$)]]
• [[Acceptance-Rejection#theorem-from-uniformity-on-a-to-target-fx|Theorem (From Uniformity on $A$ to Target $f(x)$)]]
• [[Acceptance-Rejection#theorem-a-r-generates-z-sim-f|Theorem (A-R Generates $Zsimf$)]]

Lecture 6 (1/16)

• Efficiency
• Modified A-R
• Example 1 (Sampling Standard Normal via A-R)
• Discrete Case

Lecture 7 (1/21)

• Monte-Carlo Integration
• Variance Reduction

Lecture 8 (1/23)

• Importance Sampling
• Example 1