Bobbie's Blog

Potential theory was developed a few centuries ago in part to solve the boundary value problems for partial differential equations (PDEs). It led to the so-called “indirect approach” to boundary integral equations for elliptic PDEs. The goal of this article is to give this indirect approach some physical meaning. Some basic knowledge of potential theory is assumed. (See the book by R. Kress¹ for more details.)

Consider the classical Laplace equation in a domain $\Omega\subset\mathbb{R}^2$, with Dirichlet boundary condition on the boundary $\Gamma:=\partial\Omega$, written as

The theory of integral equations solves a given boundary value problem like this by reformulating it into an integral equation on the boundary of the domain, such that the 2D spatial differential problem in $\Omega$ is reduced to a 1D boundary integral problem on $\Gamma$.

There are two main approaches to integral equations: the direct approach and the indirect approach.

Knowledge is information plus understanding. A big part of doing scientific research is to communicate knowledge with others. But knowing how to effectively communicate ideas is to some extend an art that is not so easily acquired.

One key ingredient of public communication is to know your audience well. I would like to propose a visual model that can be helpful for adjusting your presentation based on the type of your audience.

I call this model the Audience Lightcone:

The Pareto principle (a.k.a. 80/20 rule) says that for many outcomes, roughly 80% of consequences come from 20% of the causes. Qualitatively speaking, this also applies to numerical integration: most of the integral can be handled by just a few quadrature rules.

People who have learned numerical analysis all know about the Gauss quadrature rule: for any integer $n>0$, there exist $n$ nodes $x_1,x_2,\dots,x_n\in[-1,1]$ and $n$ corresponding weights $w_1,w_2,\dots,w_n$, such that the approximation

is highly accurate for any function $f$ smoothly defined on $[-1,1]$. The error of this approximation typically decays exponentially as $n$ increases. Together with scaling and shifting of variables, the Gauss quadrature efficiently handles all the regular integrals (i.e. integrals involving smooth integrands) on any interval $[a,b]\subset\mathbb{R}$.

What if the integrand is singular? How would you approximate an integral when the integrand is not smooth or even blowing up somewhere on the interval? It turns out that most of the singular integrals in practice can be handled by a few strategies (80/20 rule again!). Here they are:

… and anagrams. (How cute!)

The Reactionary Approach

The reactionary approach seems to be a default setting of human behavior: people react only when something undesirable happens. Consequently, people have to react every single time something undesirable happens. This is apparently so in politics (especially prominent in 2020). But you can observe reactionary behaviors in all other domains.

Y Combinator (YC) is a startup incubator that had launched many successful companies including Airbnb, Coinbase, Dropbox, Reddit, etc. YC’s founder Paul Graham wrote an inspiring article, “Do Things That Don’t Scale,” that gives solid advice to startup founders on what they should focus on when starting a business. I find it interesting that when I read the article through the lens of academia, Graham’s advice appears to also be useful for people heading into an academic career, such as myself.

I have noticed that there are a lot of similarities between building a startup company and building an academic career:

Yesterday was the last day of 2020 and the first day that was cold and rainy and windy all day since I resumed exercising every day. I had been expecting such a day.

At the beginning of December after reading a blog post, I finally decided to do a 365-Day Challenge for the year of 2021: running every day for at least 15 minutes a day. Then I decided to start right away even if it was still 2020, because why not. This may sound like a joke to some people because, seriously, how is this even a “challenge”?

I feel like writing something before 2020 ends. Since this has been a special year, I will write two versions of it.

Version I

This has been a special year because everybody said that it was bad. The COVID pandemic suddenly exploded and completely changed our way of living. People may still remember that this is the same year when Kobe Bryant died from an accident, but it feels like this happened an eternity ago — as with everything else that happened before lockdown.

来 Austin 一段时间了都没写什么。今天恰巧有个好故事可以讲一讲。

室友办公桌的几个插口没电，前天报修了。今天一大早回来，我发现办公室门竟敞开着，走到门口前，只见室友的办公桌前赫然坐着两位老头在聊天。跟他们聊了几句，原来是来做维修的电工，为了修这几个电插口颇费了一番工夫。

看起来年纪没那么大的老头说，这个插头不行是因为天花板上的线有问题；天花板上的线有问题呢，又是因为办公室另一头墙里的线有问题；这些问题会发生，可能就是上一个负责线路的人，因为某种原因线没接好就放着不管了。

我听他解释一大轮，只感到不明觉厉。我问，这肯定很难才诊断出来吧？他指着另外一个很老的老头说：“花了一个多小时才找出原因，都是他的功劳，他叫 George。” 我不禁肃然起敬，有种遇见少林扫地僧的感觉。我说：“非常感谢，我一定转告我的室友！”

他们走后，我一边回味一边感叹，这不就是一次电工版的 debug 吗？前人坏习惯留下的 bug，让后来的优秀程序员几经波折才修好。可见遵从 best practice 的程序员是多么可贵。

I recently switched to a new computer. The switch isn’t too messy because I basically have all my data backed up, either on Google Drive or on GitHub.

One thing I wanted to make sure is that I can still update this blog, which is hosted using Octopress. But to be honest, until recently I am not really good at using git and GitHub, so it took me some time to figure out what I need to do to resume blogging with Octopress.

Why GNU Screen?

Scenario: you finish working with an SSH session, you close your laptop to go for lunch or for a tea. Then you come back and open your laptop wanting to resume your job, but the connection is broken or the VPN breaks and forced you to disconnect. You have no choice but setup the connection once again and reopen the documents/apps before you can resume from where you left.