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 2020-05-03 
This is a post I wrote for The Aperiodical's Big Lock-Down Math-Off. You can vote for (or against) me here until 9am on Tuesday...
A few years ago, I made @mathslogicbot, a Twitter bot that tweets logical tautologies.
The statements that @mathslogicbot tweets are made up of variables (a to z) that can be either true or false, and the logical symbols \(\lnot\) (not), \(\land\) (and), \(\lor\) (or), \(\rightarrow\) (implies), and \(\leftrightarrow\) (if and only if), as well as brackets. A tautology is a statement that is always true, whatever values are assigned to the variables involved.
To get an idea of how to interpret @mathslogicbot's statements, let's have a look at a few tautologies:
\(( a \rightarrow a )\). This says "a implies a", or in other words "if a is true, then a is true". Hopefully everyone agrees that this is an always-true statement.
\(( a \lor \lnot a )\). This says "a or not a": either a is true, or a is not true
\((a\leftrightarrow a)\). This says "a if and only if a".
\(\lnot ( a \land \lnot a )\). This says "not (a and not a)": a and not a cannot both be true.
\(( \lnot a \lor \lnot \lnot a )\). I'll leave you to think about what this one means.
(Of course, not all statements are tautologies. The statement \((b\land a)\), for example, is not a tautology as is can be true or false depending on the values of \(a\) and \(b\).)
While looking through @mathslogicbot's tweets, I noticed that a few of them are interesting, but most are downright rubbish. This got me thinking: could I get rid of the bad tautologies like these, and make a list of just the "interesting" tautologies. To do this, we first need to think of different ways tautologies can be bad.
Looking at tautologies the @mathslogicbot has tweeted, I decided to exclude:
After removing tautologies like these, some of my favourite tautologies are:
You can find a list of the first 500 "interesting" tautologues here. Let me know on Twitter which is your favourite. Or let me know which ones you think are rubbish, and we can further refine the list...

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 2020-03-31 
Recently, you've probably seen a lot of graphs that look like this:
The graph above shows something that is growing exponentially: its equation is \(y=kr^x\), for some constants \(k\) and \(r\). The value of the constant \(r\) is very important, as it tells you how quickly the value is going to grow. Using a graph of some data, it is difficult to get an anywhere-near-accurate approximation of \(r\).
The following plot shows three different exponentials. It's very difficult to say anything about them except that they grow very quickly above around \(x=15\).
\(y=2^x\), \(y=40\times 1.5^x\), and \(y=0.002\times3^x\)
It would be nice if we could plot these in a way that their important properties—such as the value of the ratio \(r\)—were more clearly evident from the graph. To do this, we start by taking the log of both sides of the equation:
$$\log y=\log(kr^x)$$
Using the laws of logs, this simplifies to:
$$\log y=\log k+x\log r$$
This is now the equation of a straight line, \(\hat{y}=m\hat{x}+c\), with \(\hat{y}=\log y\), \(\hat{x}=x\), \(m=\log r\) and \(c=\log k\). So if we plot \(x\) against \(\log y\), we should get a straight line with gradient \(\log r\). If we plot the same three exponentials as above using a log-scaled \(y\)-axis, we get:
\(y=2^x\), \(y=40\times 1.5^x\), and \(y=0.002\times3^x\) with a log-scaled \(y\)-axis
From this picture alone, it is very clear that the blue exponential has the largest value of \(r\), and we could quickly work out a decent approximation of this value by calculating 10 (or the base of the log used if using a different log) to the power of the gradient.

Log-log plots

Exponential growth isn't the only situation where scaling the axes is beneficial. In my research in finite and boundary element methods, it is common that the error of the solution \(e\) is given in terms of a grid parameter \(h\) by a polynomial of the form \(e=ah^k\), for some constants \(a\) and \(k\).
We are often interested in the value of the power \(k\). If we plot \(e\) against \(h\), it's once again difficult to judge the value of \(k\) from the graph alone. The following graph shows three polynomials.
\(y=x^2\), \(y=x^{1.5}\), and \(y=0.5x^3\)
Once again is is difficult to judge any of the important properties of these. To improve this, we once again begin by taking the log of each side of the equation:
$$\log e=\log (ah^k)$$
Applying the laws of logs this time gives:
$$\log e=\log a+k\log h$$
This is now the equation of a straight line, \(\hat{y}=m\hat{x}+c\), with \(\hat{y}=\log e\), \(\hat{x}=\log h\), \(m=k\) and \(c=\log a\). So if we plot \(\log x\) against \(\log y\), we should get a straight line with gradient \(k\).
Doing this for the same three curves as above gives the following plot.
\(y=x^2\), \(y=x^{1.5}\), and \(y=0.5x^3\) with log-scaled \(x\)- and \(y\)-axes
It is easy to see that the blue line has the highest value of \(k\) (as it has the highest gradient, and we could get a decent approximation of this value by finding the line's gradient.

As well as making it easier to get good approximations of important parameters, making curves into straight lines in this way also makes it easier to plot the trend of real data. Drawing accurate exponentials and polynomials is hard at the best of times; and real data will not exactly follow the curve, so drawing an exponential or quadratic of best fit will be an even harder task. By scaling the axes first though, this task simplifies to drawing a straight line through the data; this is much easier.
So next time you're struggling with an awkward curve, why not try turning it into a straight line first.

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 2020-02-17 
This is the sixth post in a series of posts about my PhD thesis.
Before we move on to some concluding remarks and notes about possible future work, I must take this opportunity to thank my co-supervisors Timo Betcke and Erik Burman, as without their help, support, and patience, this work would never have happened.

Future work

There are of course many things related to the work in my thesis that could be worked on in the future by me or others.
In the thesis, we presented the analysis of the weak imposition of Dirichlet, Neumann, mixed Dirichlet–Neumann, Robin, and Signorini boundary conditions on Laplace's equation; and Dirichlet, and mixed Dirichlet–Neumann conditions on the Helmholtz equation. One area of future work would be to extend this analysis to other conditions, such as the imposition of Robin conditions on the Helmholtz equation. It would also be of great interest to extend the method to other problems, such as Maxwell's equations. For Maxwell's equations, it looks like the analysis will be significantly more difficult.
In the problems in later chapters, in particular chapter 4, the ill-conditioning of the matrices obtained from the method led to slow or even inaccurate solutions. It would be interesting to look into alternative preconditioning methods for these problems as a way to improve the conditioning of these matrices. Developing these preconditioners appears to be very important for Maxwell's equations: general the matrices involved when solving Maxwell's equations tend to be very badly ill-conditioned, and in the few experiments I ran to try out the weak imposition of boundary conditions on Maxwell's equations, I was unable to get a good solution due to this.

Your work

If you are a undergraduate or master's student and are interested in working on similar stuff to me, then you could look into doing a PhD with Timo and/or Erik (my supervisors). There are also many other people around working on similar stuff, including:
There are of course many, many more people working on this, and this list is in no way exhaustive. But hopefully this list can be a useful starting point for anyone interested in studying this area of maths.
Previous post in series
PhD thesis, chapter 5
This is the sixth post in a series of posts about my PhD thesis.

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PhD thesis, chapter 5
PhD thesis, chapter 4
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 2020-02-16 
This is the fifth post in a series of posts about my PhD thesis.
In the fifth and final chapter of my thesis, we look at how boundary conditions can be weakly imposed on the Helmholtz equation.

Analysis

As in chapter 4, we must adapt the analysis of chapter 3 to apply to Helmholtz problems. The boundary operators for the Helmholtz equation satisfy less strong conditions than the operators for Laplace's equation (for Laplace's equation, the operators satisfy a condition called coercivity; for Helmholtz, the operators satisfy a weaker condition called Gårding's inequality), making proving results about Helmholtz problem harder.
After some work, we are able to prove an a priori error bound (with \(a=\tfrac32\) for the spaces we use):
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$

Numerical results

As in the previous chapters, we use Bempp to show that computations with this method match the theory.
The error of our approximate solutions of a Dirichlet (left) and mixed Dirichlet–Neumann problems in the exterior of a sphere with meshes with different values of \(h\). The dashed lines show order \(\tfrac32\) convergence.

Wave scattering

Boundary element methods are often used to solve Helmholtz wave scattering problems. These are problems in which a sound wave is travelling though a medium (eg the air), then hits an object: you want to know what the sound wave that scatters off the object looks like.
If there are multiple objects that the wave is scattering off, the boundary element method formulation can get quite complicated. When using weak imposition, the formulation is simpler: this one advantage of this method.
The following diagram shows a sound wave scattering off a mixure of sound-hard and sound-soft spheres. Sound-hard objects reflect sound well, while sound-soft objects absorb it well.
A sound wave scattering off a mixture of sound-hard (white) and sound-soft (black) spheres.
If you are trying to design something with particular properties—for example, a barrier that absorbs sound—you may want to solve lots of wave scattering problems on an object on some objects with various values taken for their reflective properties. This type of problem is often called an inverse problem.
For this type of problem, weakly imposing boundary conditions has advantages: the discretisation of the Calderón projector can be reused for each problem, and only the terms due to the weakly imposed boundary conditions need to be recalculated. This is an advantages as the boundary condition terms are much less expensive (ie they use much less time and memory) to calculate than the Calderón term that is reused.

This concludes chapter 5, the final chapter of my thesis. Why not celebrate reaching the end by cracking open the following figure before reading the concluding blog post.
An acoustic wave scattering off a sound-hard champagne bottle and a sound-soft cork.
Previous post in series
PhD thesis, chapter 4
This is the fifth post in a series of posts about my PhD thesis.
Next post in series
PhD thesis, chapter ∞

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 2020-02-13 
This is the fourth post in a series of posts about my PhD thesis.
The fourth chapter of my thesis looks at weakly imposing Signorini boundary conditions on the boundary element method.

Signorini conditions

Signorini boundary conditions are composed of the following three conditions on the boundary:
\begin{align*} u &\leqslant g\\ \frac{\partial u}{\partial n} &\leqslant \psi\\ \left(\frac{\partial u}{\partial n} -\psi\right)(u-g) &=0 \end{align*}
In these equations, \(u\) is the solution, \(\frac{\partial u}{\partial n}\) is the derivative of the solution in the direction normal to the boundary, and \(g\) and \(\psi\) are (known) functions.
These conditions model an object that is coming into contact with a surface: imagine an object that is being pushed upwards towards a surface. \(u\) is the height of the object at each point; \(\frac{\partial u}{\partial n}\) is the speed the object is moving upwards at each point; \(g\) is the height of the surface; and \(\psi\) is the speed at which the upwards force will cause the object to move. We now consider the meaning of each of the three conditions.
The first condition (\(u \leqslant g\) says the the height of the object is less than or equal to the height of the surface. Or in other words, no part of the object has been pushed through the surface. If you've ever picked up a real object, you will see that this is sensible.
The second condition (\(\frac{\partial u}{\partial n} \leqslant \psi\)) says that the speed of the object is less than or equal to the speed that the upwards force will cause. Or in other words, the is no extra hidden force that could cause the object to move faster.
The final condition (\(\left(\frac{\partial u}{\partial n} -\psi\right)(u-g)=0\)) says that either \(u=g\) or \(\frac{\partial u}{\partial n}=\psi\). Or in other words, either the object is touching the surface, or it is not and so it is travelling at the speed caused by the force.

Non-linear problems

It is possible to rewrite these conditions as the following, where \(c\) is any positive constant:
$$u-g=\left[u-g + c\left(\frac{\partial u}{\partial n}-\psi\right)\right]_-$$
The term \(\left[a\right]_-\) is equal to \(a\) if \(a\) is negative or 0 if \(a\) is positive (ie \(\left[a\right]_-=\min(0,a)\)). If you think about what this will be equal to if \(u=g\), \(u\lt g\), \(\frac{\partial u}{\partial n}=\psi\), and \(\frac{\partial u}{\partial n}\lt\psi\), you can convince yourself that it is equivalent to the three Signorini conditions.
Writing the condition like this is helpful, as this can easily be added into the matrix system due to the boundary element method to weakly impose it. There is, however, a complication: due to the \([\cdot]_-\) operator, the term we add on is non-linear and cannot be represented as a matrix. We therefore need to do a little more than simply use our favourite matrix solver to solve this problem.
To solve this non-linear system, we use an iterative approach: first make a guess at what the solution might be (eg guess that \(u=0\)). We then use this guess to calculate the value of the non-linear term, then solve the linear system obtained when the value is substituted in. This gives us a second guess of the solution: we can do follow the same approach to obtain a third guess. And a fourth. And a fifth. We continue until one of our guesses is very close to the following guess, and we have an approximation of the solution.

Analysis

After deriving formulations for weakly imposed Signorini conditions on the boundary element method, we proceed as we did in chapter 3 and analyse the method.
The analysis in chapter 4 differs from that in chapter 3 as the system in this chapter is non-linear. The final result, however, closely resembles the results in chapter 3: we obtain and a priori error bounds:
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$
As in chapter 3, The value of the constant \(a\) for the spaces we use is \(\tfrac32\).

Numerical results

As in chapter 3, we used Bempp to run some numerical experiments to demonstrate the performance of this method.
The error of our approximate solutions of a Signorini problem on the interior of a sphere with meshes with different values of \(h\) for two choices of combinations of discrete spaces. The dashed lines show order \(\tfrac32\) convergence.
These results are for two different combinations of the discrete spaces we looked at in chapter 2. The plot on the left shows the expected order \(\tfrac32\). The plot on the right, however, shows a lower convergence than our a priori error bound predicted. This is due to the matrices obtained when using this combination of spaces being ill-conditioned, and so our solver struggles to find a good solution. This ill-conditioning is worse for smaller values of \(h\), which is why the plot starts at around the expected order of convergence, but then the convergence tails off.

These results conclude chapter 4 of my thesis. Why not take a break and snack on the following figure before reading on.
The solution of a mixed Dirichlet–Signorini problem on the interior of an apple, solved using weakly imposed boundary conditions.
Previous post in series
PhD thesis, chapter 3
This is the fourth post in a series of posts about my PhD thesis.
Next post in series
PhD thesis, chapter 5

Similar posts

PhD thesis, chapter 5
PhD thesis, chapter 3
PhD thesis, chapter ∞
PhD thesis, chapter 2

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