the official magazine for Scientific Malaysian

Inverse Problem

What is it and why should you care?

By Sean Lim Wei Xinq The subject of inverse problems in mathematics is an interesting one. In the mathematical community, the subject is classified under applied mathematics, and is generally regarded as a relatively new branch of mathematics. This is true in the sense that the general abstraction of an inverse problem was not considered up until possibly the early 2000s by Albert Tarantola. Tarantola’s work was motivated primarily by the seismic inversion problem, which is the task of constructing images of the subsurface of the Earth based on observations on seismic waves. His book on Inverse Problem Theorya– available freely on his webpage – is considered one of the standard texts on the theory of inverse problems today. However, particular inverse problems have been studied in the past, going back as early as the Babylonians, but more on that later.

Despite an increasing interest in inverse problems, it is a philosophically non-trivial task to pinpoint exactly what an inverse problem is. In his book, “Inverse Problems: Activities for Undergraduates”, Charles Groetsch adopts the approach of Justice Potter Stewart on the subject of defining an inverse problem: “I know it when I see it.” He then proceeds to present specific inverse problems considered in the past, and added the comment that “Inverse problems come paired with direct problems and of course the choice of which problem is called direct and which is called inverse is, strictly speaking, arbitrary.” Today, the term “forward problem” is more commonly used instead of “direct problem” as a contrast to the associated “inverse problem”.

Consider the following problem as an example: Given two integers m and n, compute its product, k. This is the forward problem – given an input, compute its output. Heuristically, the inverse problem is the opposite of this: Given its output, work out what the inputs were. In the given problem this can be rephrased as: Given an integer k, find integers m and n such that k=mn. This is a problem of factorization, and one can see that such a problem is not easy to solve, in that it is not unique. For example, if m=6 and n=4, one can easily deduce that k=mn=24. However, given that k=24, it is not easy to recover m and n since there are several combinations which produce 24:

24 = 1 × 24

24 = 2 × 12

24 = 3 × 8

24 = 4 × 6.

Mathematically speaking, we say that this problem is ill-posed, because the solution is non-unique. When it is determined that a problem has non-unique solutions, one of four things can be done. First, we try to ask a similar but different question, so that the solution is then unique. Secondly, we study the problem extensively, asking when the solutions are unique and when they are not, and classify them as completely as possible. If one of the two aforementioned tasks are unable to be executed, we resort to the remaining two solutions: We talk to someone about it, in hopes that collaborative effort may yield results. After all, two heads are better than one. If this bears no fruit, we give up and work on other problems. Sad, but such is the nature of mathematical research, not every problem has a solution, and not every solution is easy to find. One may argue that a fifth thing that mathematicians do when they hit a dead end is that they set it as a student project, but we leave the details of this as an exercise for the reader.

Coming back to the problem of factorisation, we now need a way to “fix” the problem such that it has a unique solution. Mathematically speaking, when a problem has non-unique solutions, it is considered “ill-posed”, and to “fix” an ill-posed problem such that it becomes well-posed is called the process of regularisation. In this problem, the regularisation comes in the form of the prime numbers. For the purposes of this article, let us define prime numbers to be positive integers with exactly two factors, or divisors — the number 1 and itself. Hence by definition, 1 is not prime, and 2 is the smallest prime. The problem of factorisation now is regularised into a problem of prime factorisation, stated as follows: Given a positive integer n, what is its prime factorisation?

Here are some examples:

1.  9 = 32

2.  120 = 23 x 3 x 5

3.   8208 = 24 x 33 x 19

Now, we would like to ask the question again: Is this prime factorisation unique?  In other words, given a positive integer n, is there more than one way of decomposing n into prime divisors?

True enough, it turns out that there is no other way of factorising a positive integer into its prime divisors!  In other words, prime factorisation of integers is unique, and therefore this inverse problem is well-posed, and it is solved!  In fact, this fact is of such great significance that it is called the Fundamental Theorem of Arithmetic. It tells us that every positive integer greater than 1 is either a prime or a product of primes, and that the factorisation is unique (up to rearrangements, so that 3×5 is the same as 5×3). Of course, the next question one may ask is, what is so “fundamental” about the Fundamental Theorem of Arithmetic?  In other words, given an integer, we now know that we can write it as a product of primes, if it itself is not a prime, but so what?

To start off, this theorem tells us that primes are the basic building blocks of the integers. This means that in order to construct any integer, all we need is to take our building blocks, i.e. the primes, and multiply them, and that gives us a new integer. The primes are to integers whatthe periodic table of elements are to chemistry, and pawns to chess!

Therefore, to understand the integers, all we need to do is to study the primes. Once we understand the primes, then we can understand the integers. The study of integers and primes form one of the oldest subjects in mathematics called number theory (i.e., the study of positive integers, including their properties). Some of you may recognise the field of number theory as one of the hardest branches in mathematics. Indeed, one of the reasons why questions in number theory have confounded people for hundreds of years (dating back as early as Pythagoras) is because the behaviour of primes are not so easily understood. A host of questions regarding primes and integers have been asked over the years. Today, some have been answered, but there are many problems that remain open even until today.

So there you have it – the reason why we should we care about inverse problems. Turns out it is the inverse problem, and not the forward problem, that gave rise to the study of the numbers that we so often use everyday!

Footnote

a Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics. http://www.ipgp.fr/~tarantola/Files/Professional/Books/InverseProblemTheory.pdf