Solving the Unsolvable

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Have you ever wondered about the math behind a radio signal? Or telephones? Scientists who study electromagnetic physics and mathematics constantly are: their work is founded upon partial differential equations. These equations are unlike the differential equations we encounter in calculus: they are three-dimensional and extremely difficult to solve because they vary with both time and space, even for simple shapes. As a result, designing cell phone antennas and radio telescopes is extremely challenging. To solve these wave equations, scientists use a technique that separates the part which varies with time and the part which varies with space, also known as “Helmholtz’s equation.” However, Helmholtz’s equation is still a differential equation, which is hard to solve. Scientists have been using a trick called Green’s functions to solve these sorts of problems for over a hundred years. Green’s functions are powerful techniques that let you solve a differential equation by computing an integral, making the solution easier to attain. However, even though they can convert these problems from differential equations into integral equations, they still have to do it in 3D, which is extremely hard. 

However, for certain types of symmetric objects, like a radar dish, there is a trick that turns a 3D problem into a 2D problem. However, it comes at a price. The Green’s function would have to be split up into its individual frequencies using the Fourier series technique. For each frequency, a very difficult integral, called “the modal Green’s function,” would have to be computed. This Green’s function is tricky: it oscillates incredibly quickly between massive positive and negative values, but somehow, they all end up adding up to something tiny. Any attempt to estimate the integral by “adding it up” essentially adds and subtracts infinity trillions of times, causing so much error that the estimate is useless. Scientists have been struggling with how to compute this integral since the 1960s.

To handle difficult integrals, mathematicians will often solve them using complex analysis, the field of math that studies imaginary numbers. Real numbers are plotted on the “real line,” ranging from negative to positive infinity. In contrast, complex numbers have real and imaginary components, meaning they live in a 2D space called the “complex plane.” Amazingly, for many functions, the integral can either be computed along the real line or as a “contour integral” through the 2D complex plane, a technique invented by the mathematician Augustin-Louis Cauchy in the nineteenth century. By carefully choosing the contour, sometimes those integrals can be made very simple. For example, some functions oscillate between negative one and positive one on the real line, but in the complex plane, they never oscillate.  

For decades, scientists had written off using this contour trick because no matter which contour they picked, part of the Green’s function wildly exploded and oscillated. James Garritano, along with his mentors Yuval Kluger, Vladimir Rokhlin, and Kirill Serkh at the Kluger Lab in the Yale School of Medicine, recently overcame the oscillations of Green’s function by ignoring the real line and integrating it into the complex plane by using a contour invented in 2010 by a scientist named Mats Gustafsson. The key idea of their paper was to replace part of the Green’s function with an approximation that does not grow in the complex plane. Then, they could use Gustafsson’s contour and avoid wild oscillations.

“Dr. Kluger and Dr. Rokhlin, who I work for, are both famous for creating fast algorithms to solve problems in physics and genomics,” Garritano said. For example, with his student Leslie Greengard, Rokhlin created the fast multipole algorithm, named one of the top ten algorithms of the twentieth century. It became the basis for a wide range of physics simulations ranging from modeling gravitational bodies to electrons. Kluger recently developed the fastest method for clustering data in single-cell experiments by using insights from computational physics to accelerate one of the key algorithms of data science.

This work has paved the way to create ultra-fast physics solvers for rotationally symmetric objects such as antennas and radar dishes. Also, their work showed that Cauchy’s contour-trick could be applied to more problems than previously thought.