A Faster, More Efficient Way to Solve Partial Differential Equations 

| January 24, 2023

A new method from Wade Hsu’s group is up to 30,000,000 times faster than existing methods – a game-changer for nanophotonics research and many other disciplines 

Scattering matrix

Professor Wade Hsu’s research was featured on the cover of Nature Computational Science in December 2022

Imagine a medical scope that could image the cells beneath your skin with such high resolution that doctors could diagnose cancer without having to biopsy (surgically extract) any tissue.

That’s just one of the many applications that may one day be possible from advances in nanophotonics, which involves the study of light on the nanometer scale.

Wade Hsu — an assistant professor in the USC Ming Hsieh Department of Electrical and Computer Engineering — and his team in the Optics in Complex Systems lab are working on a variety of nanophotonics projects that aim to make optical systems smaller and more powerful.

But even as the research may help miniaturize devices like cameras and endoscopes, it relies on large-scale nanophotonic simulations that model systems with light coming from thousands of different angles. These simulations require solving Maxwell’s equations — a set of partial differential equations that govern the behavior of electromagnetic waves.

It’s a time-intensive process, and conventional software systems aren’t designed for the scope of such computations. In developing technologies to peer beneath human skin and other opaque surfaces, “We had a computational problem that no existing software could handle,” says Hsu.

So, he and his team set out to design their own computational method. They stumbled on a new way to solve partial differential equations.

Their results, published in Nature Computational Science and featured on the cover of the journal’s December 2022 issue, are a game-changer not only for nanophotonics research, but for any discipline that depends on solving systems of linear equations, from acoustics to economics.

The method proposed by Hsu and PhD students Ho-Chun Lin and Zeyu Wang is called augmented partial factorization (APF). Applied to nanophotonics problems, APF can compute quantities of interest 1,000 to 30,000,000 times faster than existing methods, enabling research that was previously impossible.

Discovering a shortcut

In nanophotonics simulations, Maxwell’s equations are typically solved in full for each input of light. Multiple inputs require multiple simulations.

Using this standard approach, “It can take hours just to do a simulation for one incident angle,” says Hsu. “But we need to have light coming in over thousands of different angles, so now there are thousands of simulations, each of them taking many hours.”

The process is not only time-consuming but also inefficient. “Doing the full computation is, in fact, very wasteful,” notes Hsu. “We may compute millions of quantities but only extract a few bits of information from those.”

Hsu’s new APF method bypasses full solutions of Maxwell’s equations and takes a savvy shortcut to compute only the desired quantities. It involves first augmenting the differential operator with the inputs and outputs of interest and then partially factorizing the augmented matrix.

“We don’t factorize the whole matrix,” explains Hsu. “We stop the factorization halfway. And when we stop halfway, we get a byproduct called the Schur complement. That byproduct turns out to be exactly the quantities we want to compute.”

Though it may sound complex to the uninitiated, Hsu notes that anyone who has taken a linear algebra class will be able to understand the underlying math. The solution is so simple, in fact, that he’s surprised no one else had previously proposed it.

The approach dawned on Hsu while he was reading the user manual for MUMPS, a software package used to solve linear equations. He was exploring the software’s features when he came across a section about how to calculate the Schur complement using MUMPS. “That was when it clicked and I realized, ‘Oh, this Schur complement is exactly what I want,’” he says.

Sharing the code

The APF method is not specific to Maxwell’s equations that describe electromagnetic waves. “After developing this method, we realized actually many different problems face the same issue,” says Hsu. “It’s not just the imaging problem that motivated us, but many other applications.”

Hsu points to nanophotonic designs, stealth aircraft modeling, seismic imaging, acoustic insulation, and quantum transport simulations as a few of the applications beyond optical imaging that can utilize APF. “Any linear system may adopt this approach,” he notes.

His team has developed an open-source, general-purpose software called MESTI (Maxwell’s Equations Solver with Thousands of Inputs) that implements the APF method. It’s available on GitHub.

“By making it open source, we hope more people can use it and benefit from this method without having to write their own code,” says Hsu, who prior to this project had only been a user, not a developer, of computational methods.

“We are quite excited about this work,” he adds. “It has already enabled us and our collaborators to model many systems that were impossible to model in the past.”

Wade Hsu, Mengjie Yu, and Andrew Hires receive Chan Zuckerberg Initiative Grant to Study Neural Imaging

Professors Andrew Hires, Wade Hsu, and Mengjie Yu

Professors Andrew Hires, Wade Hsu, and Mengjie Yu

One exciting application of the nanophotonics research being conducted by Wade Hsu and his team is the development of high-resolution optical imaging for neural activity in the brain.

Knowing how neurons communicate with each other is essential for figuring out how our brain functions. Typically, to observe the function of neurons, one has to “label” the neurons by genetically engineering them to be fluorescent. However, such labeling may alter the brain’s behavior and is also not suitable for human subjects.

Now, Hsu — along with Mengjie Yu, assistant professor of Electrical and Computer Engineering, and Andrew Hires, assistant professor of Biological Sciences in the USC Dornsife College of Letters, Arts and Sciences — has been awarded a $1.25 million grant from the Chan Zuckerberg Initiative (CZI) to develop a new approach to imaging neural activity non-invasively, without labeling the neurons.

Instead of looking at fluorescence, the proposed approach will detect the transient change of shape when a neuron fires. To achieve such spatial-temporally resolved detection, Hsu will employ a 3D high-resolution optical imaging scheme his group is developing, using light sources developed by the Yu Group and the Chen Group for high-speed measurements. Hires’ Lab will provide biological expertise in the realm of neural activity and cortical circuit function.

The Chan Zuckerberg Initiative makes grants to organizations working in support of its mission to help solve some of society’s toughest challenges — from eradicating disease to improving education.

Published on January 24th, 2023

Last updated on May 16th, 2024

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