A new era in numerical computation

Rockets carry out critical missions. They carry satellites into orbit that deliver important information, and they’ve recently taken humans farther than we’ve ever gone before. Rockets are also expensive and extremely heavy, and they require reliable design to ensure they are structurally sound and can accurately make their journey. By increasing the total payload, or weight the rocket can carry, while minimizing the structural weight of the rocket, we can save on manufacturing cost and launch heavier satellites with better and more comprehensive functionality.

A process called numerical optimization can find the best way to reduce the structural weight of a rocket while maintaining its power and structural integrity. Numerical optimization involves mathematical algorithms that identify the best solution to a problem, and it has applications across many science and engineering fields. 

Not only does numerical optimization support aerospace engineering, but it can help develop life-saving healthcare by determining exact drug dosages to properly treat an illness, as opposed to under-treatment or even potential adverse effects. Other potential use cases range from improving the fuel efficiency of automobiles to minimizing the operational costs of chemical plants.

However, existing numerical optimization methods are not guaranteed to find the best solution in general. For realistic problems, they often come up with different solutions when run multiple times, leaving it up to engineers to choose the solution they believe may be the best or most desirable. In addition, these methods often don’t encapsulate all possible solutions to the problem.

Many modern computers now operate on graphics processing units, or GPUs, that are originally designed for digital image processing and prioritize computer graphics. GPU computing offers unprecedented computational power to us, opening new opportunities for numerical optimization.

This is the beginning of a new era

Guanglu Zhang, Research Scientist, Mechanical Engineering

Guanglu Zhang, a research scientist in Jonathan Cagan’s lab, says that we generally use less than half of the GPU computing power currently available to us. “When our computational hardware changes, our computational software—especially numerical methods and algorithms—need to change accordingly,” says Zhang. In other words, the software we use should match the capabilities and architecture of the hardware it runs on.

Often working alongside a central processing unit (CPU), which is designed to take care of one computational task at a time, GPUs run different processes parallel to each other. They can intake and handle a set of instructions and carry each task out simultaneously. However, many of the systems and software we have access to, especially existing numerical optimization methods, can’t efficiently employ GPU processing. They are trying to accelerate computation through methods that are originally designed for CPU, leading to an underutilization of the GPU’s computational power.

Zhang is working alongside Cagan, head and professor of mechanical engineering, and Ph.D. student Qihang Shan to identify guaranteed best solutions for optimization problems in practice, utilizing both the computational capability and architecture of modern GPUs. The group has authored papers published in PNAS Nexus and JCISE that explain how to efficiently utilize GPUs to more accurately solve optimization problems. The paper in PNAS Nexus is a more technical exploration of the topic, while the paper in JCISE focuses more on applications of the method in engineering design.

The new methodology only needs to be run once, and it definitively reveals the best solution. It works by first ruling out the regions where the best solution can not be, then providing a finite set of remaining regions where it subsequently must be. By using interval arithmetic, a technique that defines a value as a range rather than a single number, the new methodology accounts for rounding errors, paving the way for more efficient and accurate results.

The method is still only in its beginning stages, and as the research continues the lab expects to see even more use cases, extending to very different fields like artificial intelligence and machine learning. So far, they have solved multiple well-known benchmark functions and engineering problems using a new parallel programming style they developed based on GPU architecture, among other innovations that allow for solving larger-scale optimization problems with many variables.