Solving Real-World Problems with the Power of Math

At the 42nd Student Research Conference, Tran Sy Toan and Nguyen Quoc Anh from Cohort 66, Faculty of Mathematics and Informatics, HUST, presented a new mathematical method for finding approximate solutions to complex problems in fields like medicine, artificial intelligence, and image processing. Their research, titled “Two-Step Inertial Relaxation Algorithm for Solving a Class of Split Variational Inequality Problems and Applications,” was supervised by Assoc. Prof. Nguyen Thi Thu Thuy.

Using approximate math methods to solve real-world problems

Tran Sy Toan and Nguyen Quoc Anh began their research driven by a deep passion for mathematics. Not wanting their love for math to stay confined to the classroom, the two friends teamed up to apply math to real-world problems.

After presenting their idea to Assoc. Prof. Nguyen Thi Thu Thuy, Faculty of Mathematics and Informatics, HUST, they received full support and guidance. The duo started their research project in mid-2024, focusing on solving optimization and nonlinear equation problems with new mathematical methods.

The research project "A Relaxed Two-Step Inertial Algorithm for Solving a Class of Split Variational Inequality Problems and Applications" focuses on developing a method to solve complex optimization problems in real Hilbert spaces.

Sy Toan and Quoc Anh applied their method to real-world problems, including cancer treatment planning (IMRT) and image restoration. For image restoration, they modeled a blurred image as the result of a convolution with a kernel matrix. The goal was to recover a high-quality version, measured by the Signal-to-Noise Ratio (SNR), rather than the exact original.

They framed the task as a Split Feasibility Problem (SFP): finding an image 𝑥 such that both 𝑥 and its transformed version 𝐹(𝑥) fall within the pixel range [0, 255]. This setup forms a closed convex set, making it ideal for applying modern optimization algorithms. The transformation 𝐹 simulates the blurring effect and is mathematically represented by a matrix multiplication. This approach turns a practical image processing challenge into a solvable mathematical problem using approximation methods.
 

In cancer treatment, IMRT is an advanced technique that allows precise control of radiation doses delivered to the body. The body is divided into two main regions: Planning Target Volumes (PTV), which need high radiation to destroy cancer cells, and Organs At Risk (OAR), which must be protected by minimizing exposure.

The planning process is modeled by a linear equation 𝑦 = 𝐷𝑥, where 𝐷 is the dose matrix showing how each beam affects different body voxels, 𝑥 is the vector of beam intensities, and 𝑦 is the resulting dose vector. The goal is to find an 𝑥 that ensures PTVs receive enough radiation while OARs receive as little as possible.

Sy Toan and Quoc Anh used approximation methods with simple vector-based iterations, making the solution computationally efficient and practical for real-world CPU processing. 
 

"One for all - All for one" mindset

In their research team, Nguyen Quoc Anh focused on programming, testing, and writing technical descriptions for the models, while Tran Sy Toan handled the theoretical proofs, presentation, and formatting of the research paper. Together, they co-authored the scientific article, reviewed each other’s work, and aimed to produce a polished, high-quality academic product.

“Our biggest challenge was managing a large volume of knowledge and data in a short time,” Quoc Anh shared. They also had to self-learn various programming tools, domain-specific software, and model evaluation methods.

There were times when results didn’t meet expectations. Some models didn’t converge, and some algorithms failed, but the duo didn’t give up.

“If something went wrong, we’d recheck every assumption, tweak parameters, and dig deeper into the original algorithm until it worked,” Quoc Anh said with a smile, as if rebuilding a study from scratch was just routine.

With a “One for all - All for one” mindset, the two divided tasks based on their strengths and supported each other throughout. They also sought guidance from professors and drew on previous studies to stay on track.
 

Speaking about their advisor, Assoc. Prof. Nguyen Thi Thu Thuy, Tran Sy Toan shared: “She’s incredibly dedicated and inspires us to keep pushing forward. With her, every question and proof must be justified, accurate, and clean. Watching her work made us take our research more seriously.”

Their hard work paid off. The team’s research proved that the relaxed two-step inertial algorithm can effectively solve complex real-world problems with constraints. Their project won First Prize in the Applied Mathematics and Informatics category at HUST 42nd Student Research Conference.

Wherever there’s data and a mathematical model, their algorithm could be applied. Building on that potential, Quoc Anh and Sy Toan are planning to expand their work into interdisciplinary areas like data science, economics, and transportation.