## Available projects for undergraduate and master-level Engineering students.

### Can you hear the shape of a tumor

• In 1966 the Polish American mathematician Mark Kac asked the question: Can One Hear the Shape of a Drum? Using today medical imaging techniques, we can hear tumors! Photoacoustic tomography is an emerging medical imaging modality that uses light and sound to create 3D images of hemoglobin concentration and oxygen saturation within a target tissue. Since tumors consume a lot of energy to grow, regions of the image showing high hemoglobin concentration and low oxygen saturation may hint to the presence of tumor.

• In this project, we will use partial differential equations to model light and sound propagation, large-scale optimization methods to reconstruct images of hemoglobin concentration, machine learning to further refine those images.

• Suggested background: Python, TensorFlow, numerical analysis, partial differential equations, and finite element methods.

### Make each measurement count: smaller data, richer information

• Where should I measure? For how long? How often? Optimal design of experiments is an information-theoretic framework to guide how data are collected to maximize accuracy and reduce uncertainty in parameter estimation and image reconstruction problems.
• In this project, we will use numerical simulations and partial differential equations to model the measurement process, statistics and information theory to quantify the information gain using a particular set of measurements, and large-scale optimization to find the best design.
• Suggested background: Python, numerical optimization, linear algebra, partial differential equations.

### ADLA: Automatic differentiation and local assembly for finite element methods

• MFEM is a free, lightweight, scalable C++ library for finite element methods that can solves partial differential equations with (bi)millions of parameters on the fastest supercomputer of the world! The goal of this project is to develop a general, lightweight and user-friendly framework in MFEM to evaluate user-defined finite element functionals and their first/second-order variations using automatic differentiation.
• Suggested background: scientific computing, finite element methods
• Required skill: Fluent in C++ programming