The PhD program in Applied and Computational Mathematics at Caltech focuses on solving complex scientific and engineering problems using analytical rigor and computational methods. Below are advanced research topics for exploration.
Numerical Methods for Solving High-Dimensional Partial Differential Equations
Computational Fluid Dynamics Using Finite Volume Schemes
Multiscale Modeling in Materials Science
Inverse Problems in Medical Imaging and Signal Processing
Mathematical Foundations of Machine Learning Algorithms
Sparse Matrix Computations for Scientific Simulations
Optimal Transport Theory in Machine Learning Applications
Bayesian Inference in Uncertain Dynamical Systems
Graph Neural Networks and Their Mathematical Properties
Adaptive Mesh Refinement for Hyperbolic PDEs
Stochastic Differential Equations in Finance and Physics
Quantum Computing Algorithms for Matrix Computation
Topological Data Analysis for Complex Systems
Reduced Order Modeling of Turbulent Flow Fields
High-Performance Computing for Real-Time Simulations
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