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Applied Mathematics Research

Fluid dynamics 

The Fluid Dynamics group in the Applied Mathematics Department at the University of California, Santa Cruz, combines research interests and strengths in mathematical and computational modeling of nonlinear dynamics, turbulence, climate dynamics, and applied astrophysical and geophysical fluid dynamics using analytical methods, high-performance scientific computing, and scientific machine learning. Below we highlight in more detail some of the subtopics in the area. 


Mathematical and numerical modeling applied to the discovery and understanding of astrophysical and geophysical fluid dynamics, especially the magnetohydrodynamics of stars (Pascale GaraudNicholas Brummell). Development of machine learning models for turbulent flows in engineering and natural sciences, geophysical fluid dynamics, and climate modeling with reduced computational costs (Ashesh Chattopadhyay).

High-performance scientific computing

Application of high performance computing and design of numerical experiments to explore flow dynamics (Pascale GaraudNicholas Brummell). Developing numerical schemes of high-order shock-capturing methods for computational fluid dynamics on large-scale computing architectures to investigate nonlinear flow problems (Dongwook Lee).

Uncertainty quantification (UQ) methods in fluid dynamics

Theoretical and computational methods to quantify uncertainty in fluid dynamic models subject to random initial conditions, random boundary conditions, or random physical parameters (Daniele Venturi).

Scientific Computing and Scientific Machine Learning 

Our department holds expertise in the development of numerical methods for system learning and solutions to nonlinear, high-dimensional partial differential equations. Our department has established foundations in this area from analytical to computational methods with applications to a multitude of physical systems.

Data-driven modeling 

Theoretical scientific machine learning for modeling multi-scale complex systems, e.g., geophysical turbulence in atmospheric/oceanic dynamics and climate modeling (Ashesh Chattopadhyay).  Data-driven multi-fidelity stochastic modeling (Daniele Venturi) with applications to nonlinear dynamics and control (Daniele VenturiQi Gong). 

High-dimensional dynamical systems 

Developing theoretical and numerical methods for high-dimensional dynamical systems and high-dimensional partial differential equations arising in control theory, statistical mechanics, and quantum field theory (Daniele VenturiQi GongAshesh Chattopadhyay).

Numerical Methods for Machine Learning

Developing stable and efficient numerical methods for machine learning leveraging modern deep learning theory, advanced linear algebra techniques, parallel computing, and numerical optimization (Dongwook LeeAshesh ChattopadhyayQi GongDaniele VenturiNicholas Brummell).

Optimization and control of complex systems

Natural systems are complex and challenging to model. Our department houses expertise in the application of data-informed methods from statistics, data science, and machine learning to problems in control, engineering, and scientific discovery. In many cases, the applications motivate new foundational approaches and modeling frameworks. 

Modeling and prediction of complex biological and physical systems

Development of aerial robotic systems that are automated, interconnected, and reliable for sensing and predicting atmospheric transport in real-time. (Javier Gonzalez-Rocha)

Data-driven methods for advancing work towards engineering controlled biological responses from the single-cell to tissue level. (Marcella Gomez)

Data-driven methods for genomics in cancer immunology to comprehend the complex interplay between the immune system and cancer, with the ultimate goal of developing innovative therapies for the treatment of Malignancies. (Vanessa Jonsson)

Stochastic modeling and analysis of biological motors, chemical reactions, polymer materials, and human reactions to noxious stimuli. (Hongyun Wang)

Data-driven Computational Optimal Control 

Computational optimal control, machine learning-based control, nonlinear control theory, and engineering control applications (Qi GongDaniele Venturi). Controls and state estimation for drone systems (Javier Gonzalez-RochaQi Gong).

Stochastic Systems and Uncertainty Quantification

Computing the statistical properties of nonlinear random systems is of fundamental importance in many areas of science and engineering. Our department holds expertise in state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of stochastic ordinary and partial differential equations. (Daniele VenturiQi GongHongyun Wang)