Applied Mathematics

Ph.D., Massachusetts Institute of Technology, 1987

Teaching

APMA1170 - Introduction to Computational Linear Algebra 

APMA2550 - Numerical Solution of Partial Differential Equations I

APMA 2560 - Numerical Solution of Partial Differential Equations II

APMA2580 - Multiscale Computational Fluid Dynamics

Biography

George Karniadakis received his S.M. (1984) and Ph.D. (1987) from Massachusetts Institute of Technology. He was appointed Lecturer in the Department of Mechanical Engineering at MIT in 1987 and subsequently he joined the Center for Turbulence Research at Stanford / Nasa Ames. He joined Princeton University as Assistant Professor in the Department of Mechanical and Aerospace Engineering and as Associate Faculty in the Program of Applied and Computational Mathematics. He was a Visiting Professor at Caltech (1993) in the Aeronautics Department. He joined Brown University as Associate Professor of Applied Mathematics in the Center for Fluid Mechanics on January 1, 1994. He became a full professor on July 1, 1996.  He has been a Visiting Professor and Senior Lecturer of Ocean/Mechanical Engineering at MIT since September 1, 2000. He was Visiting Professor at Peking University (Fall 2007 & 2013). He has a joing appointment with PNNL since 2013. He is a Fellow of the Society for Industrial and Applied Mathematics (SIAM, 2010-), Fellow of the American Physical Society (APS, 2004-), Fellow of the American Society of Mechanical Engineers (ASME, 2003-) and Associate Fellow of the American Institute of Aeronautics and Astronautics (AIAA, 2006-). He received the Ralf E Kleinman award from SIAM (2015), the (inaugural) J. Tinsley Oden Medal (2013), and the CFD award (2007) by the US Association in Computational Mechanics.  His h-index is 102 and he has been cited about 50,500 times (see his google scholar citations).  You can also check out his Geneaology Tree (remember to zoom in!). Here is his wikepedia profile. 

Karniadakis is currently the lead PI of an OSD/ARO/MURI on Fractional PDEs, and previously the lead PI of an OSD/AFOSR MURI on Uncertainty Quantification.  He is currently the Director of the DOE center PhILMS on Physics-Informed Learning Machines and previously he was also the Director of  the DOE Center of Mathematics for Mesoscale Modeling of Materials (CM4). 

Research Interests

His current research interests are on machine learning for scientific computing (Scientific Machine Learning), that is how to solve and discover new PDEs via deep learning, hence removing the tyranny of grids and using gappy data only. Check here to find how to use deep neural networks to solve the Schrodinger equations with only 10 lines of Tensorflow code!  Check here to see how to use deep CNN to classify different types of red blood cells in sickle cell anemia.

His broad research interests focus on stochastic multiscale mathematics and modeling of physical and biological systems.  Current thrusts include probabilistic numerics, stochastic simulation (in the context of uncertainty quantification and beyond), fractional PDEs, and multiscale modeling of complex systems (e.g., the brain). Can you believe that we solve problems in 100 dimensions - check this out! Read here about our work on sickle cell anemia and also on modeling malaria from first principles, which was also featured on the web site of the National Public Radio. Read also about our work on the first large multiscale modeling of a brain aneurysm (finalist in the Gordon Bell Award, Supercomputing'11). Our new area is neurovascular coupling in the brain, i.e., bridging the gap between neuroscience and vascular mechanics. New experimental evidence suggests the intriguing possibility that by slightly modulating the brain blood flow one can control information processing -- read our paper here! Recent feature article of our work ("Blood in Motion") in American Scientist.  Particular aspects include:

Fractional PDEs: A breakthrough paper!

Numerical solution of stochastic differential equations: SISC article, also PNAS article

Modeling uncertainty with polynomial chaos: PNAS article, CiSE, JCP

Biophysics - Multiscale modeling of biological systems: PNAS (sickle cell anemia); PNAS (blood viscosity); PNAS (malaria) article; PNAS (thrombosis) article, PRS article

Atomistic/Mesoscopic modeling - Dissipative Particle Dynamics: JCP (triple-decker); PRL (adaptive BCs)

Low Dimensional Modeling - Gappy Data - Data assimilation: JCP article 

Spectral/hp Element and Discontinuous Galerkin methods:  OUP Book

Turbulent Drag Reduction: Science article

DNS/LES of turbulence in complex geometries: JFM article

Flow-structure interactions: PRL article

Micro-transport and Dynamic self-assembly: Springer Book

Flow and heat control applications: JFM article

Parallel computing; Interactive/virtual reality computer graphics: CUP Book

A new report by the National Research Council points to the philosophy and direction of my group, practiced over the last 25 years.  (Thank you NRC!)  Here's a brief sample: "...But the value of the mathematical sciences to the overall science and engineering enterprise and to the nation would be heightened if the number of mathematical scientists who share the following characteristics could be increased: (1) They are knowledgeable across a broad range of the discipline, beyond theirown area(s) of expertise; (2) They communicate well with researchers in other disciplines; (3) They understand the role of the mathematical sciences in the wider world of science, engineering, medicine, defense, and business; and (4) They have some experience with computation..."

Links 

SHORT COURSE:  AN INTRODUCTION TO FRACTIONAL CALCULUS

Professor Francesco Mainardi

SHORT COURSE ON FRACTIONAL PDES

MAY 22 - 31, 2013

INTERNATIONAL SYMPOSIUM ON FRACTIONAL PDES: 

THEORY, NUMERICS AND APPLICATIONS

JUNE 3 - 5, 2013

 CRUNCH Group

Faculty Research Profile

Uncertainty  Quantification AFOSR MURI

Fractional PDEs ARO/MURI