My research relies on the art of model formulation and scientific computation to predict, reconstruct, and describe physical processes. The aim is to produce a model that is realistic enough to reflect the essential aspects of the phenomena, but simple enough so that it can be analyzed and solved through the construction of a numerical algorithm.

The physical problems I study in my research program pertain to flows of biological and complex fluids, coating processes, and rare events.

Dynamics of the Human Tear Film:

Each time someone blinks, a thin multilayered film of fluid must reestablish itself, within a second or so, on the front of the eye. This thin film is essential for both the health and optical quality of the human eye. An important first step towards effectively managing eye syndromes, like dry eye, is understanding the fluid dynamics of the tear film. In close collaboration with the University of Delaware tear film group, optometrist P. Ewen King-Smith, and national laboratory computational scientist Bill Henshaw, I have focused on understanding the movement of the tear film on the eye. To do so, mathematical models for the tear film thickness are derived from the Navier-Stokes equations using lubrication theory. The highly nonlinear governing evolution equations are simulated with overset grid based computational methods in the Overture framework.

For more information,

K. L. Maki, R. J. Braun, P. Ucciferro, W. D. Henshaw, and P. E. King-Smith. Tear film dynamics on an eye-shaped domain II: Flux Boundary Conditions. Journal of Fluid Mechanics 165 (2010), 1373-1385.

K. L. Maki, R. J. Braun, W. D. Henshaw, and P. E. King-Smith. Tear film dynamics on an eye-shaped domain I: Pressure Boundary Conditions. Mathematical Medicine and Biology 27 (2010), 227-254.

K. L. Maki, R. J. Braun, T. A. Driscoll, and P. E. King-Smith. An overset grid method for the study of reflex tearing. Mathematical Medicine and Biology 25 (2008), 187-214.

A. Heryudono, R. J. Braun, T. A. Driscoll, L. P. Cook, K. L. Maki and P. E. King-Smith. Single-equation models for the tear film in a blink cycle: Realistic lid motion. Mathematical Medicine and Biology 24 (2007), 347-377.

Rarity of Large Growth Factors:

I have worked on simulating rare events. In particular, on reconstructing the probability distribution functions for growth factors of random matrices. The growth factor of a matrix, denoted by rho, quantifies potential error growth when a linear system is solved by Gaussian elimination with partial pivoting. While the growth factor has a maximum of 2^(n-1) for an nxn matrix, the occurrences of matrices with exponentially large growth factors is extremely rare. We implemented a multicanonical Monte Carlo method to explore the tails of growth factor probability distributions for random matrices. Our results attain a probability level of 10^(-12) and suggests the occurrence of an 8x8 matrix with a growth factor of 40 is on the order of a once-in-the-age-of-the-universe event.

For more information, see T. A. Driscoll and K. L. Maki in the Education section of SIAM Review, Searching for Rare Growth Factors Using Multicanonical Monte Carlo Methods.