@article{kuster_gremaud_touzani_2007, title={Fast numerical methods for Bernoulli free boundary problems}, volume={29}, ISSN={["1064-8275"]}, DOI={10.1137/06065444X}, abstractNote={The numerical solution of the free boundary Bernoulli problem is addressed. An iterative method based on a level-set formulation and boundary element method is proposed. Issues related to the implementation, the accuracy, and the generality of the method are discussed. The efficiency of the approach is illustrated by numerical results.}, number={2}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Kuster, Christopher M. and Gremaud, Pierre A. and Touzani, Rachid}, year={2007}, pages={622–634} }
 @article{gremaud_kuster_li_2007, title={A study of numerical methods for the level set approach}, volume={57}, ISSN={["0168-9274"]}, DOI={10.1016/j.apnum.2006.07.022}, abstractNote={The computation of moving curves by the level set method typically requires reinitializations of the underlying level set function. Two types of reinitialization methods are studied: a high order “PDE” approach and a second order Fast Marching method. Issues related to the efficiency and implementation of both types of methods are discussed, with emphasis on the tube/narrow band implementation and accuracy considerations. The methods are also tested and compared. Fast Marching reinitialization schemes are faster but limited to second order, PDE based reinitialization schemes can easily be made more accurate but are slower, even with a tube/narrow band implementation.}, number={5-7}, journal={APPLIED NUMERICAL MATHEMATICS}, author={Gremaud, Pierre A. and Kuster, Christopher M. and Li, Zhilin}, year={2007}, pages={837–846} }
 @article{finkel_kuster_lasater_levy_reese_ipsen_2006, title={Communicating Applied Mathematics: Four Examples}, volume={48}, ISSN={0036-1445 1095-7200}, url={http://dx.doi.org/10.1137/s0036144504443523}, DOI={10.1137/S0036144504443523}, abstractNote={Communicating Applied Mathematics is a writing- and speaking-intensive graduate course at North Carolina State University. The purpose of this article is to provide a brief description of the course objectives and the assignments. Parts A--D of of this article represent the class projects and illustrate the outcome of the course: 
* The Evolution of an Optimization Test Problem: From Motivation to Implementation, by Daniel E. Finkel and Jill P. Reese 
* Finding the Volume of a Powder from a Single Surface Height Measurement, by Christopher Kuster 
* Finding Oscillations in Resonant Tunneling Diodes, by Matthew Lasater 
* A Shocking Discovery: Nonclassical Waves in Thin Liquid Films, by Rachel Levy 
We introduce a water-supply problem considered by the optimization and hydrology communities for benchmarking purposes. The objective is to drill five wells so that the cost of pumping water out of the ground is minimized. Using the implicit filtering optimization algorithm to locate the wells, we save approximately $2,500 over the cost of a given initial well configuration. 
 
The volume of powder poured into a bin with obstructions is found by calculating the height of the surface at every point. This is done using the fast marching algorithm. We look at two different bin geometries and determine the volumes as a function of the powder height under the spout. The surface of the powder satisfies a two-dimensional eikonal equation. This equation is solved using the fast marching method. 
 
Resonant tunneling diodes (RTDs) are ultrasmall semiconductor devices that have potential as very high-frequency oscillators. To describe the electron transport within these devices, physicists use the Wigner--Poisson equations which incorporate quantum mechanics to describe the distribution of electrons within the RTD. Continuation methods are employed to determine the steady-state electron distributions as a function of the voltage difference across the device. These simulations predict the operating state of the RTD under different applied voltages and will be a tool to help physicists understand how changing the voltage applied to the device leads to the development of current oscillations. 
 
When a thin film flows down an inclined plane, a bulge of fluid, known as a capillary ridge, forms on the leading edge and is subject to a fingering instability in which the fluid is channeled into rivulets. This process is familiar to us in everyday experiments such as painting a wall or pouring syrup over a stack of pancakes. It is also observed that changes in surface tension due to a temperature gradient can draw fluid up an inclined plane. Amazingly, in this situation the capillary ridge broadens and no fingering instability is observed. Numerical and analytical studies of a mathematical model of this process led to the discovery that these observations are associated with a nonclassical shock wave previously unknown to exist in thin liquid films.}, number={2}, journal={SIAM Review}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Finkel, Daniel E. and Kuster, Christopher and Lasater, Matthew and Levy, Rachel and Reese, Jill P. and Ipsen, Ilse C. F.}, year={2006}, month={Jan}, pages={359–389} }
 @article{gremaud_kuster_2006, title={Computational study of fast methods for the eikonal equation}, volume={27}, ISSN={["1064-8275"]}, DOI={10.1137/040605655}, abstractNote={A computational study of the fast marching and the fast sweeping methods for the eikonal equation is given. It is stressed that both algorithms should be considered as "direct" (as opposed to iterative) methods. On realistic grids, fast sweeping is faster than fast marching for problems with simple geometry. For strongly nonuniform problems and/or complex geometry, the situation may be reversed. Finally, fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented.}, number={6}, journal={SIAM JOURNAL ON SCIENTIFIC COMPUTING}, author={Gremaud, PA and Kuster, CM}, year={2006}, pages={1803–1816} }
 @article{ahmed_buckingham_gremaud_hauck_kuster_prodanovic_royal_silantyev_2004, title={Volume determination for bulk materials in bunkers}, volume={61}, ISSN={["0029-5981"]}, DOI={10.1002/nme.1144}, abstractNote={AbstractA simple model for the determination of the shape of large granular piles in complicated geometries is discussed. An eikonal formulation of the problem is proposed. Two distinct cases arise. In cylindrical geometries, i.e., if both container and possible obstacles have vertical walls, the problem is equivalent to a two‐dimensional travel time problem with obstacles, while in general geometries, this analogy breaks down. In the first case, classical one‐sided discretizations are generalized to handle obstacles without loss in accuracy. In the second case, a fast and efficient numerical method is proposed, implemented and tested. The discrete problems are solved through fast marching. Copyright © 2004 John Wiley & Sons, Ltd.}, number={13}, journal={INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING}, author={Ahmed, SA and Buckingham, R and Gremaud, PA and Hauck, CD and Kuster, CM and Prodanovic, M and Royal, TA and Silantyev, V}, year={2004}, month={Dec}, pages={2239–2249} }