Discover and explore modern applied mathematics. Apply new methods to interesting open problems.
Work individually or in teams to develop new algorithms, hone soft skills, and enhance your scientific background.
Get hands-on experience in modern research under faculty guidence and explore opportunities in STEM.
Machine learning, data-driven methods, and optimization are useful tools for studying structures and trends in high dimensional datasets.
ODE and PDE are fundemental models in the physical sciences. Modern numerical solvers utilize solution behavior and data to increase efficiency and reduce complexity.
Compression, noise, and data corruption are common issues in image processing. With optimization and PDE, rigourous models can be used to recover a suitable approximation.
Image segmentation (also known as object/edge detection) is the process of dividing an image into its constituent parts using information about the boundaries between objects, edges within objects, variations in intensity, etc. The human eye can easily recognize salient information from an image; however, background variations in intensity, noise, and other highly oscillatory features make the process of image segmentation computationally challenging. This work is unique because we propose using a cartoon-texture-noise separation to remove highly oscillatory features from the image prior to segmentation. A new numerical implementation is provided for one of the two decompositions used as well as various experimental results. The method is applied to the classic example of finding a needle in a haystack, as well as real images where the texture component and noise causes problems for standard techniques.
This work presented a method to construct an aggregation function, reflecting a complex set of initial user preferences, which can be used in the framework of multi-criteria decision making. We consider problems where the decision maker can provide information about the importance and interactions between criteria, as well as a desired portion of criteria to be satisfied. The proposed aggregation process is a vague Choquet integral, whose parameters are constructed in two steps. First, we solve a convex constrained L1 optimization problem to obtain a fuzzy measure reflecting the importances and interactions between the criteria. Then the measure is transformed by a monotonic mapping to include vague information on what portion of criteria has to be satisfied. The proposed approach provides an automated construction of an aggregation function, which is completely free of data learning and manual processing. In addition, this method provides a novel fuzzy measure that integrates two different classes of information: importance/interactions of criteria and vague statements. With Dr. Olivier Thonnard.