Applied Mathematics and Computation Seminars

Convex duality and glacier momentum balance

Speaker: Daniel Shapero

ABSTRACT: The momentum balance equation for glacier flow can be expressed as a convex optimization problem for the ice velocity. The geometry of real glaciers, however, evolves in time as the terminus advances and retreats. The orthodox approaches to solving this free boundary problem are to either (1) set a minimum thickness and accept the resulting mass balance errors, or (2) apply the level-set method or related approaches that "turn off" the physics where the ice thickness is zero. Both of these approaches have their drawbacks. Here, I'll show an alternative formulation of the momentum balance equation that I derived using the principle of convex duality. This dual form adds the stresses as unknowns, but it remains solvable even when the ice thickness is equal to 0. Solvability at zero thickness allows us to simulate terminus advance and retreat -- including iceberg calving -- in a simple way. I'll show some numerical demonstrations and discuss what solution methods were viable. I… Read more.

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Data-Driven Kernel Matrix Computations: Geometric Analysis and Scalable Algorithms

Speaker: Cai Difeng

ABSTRACT: Dense kernel matrices arise in a broad range of disciplines, such as potential theory, molecular biology, statistical machine learning, etc. To reduce the computational cost, low-rank or hierarchical low-rank techniques are often used to construct an economical approximation to the original matrix. In this talk, we consider general kernel matrices associated with possibly high dimensional data. We perform analysis to provide a straightforward geometric interpretation that answers a central question: what kind of subset is preferable for skeleton low-rank approximations. Based on the theoretical findings, we present scalable and robust algorithms for black-box dense kernel matrix computations. The efficiency and robustness will be demonstrated through experiments for various datasets, kernels, and dimensions, including benchmark comparison to the state-of-the-art packages for N-body simulations.BIO: Difeng Cai received his BS in math from University of Science and Technology… Read more.

Sink or Soar: the interplay between buoyant bubbles and sinking sediments inenergizing turbulence near the ice-ocean boundary

Speaker: Megan Wengrove

ABSTRACT: At the terminus of tidewater glaciers an interplay of connected processes result in the melt of ice. From both field and laboratory observations, it has been suggested that both bubbles and sediments could be important yet neglected contributors to ice melt at the submarine tidewater glacier terminus. In the laboratory it has been shown that as glacier ice melts, air trapped in pores inside of the ice is released creating flow transpiration at the boundary and buoyant bubble rise at the ice-ocean interface, leading to increased melt (Wengrove et al.,2023). Additionally, during separate laboratory experiments, sediments entrained in the subglacial discharge plume are shown to increase the entrainment of warm ocean water toward the ice leading to higher melt rates (McConnochie andCenedese, 2023). In July 2024, we made the first ever video observations of both bubbles rising and sediments mixing and falling from a stationary-bolted platform to an Alaskan tidewater glacier… Read more.