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Organizing Committee
Yale University
Yale University
University of Minnesota
University of Texas at Austin
University of Massachusetts Amherst
University of Toronto
University of Bonn
University of Michigan
Abstract
The mathematical and computational toolbox for modern experimental and engineering problems has become more diverse than ever before, with a flurry of new challenges in inverse problems and successful practical solutions that present further theoretical questions. In the spirit of the 2012 “Challenges in Geometry, Analysis, and Computation: High-Dimensional Synthesis” workshop at Yale, the “Modern Applied and Computational Analysis” workshop will be a celebration of different perspectives on inverse problems, models, inference, and harmonic analysis and a debate about the challenges and opportunities in the next decade of applied analysis. The topics include inverse problems, randomized linear algebra, machine learning in applied analysis, and tensor networks.
The organizers would like to thank James Bremer, Ronald Coifman, Jingfang Huang, Peter Jones, Mauro Maggioni, Yair Minsky, Vladimir Rokhlin, Wilhelm Schlag, John Schotland, Amit Singer, Stefan Steinerberger, and Mark Tygert for their help.
The organizers would like to thank James Bremer, Ronald Coifman, Jingfang Huang, Peter Jones, Mauro Maggioni, Yair Minsky, Vladimir Rokhlin, Wilhelm Schlag, John Schotland, Amit Singer, Stefan Steinerberger, and Mark Tygert for their help.
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Workshop Schedule
Monday, June 26, 2023
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
Weil-Petersson curves, traveling salesman theorems, and minimal surfaces
11th Floor Lecture Hall
Speaker
Chris Bishop, Stony Brook University
Session Chair
Amir Sagiv, Columbia University
Session Chair
Raanan Schul, stony brook university
Abstract
Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2006. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman condition characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature. I will discuss these and several other characterizations and sketch why they are all equivalent to each other.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
New perspectives on inverse problems: stochasticity and Monte Carlo method
11th Floor Lecture Hall
Speaker
Li Wang, University of Minnesota
Session Chair
Amir Sagiv, Columbia University
Session Chair
Raanan Schul, stony brook university
Abstract
In this talk, we introduce two new aspects of inverse problems formulated as PDE-constrained optimization. Firstly, while current approaches assume deterministic parameters, many real-world problems exhibit stochastic behavior. We present a novel approach that treats the PDE solver as a push-forward map to recover the full distribution of unknown random parameters. We introduce a gradient-flow equation to estimate the ground-truth parameter probability distribution. Secondly, as problem dimensions increase, Monte Carlo methods regain relevance. However, directly applying them to gradient-based PDE-constrained optimization poses challenges due to the product of forward and adjoint solutions involving Dirac deltas. We propose strategies to rescue Monte Carlo methods and make them compatible with gradient-based optimization.
11:30 AM - 12:15 PM EDT
Quantum Signal Processing
11th Floor Lecture Hall
Speaker
Lin Lin, University of California - Berkeley
Session Chair
Amir Sagiv, Columbia University
Session Chair
Raanan Schul, stony brook university
Abstract
Quantum Signal Processing (QSP) is a revolutionary technique that uses a product of unitary matrices to represent polynomials, with numerous applications in quantum computing. In this talk, I will introduce QSP in a fashion that does not require prior knowledge on quantum computing. We introduce optimization-based algorithms that can efficiently find the ""phase factors"" used to represent a given polynomial. We also identify a surprising connection between the smoothness of the target function and the decay properties of a specific branch of the phase factors.
Y. Dong, L. Lin, H. Ni, J. Wang, Infinite quantum signal processing [arXiv:2209.10162]
J. Wang, Y. Dong, L. Lin, On the energy landscape of symmetric quantum signal processing, Quantum 6, 850, 2022
Y. Dong, X. Meng, K. B. Whaley, L. Lin, Efficient phase factor evaluation in quantum signal processing, Phys. Rev. A 103, 042419, 2021
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Nonlocal PDEs and Quantum Optics
11th Floor Lecture Hall
Speaker
John Schotland, Yale University
Session Chair
Per-Gunnar Martinsson, University of Texas at Austin
Session Chair
Manas Rachh, Flatiron Institute
Abstract
Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Wigner-Smith Methods for Computational Electromagnetics and Acoustics
11th Floor Lecture Hall
Speaker
Eric Michielssen, University of Michigan
Session Chair
Per-Gunnar Martinsson, University of Texas at Austin
Session Chair
Manas Rachh, Flatiron Institute
Abstract
Wigner-Smith (WS) time delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This presentation will formally extend WS time delay theory to Maxwells equations and explores its potential applications in electromagnetics. The WS time delay matrix relates a lossless and reciprocal systems scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. Numerous applications in electromagnetics will be highlighted, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices. Extensions of WS methods towards lossy and dispersive systems will be analyzed as well, and avenues for leveraging WS concepts in computational electromagnetics will be discussed.
5:00 - 6:30 PM EDT
Reception
11th Floor Collaborative Space
Tuesday, June 27, 2023
9:00 - 9:45 AM EDT
Of Crystals and Corals
11th Floor Lecture Hall
Speaker
Stanislav Smirnov, Université de Genève
Session Chair
Gilad Lerman, University of Minnesota
Session Chair
Gal Mishne, University of California San Diego
Abstract
There are many real-world processes exhibiting fractal growing shapes - from mineral deposition and coral growth to lightning strikes, and in many of them growth is related to diffusion properties. We will discuss two seminal models: Diffusion Limited Aggregation was introduced by Witten and Sanders in 1981 and was generalized to Dielectric Breakdown Model by Niemayer et al shortly afterwards. Numerically they approximate very well a wide range of physical phenomena.
However, despite a very simple definition (DLA cluster grows by attaching particles undergoing Brownian motion when they hit the aggregate), very little is understood today, and even less is known rigorously - essentially, only the famous Harry Kesten upper bound on the DLA growth.
We will try to show the flavor of these models and present some new results. Based on a joint preprint with Ilya Losev and some further work.