SIAM Student Chapter Seminar: Difference between revisions
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!Title | !Title | ||
|- | |- | ||
| | |10 AM 10/4 | ||
| | |Birge 346 | ||
| | |Federica Ferrarese (University of Ferrara, Italy) | ||
| | |Control plasma instabilities via an external magnetic field: deterministic and uncertain approaches | ||
|- | |- | ||
| | |11 AM 10/18 | ||
| | |9th floor | ||
| | |Martin Guerra (UW-Madison) | ||
| | |Swarm-Based Gradient Descent Meets Simulated Annealing | ||
|- | |- | ||
| | |12:30 PM 10/31 | ||
| | |VV 901 | ||
| | |Chuanqi Zhang (University of Technology Sydney) | ||
| | |Faster isomorphism testing of p-groups of Frattini class-2 | ||
|- | |- | ||
| | |11/8 | ||
| | |9th floor | ||
| | |Borong Zhang (UW-Madison) | ||
| | |Solving the Inverse Scattering Problem: Leveraging Symmetries for Machine Learning | ||
|- | |- | ||
| | |11/15 | ||
| | |9th floor | ||
| | (zoom) | ||
| | |Yantao Wu (Johns Hopkins University) | ||
|Conditional Regression on Nonlinear Variable Model | |||
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==Abstracts== | ==Abstracts== | ||
''' | '''October 4th, Federica Ferrarese (University of Ferrara, Italy)''': The study of the problem of plasma confinement in huge devices, such as for example Tokamaks and Stellarators, has attracted a lot of attention in recent years. Strong magnetic fields in these systems can lead to instabilities, resulting in vortex formation. Due to the extremely high temperatures in plasma fusion, physical materials cannot be used for confinement, necessitating the use of external magnetic fields to control plasma density. This approach involves studying the evolution of plasma, made up of numerous particles, using the Vlasov-Poisson equations. In the first part of the talk, the case without uncertainty is explored. Particle dynamics are simulated using the Particle-in-Cell (PIC) method, known for its ability to capture kinetic effects and self-consistent interactions. The goal is to derive an instantaneous feedback control that forces the plasma density to achieve a desired distribution. Various numerical experiments are presented to validate the results. In the second part, uncertainty is introduced into the system, leading to the development of a different control strategy. This method is designed to steer the plasma towards a desired configuration even in the presence of uncertainty. The presentation concludes with a comparison of the two control strategies, supported by various numerical experiments. | ||
''' | |||
This is | |||
''' | '''October 18th, Martin Guerra (UW-Madison)''': In generic non-convex optimization, one needs to be able to pull samples out of local optimal points to achieve global optimization. Two common strategies are deployed: adding stochasticity to samples such as Brownian motion, as is done in simulated annealing (SA), and employing a swarm of samples to explore the whole landscape, as is done in Swarm-Based Gradient Descent (SBGD). The two strategies have severe drawbacks but complement each other on their strengths. SA fails in the accuracy sense, i.e., finding the exact optimal point, but succeeds in always being able to get close, while SBGD fails in the probability sense, i.e., it has non-trivial probability to fail, but if succeeds, can find the exact optimal point. We propose to combine the strength of the two and develop a swarm-based stochastic gradient method with samples automatically adjusting their annealing. Using mean-field analysis and long-time behavior PDE tools, we can prove the method to succeed in both the accuracy sense and the probability sense. Numerical examples verify these theoretical findings. | ||
'''October 31st, Chuanqi Zhang''' (University of Technology Sydney): The finite group isomorphism problem asks to decide whether two finite groups of order N are isomorphic. Improving the classical $N^{O(\log N)}$-time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that p-groups of class 2 and exponent p form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun (STOC '23) presents an $N^{O((\log N)^{5/6})}$-time algorithm for this group class. Our work sharpens the key technical ingredients in Sun's algorithm and further improves Sun's result by presenting an $N^{\tilde O((\log N)^{1/2})}$-time algorithm for this group class. Besides, we also extend the result to the more general p-groups of Frattini class-2, which includes non-abelian 2-groups. In this talk, I will present the problem background and our main algorithm in detail, and introduce some connections with other research topics. For example, one intriguing connection is with the maximal and non-commutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of Tensor Isomorphism complexity class (Grochow--Qiao, SIAM J. Comput. '23) are utilized to simplify the algorithm and achieve the extension to p-groups of Frattini class-2. | |||
'''November 8th, Borong Zhang''' (UW-Madison): The inverse scattering problem—reconstructing the properties of an unknown medium by probing it with waves and measuring the medium's response at the boundary—is fundamental in physics and engineering. This talk will focus on how leveraging the symmetries inherent in this problem can significantly enhance machine learning methods for its solution. By incorporating these symmetries into both deterministic neural network architectures and probabilistic frameworks like diffusion models, we achieve more accurate and computationally efficient reconstructions. This symmetry-driven approach reduces the complexity of the models and improves their performance, illustrating how physical principles can inform and strengthen machine learning techniques. Applications demonstrating these benefits will be briefly discussed. | |||
'''November 15th, Yantao Wu''' (Johns Hopkins): We consider the problem of estimating the intrinsic structure of composite functions of the type $\mathbb{E} [Y|X] = f(\Pi_\gamma X) $ where $\Pi_\gamma:\mathbb{R}^d\to\mathbb{R}^1$ is the closest point projection operator onto some unknown smooth curve $\gamma: [0, L]\to \mathbb{R}^d$ and $f: \mathbb{R}^1\to \mathbb{R}^1$ is some unknown {\it link} function. This model is the generalization of the single-index model where $\mathbb{E}[Y|X]=f(\langle v, X\rangle)$ for some unknown {\it index} vector $v\in\mathbb{S}^{d-1}$. On the other hand, this model is a particular case of function composition model where $\mathbb{E}[Y|X] = f(g(x))$ for some unknown multivariate function $g:\mathbb{R}^d\to\mathbb{R}$. In this paper, we propose an algorithm based on conditional regression and show that under some assumptions restricting the complexity of curve $\gamma$, our algorithm can achieve the one-dimensional optimal minimax rate, plus a curve approximation error bounded by $\mathcal{O}(\sigma_\zeta^2)$. We also perform numerical tests to verify that our algorithm is robust, in the sense that even without some assumptions, the mean squared error can still achieve $\mathcal{O}(\sigma_\zeta^2)$. | |||
==Past Semesters== | ==Past Semesters== | ||
Latest revision as of 22:40, 13 November 2024
- When: Fridays at 1 PM unless noted otherwise
- Where: 9th floor lounge (we will also broadcast the virtual talks on the 9th floor lounge with refreshments)
- Organizers: Yahui Qu, Peiyi Chen and Zaidan Wu
- Faculty advisers: Jean-Luc Thiffeault, Steve Wright
- To join the SIAM Chapter mailing list: email siam-chapter+join@g-groups.wisc.edu.
- Zoom link: https://uwmadison.zoom.us/j/97976615799?pwd=U2xFSERIcnR6M1Y1czRmTjQ1bTFJQT09
- Passcode: 281031
Fall 2024
| Date | Location | Speaker | Title |
|---|---|---|---|
| 10 AM 10/4 | Birge 346 | Federica Ferrarese (University of Ferrara, Italy) | Control plasma instabilities via an external magnetic field: deterministic and uncertain approaches |
| 11 AM 10/18 | 9th floor | Martin Guerra (UW-Madison) | Swarm-Based Gradient Descent Meets Simulated Annealing |
| 12:30 PM 10/31 | VV 901 | Chuanqi Zhang (University of Technology Sydney) | Faster isomorphism testing of p-groups of Frattini class-2 |
| 11/8 | 9th floor | Borong Zhang (UW-Madison) | Solving the Inverse Scattering Problem: Leveraging Symmetries for Machine Learning |
| 11/15 | 9th floor
(zoom) |
Yantao Wu (Johns Hopkins University) | Conditional Regression on Nonlinear Variable Model |
Abstracts
October 4th, Federica Ferrarese (University of Ferrara, Italy): The study of the problem of plasma confinement in huge devices, such as for example Tokamaks and Stellarators, has attracted a lot of attention in recent years. Strong magnetic fields in these systems can lead to instabilities, resulting in vortex formation. Due to the extremely high temperatures in plasma fusion, physical materials cannot be used for confinement, necessitating the use of external magnetic fields to control plasma density. This approach involves studying the evolution of plasma, made up of numerous particles, using the Vlasov-Poisson equations. In the first part of the talk, the case without uncertainty is explored. Particle dynamics are simulated using the Particle-in-Cell (PIC) method, known for its ability to capture kinetic effects and self-consistent interactions. The goal is to derive an instantaneous feedback control that forces the plasma density to achieve a desired distribution. Various numerical experiments are presented to validate the results. In the second part, uncertainty is introduced into the system, leading to the development of a different control strategy. This method is designed to steer the plasma towards a desired configuration even in the presence of uncertainty. The presentation concludes with a comparison of the two control strategies, supported by various numerical experiments.
October 18th, Martin Guerra (UW-Madison): In generic non-convex optimization, one needs to be able to pull samples out of local optimal points to achieve global optimization. Two common strategies are deployed: adding stochasticity to samples such as Brownian motion, as is done in simulated annealing (SA), and employing a swarm of samples to explore the whole landscape, as is done in Swarm-Based Gradient Descent (SBGD). The two strategies have severe drawbacks but complement each other on their strengths. SA fails in the accuracy sense, i.e., finding the exact optimal point, but succeeds in always being able to get close, while SBGD fails in the probability sense, i.e., it has non-trivial probability to fail, but if succeeds, can find the exact optimal point. We propose to combine the strength of the two and develop a swarm-based stochastic gradient method with samples automatically adjusting their annealing. Using mean-field analysis and long-time behavior PDE tools, we can prove the method to succeed in both the accuracy sense and the probability sense. Numerical examples verify these theoretical findings.
October 31st, Chuanqi Zhang (University of Technology Sydney): The finite group isomorphism problem asks to decide whether two finite groups of order N are isomorphic. Improving the classical $N^{O(\log N)}$-time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that p-groups of class 2 and exponent p form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun (STOC '23) presents an $N^{O((\log N)^{5/6})}$-time algorithm for this group class. Our work sharpens the key technical ingredients in Sun's algorithm and further improves Sun's result by presenting an $N^{\tilde O((\log N)^{1/2})}$-time algorithm for this group class. Besides, we also extend the result to the more general p-groups of Frattini class-2, which includes non-abelian 2-groups. In this talk, I will present the problem background and our main algorithm in detail, and introduce some connections with other research topics. For example, one intriguing connection is with the maximal and non-commutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of Tensor Isomorphism complexity class (Grochow--Qiao, SIAM J. Comput. '23) are utilized to simplify the algorithm and achieve the extension to p-groups of Frattini class-2.
November 8th, Borong Zhang (UW-Madison): The inverse scattering problem—reconstructing the properties of an unknown medium by probing it with waves and measuring the medium's response at the boundary—is fundamental in physics and engineering. This talk will focus on how leveraging the symmetries inherent in this problem can significantly enhance machine learning methods for its solution. By incorporating these symmetries into both deterministic neural network architectures and probabilistic frameworks like diffusion models, we achieve more accurate and computationally efficient reconstructions. This symmetry-driven approach reduces the complexity of the models and improves their performance, illustrating how physical principles can inform and strengthen machine learning techniques. Applications demonstrating these benefits will be briefly discussed.
November 15th, Yantao Wu (Johns Hopkins): We consider the problem of estimating the intrinsic structure of composite functions of the type $\mathbb{E} [Y|X] = f(\Pi_\gamma X) $ where $\Pi_\gamma:\mathbb{R}^d\to\mathbb{R}^1$ is the closest point projection operator onto some unknown smooth curve $\gamma: [0, L]\to \mathbb{R}^d$ and $f: \mathbb{R}^1\to \mathbb{R}^1$ is some unknown {\it link} function. This model is the generalization of the single-index model where $\mathbb{E}[Y|X]=f(\langle v, X\rangle)$ for some unknown {\it index} vector $v\in\mathbb{S}^{d-1}$. On the other hand, this model is a particular case of function composition model where $\mathbb{E}[Y|X] = f(g(x))$ for some unknown multivariate function $g:\mathbb{R}^d\to\mathbb{R}$. In this paper, we propose an algorithm based on conditional regression and show that under some assumptions restricting the complexity of curve $\gamma$, our algorithm can achieve the one-dimensional optimal minimax rate, plus a curve approximation error bounded by $\mathcal{O}(\sigma_\zeta^2)$. We also perform numerical tests to verify that our algorithm is robust, in the sense that even without some assumptions, the mean squared error can still achieve $\mathcal{O}(\sigma_\zeta^2)$.