# Difference between revisions of "Applied/ACMS/absF21"

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This is a joint work with Barak Sober. | This is a joint work with Barak Sober. | ||

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+ | === Yuhua Zhu (Stanford) === | ||

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+ | Title: Fokker-Planck Equations and Machine Learning | ||

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+ | Abstract: As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm's behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast. In the last one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm. |

## Revision as of 10:17, 1 October 2021

## Contents

# ACMS Abstracts: Fall 2021

### Jiuhua Hu (TAMU and UW)

Title: Wavelet-based Edge Multiscale Parareal Algorithm for Parabolic Equations with Heterogeneous Coefficients

Abstract: In this talk, I will talk about the Wavelet-based Edge Multiscale Parareal Algorithm to solve parabolic equations with heterogeneous coefficients. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spatial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems. We derive the convergence rate of this algorithm and present extensive numerical tests to demonstrate the performance of our algorithm. This is a joint work with Guanglian Li (The University of Hong Kong).

### Di Fang (UC-Berkeley)

Title: Time-dependent unbounded Hamiltonian simulation with vector norm scaling

Abstract: Hamiltonian simulation is a basic task in quantum computation. The accuracy of such simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.

### L. Ruby Leung (PNNL)

Title: Mesoscale convective systems in observations and a hierarchy of models

Abstract: Mesoscale convective systems (MCSs) account for 50-70% of warm season precipitation in the central U.S. With high intensity rainfall covering large area, MCSs are also responsible for most of the slow-rising and hybrid floods in the U.S. east of the Rocky Mountains. MCSs develop under different environments featuring frontal systems and the Great Plains low-level jet providing a lifting mechanism and moist environment for initiation of MCSs. During summer, eastward propagating sub-synoptic perturbations are crucial for MCS initiation under unfavorable large-scale circulations. MCSs have been producing more intense precipitation and lasting longer in the last 35 years, motivating the need to understand how they may change in the future. However, MCSs are notoriously difficult to simulate, as even convection permitting simulations underestimate MCS number and precipitation in the central U.S., particularly during summer. Using observation data and a hierarchy of models including a Lagrangian parcel model, regional and global convection permitting models, global climate models, and a tracer-enabled land surface model, we study MCSs, their large-scale environments, their role in land-atmosphere interactions, and the mechanisms of their response to global warming.

### Yariv Aizenbud (Yale)

Title: Non-parametric estimation of manifolds from noisy data

Abstract: A common task in many data-driven applications is to find a low dimensional manifold that describes the data accurately. Estimating a manifold from noisy samples has proven to be a challenging task. Indeed, even after decades of research, there is no (computationally tractable) algorithm that accurately estimates a manifold from noisy samples with a constant level of noise.

In this talk, we will present a method that estimates a manifold and its tangent in the ambient space. Moreover, we establish rigorous convergence rates, which are essentially as good as existing convergence rates for function estimation.

This is a joint work with Barak Sober.

### Yuhua Zhu (Stanford)

Title: Fokker-Planck Equations and Machine Learning

Abstract: As the continuous limit of many discretized algorithms, PDEs can provide a qualitative description of algorithm's behavior and give principled theoretical insight into many mysteries in machine learning. In this talk, I will give a theoretical interpretation of several machine learning algorithms using Fokker-Planck (FP) equations. In the first one, we provide a mathematically rigorous explanation of why resampling outperforms reweighting in correcting biased data when stochastic gradient-type algorithms are used in training. In the second one, inspired by an interactive particle system whose mean-field limit is a non-linear FP equation, we develop an efficient gradient-free method that finds the global minimum exponentially fast. In the last one, we propose a new method to alleviate the double sampling problem in model-free reinforcement learning, where the FP equation is used to do error analysis for the algorithm.