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Applied and Computational Mathematics Seminar

Applied Math Seminar

Title: Self-Correcting Discriminator Optimization for Image and Speech Enhancement GANs

Abstract: Generative adversarial network (GAN) has become one of the most important neural network models for classical unsupervised machine learning. Various discriminator loss functions have been developed to train GAN's discriminators, most of which have a common structure: a sum of real and fake losses that depend only on the actual and generated data, respectively. One challenge associated with an equally weighted sum of two losses is that the training may benefit one loss but harm the other. We present self-correcting optimization for training a GAN discriminator, which helps avoid "harmful" training directions for parts of the discriminator loss function. Experiments validated the effectiveness of our loss functions on conditional and unconditional image generation tasks as well as speech enhancement tasks.

Date:
-
Location:
POT 745
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Applied Math Seminar

Title: Robust Estimation of Smooth Graph Signals from Randomized Space-time Samples

Abstract: Heat diffusion processes have found wide applications in modeling dynamical system over graphs. In this talk, I will talk about the recovery of a k-bandlimited graph signal that is an initial signal of a heat diffusion process from its space-time samples. In this work, we have proposed three random space-time sampling regimes, termed dynamical sampling techniques, that consist in selecting a small subset of space-time nodes at random according to some probability distribution. We show that the number of space-time samples required to ensure stable recovery for each regime depends on a parameter called the spectral graph weighted coherence, that depends on the interplay between the dynamics over the graphs and sampling probability distributions. Then, we propose a computationally efficient method to reconstruct k-bandlimited signals from their space-time samples. We prove that it yields accurate reconstructions and that it is also stable to noise. Finally, we test dynamical sampling techniques on a wide variety of graphs. The numerical results on support our theoretical findings and demonstrate the efficiency.

Date:
-
Location:
POT 745
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Applied Math Seminar

Title: A generic framework to coarse-grain stochastic reaction networks by Abstract Interpretation

Abstract: In the last decades, logical or discrete models have emerged as a successful paradigm for capturing and predicting the behaviors of systems of molecular interactions. Intuitively, they consist in sampling the abundance of each kind of biochemical entity within finite sets of intervals and deriving transitions accordingly. On one hand, formally-proven sound derivation from more precise descriptions (such as from reaction networks) may include many fictitious behaviors. On the other hand, direct modeling usually favors dominant interactions with no guarantee on the behaviors that are neglected.

In this paper, we formalize a sound coarse-graining approach for stochastic reaction networks. Its originality relies on two main ingredients. Firstly, we abstract values by intervals that overlap in order to introduce a minimal effort for the system to go back to the previous interval, hence limiting fictitious oscillations in the coarse-grained models. Secondly, we compute for pairs of transitions (in the coarse-grained model) bounds on the probabilities on which one will occur first. 

We illustrate our ideas on two case studies and demonstrate how techniques from Abstract Interpretation can be used to design more precise discretization methods, while providing a framework to further investigate the underlying structure of logical and discrete models. 

Date:
-
Location:
Online
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Applied Math Seminar

Title:  Optimal Decision-Making in Social Networks
 
Abstract:

To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. We then consider large networks under the same framework. Using a combination of asymptotic methods and first passage time calculations, we find that when the network is sufficiently large, most agents decide correctly irrespective of whether the first agent’s decision is right or wrong. Interestingly, individuals in networks with both hasty and deliberate agents can make the right choice more quickly and more often than in networks of identical agents: Observing the choices of a small group of hasty agents can allow the more deliberate agents to make accurate decisions. Our model is tractable and readily generalizable, paving the way for the future study of different social network topologies. We conclude that diverse groups make quicker, more accurate decisions than homogenous groups.

Date:
-
Location:
POT 745
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Applied Math Seminar

Title: Extrapolation for eigenvalue problems: Upcycling data for faster convergence

Abstract: We will discuss accelerating convergence to numerical solutions of eigenvalue problems using a simple post-processing step applied to standard eigensolver techniques. First we will consider accelerating the standard power iteration, one of the most basic and powerful but sometimes very slow iterative methods. We will review some recent results on how we can make the power iteration faster by recombining previous iterates to form our next approximation to a solution; and, we will discuss why this works. We can also apply a similar technique to a restarted Arnoldi method to boost its performance with little additional computational cost. Numerical examples will illustrate the theory.

Date:
-
Location:
POT 745
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Applied Math Seminar

Title: A C0 finite element method for the biharmonic problem with Dirichlet boundary conditions in a polygonal domain
 
Abstract: 
In this talk, we discuss the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the fourth-order problem into a system of two Poison equations and one Stokes equation, or a system of one Stokes equation and one Poisson equation. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are in turn proposed to solve the decoupled systems. In addition, we show the regularity of the solutions in each decoupled system in both the Sobolev space and the weighted Sobolev space, and we derive the optimal error estimates for the numerical solutions on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings.
Date:
-
Location:
POT 745
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Applied Math Seminar

Title: Theory and Algorithms for Nonlinear Eigenvector Problems with Affine-Linear Structures

Abstract: Eigenvector-dependent Nonlinear Eigenvalue Problems (NEPv) have long played critical roles in computational physics and chemistry and are becoming increasingly important in numerous data science applications. In this talk, we consider a class of NEPv where the coefficient matrices have a special affine-linear structure. One important origin of affine-linear NEPv is the Rayleigh-quotient-related optimization, including the trace-ratio optimization for dimension reduction and robust Rayleigh-quotient optimization for handling data uncertainties. We will establish variational characterizations for particular affine-linear NEPv, and then provide a geometric interpretation of a Self-Consistent Fields (SCF) iteration for solving the NEPv. The geometric interpretation reveals the global convergence of SCF in many cases and explains its potential non-convergence issues in others. New improvements to SCF, including the local acceleration schemes and the global verification techniques, are also discussed. Numerical experiments demonstrate the effectiveness of our approach.

Date:
-
Location:
POT 745
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Applied Math Seminar

Title: An Adaptive Formation Control Architecture for A Team of Quadrotors with Performance and Safety Constraints

Abstract: We propose a novel adaptive formation control architecture for a group of quadrotor systems, under line-of-sight (LOS) distance and relative distance constraints, where the constraint requirements can be both asymmetric and time-varying in nature. Universal barrier functions are adopted in the controller design and analysis, which is a generic framework that can address system with different types of constraints in a unified controller architecture. Furthermore, each quadrotor’s mass is unknown, and the system dynamics are subjected to time-varying external disturbance. Through rigorous analysis, an exponential convergence rate can be guaranteed on the distance tracking errors, while the constraints are satisfied during the operation. A simulation example further demonstrates the efficacy of the proposed control framework.

Date:
-
Location:
POT 745
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Type of Event (for grouping events):

Applied Math Seminar

Title: Optimal Decision-Making in Social Networks
 
Abstract: To make decisions we are guided by the evidence we collect and the opinions of friends and neighbors. How do we combine our private beliefs with information we obtain from our social network? To understand the strategies humans use to do so, it is useful to compare them to observers that optimally integrate all evidence. Here we derive network models of rational (Bayes optimal) agents who accumulate private measurements and observe the decisions of their neighbors to make an irreversible choice between two options. The resulting information exchange dynamics has interesting properties: When decision thresholds are asymmetric, the absence of a decision can be increasingly informative over time. In a recurrent network of two agents, the absence of a decision can lead to a sequence of belief updates akin to those in the literature on common knowledge. We then consider large networks under the same framework. Using a combination of asymptotic methods and first passage time calculations, we find that when the network is sufficiently large, most agents decide correctly irrespective of whether the first agent’s decision is right or wrong. Interestingly, individuals in networks with both hasty and deliberate agents can make the right choice more quickly and more often than in networks of identical agents: Observing the choices of a small group of hasty agents can allow the more deliberate agents to make accurate decisions. Our model is tractable and readily generalizable, paving the way for the future study of different social network topologies. We conclude that diverse groups make quicker, more accurate decisions than homogenous groups.
Date:
-
Location:
POT 745
Tags/Keywords:
Type of Event (for grouping events):
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