Math Frontier Seminar

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/non-linear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) ε-KKT point within O(ε−2) outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of eO ε−2.5 to produce an ε-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is eO ε−3 to produce a near ε-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.

Qubit control is one of the main challenges of building a scalable Quantum computer with superconducting qubits. Current technologies are based on room temperature microwave generators. Scaling microwave control to millions of qubits is more than an engineering challenge due to the excessive heat delivered to the cryostat and the hardware cost. This talk will introduce the basics of microwave qubit control and some of its issues. Then we will discuss the theory and experimental realization of a control technology using only superconducting (cold) digital (cheap) circuits, which opens a path towards scalable qubit control.

In this talk, I will present a Julia package, HypersurfaceRegions.jl, for computing all connected components in the complement of an arrangement of real algebraic hypersurfaces in $R^n$. The package is based on a modified implementation of the algorithm from the paper "Smooth Connectivity in Real Algebraic Varieties" by Cummings et al. I will outline the theory behind the algorithm and our implementation. I will demonstrate the use of the package through various examples.

As described by quantum mechanics, energy is absorbed and emitted from atoms in the form of photons with discrete energy values. When an atom absorbs or emits a photon, electrons transition up or down energy levels. The energy associated with these transition determines the frequency, i.e. color, of the absorbed or emitted light. Using Schrodinger’s equation to describe the atomic structure, and Maxwell’s equations to describe the light, a system of equations that fully describes the behavior of the light matter interaction can be derived. This system is known as the Maxwell-Bloch equations. In this talk, an algorithm for approximating solutions to the Maxwell-Bloch equations is discussed. Algorithms for an atom with 2 and 3 energy levels are discussed, as well as an algorithm for the more general M level problem. We also use these algorithms to probe the behavior of soliton solutions for the 2 and 3 energy level systems.

A novel EigenWave algorithm is described to compute eigenvalues and eigenfunctions of elliptic boundary value problems. Based on the recently developed WaveHoltz scheme, the algorithm solves a related time-dependent wave equation as part of an iteration. At each iteration, the solution is filtered in time. After filtering, the solution mainly contains eigenmodes whose eigenvalues are near the target frequency of the filter. The iteration is embedded within a matrix-free Arnoldi algorithm, allowing the efficient computation of multiple eigenpairs near the target frequency. The approach allows the computation of eigenvalues anywhere in the spectrum without the need to invert an indefinite matrix, as is common with other approaches. The approach is demonstrated by finding eigenpairs of the Laplacian in complex geometry using overset grids. Results in two and three-space dimensions are presented. For large enough problems, it is demonstrated that EigenWave can outperform modern Arnoldi-type eigenvalue algorithms.

Helicopter aerial refueling refers to the process of refueling a helicopter in mid-flight with the aid of a tanker aircraft. This maneuver is particularly challenging due to 1) complex aerodynamic interactions between the helicopter, the tanker, and the refueling hose-drogue system, 2) high pilot workload, 3) strict safety constraints, and 4) the contact-critical nature of the operation. To address these challenges, we propose a novel autonomous control methodology that combines model-based control with data-driven approaches such as reinforcement learning (RL). Given that aerial refueling is a safety-critical operation, it is essential to guarantee stability and performance for this hybrid control system. This talk will explain how Lyapunov stability and ultimate boundedness can be leveraged to establish rigorous stability and performance guarantees while accounting for controller parameters and uncertainties in the drogue dynamics.

By admixing a fraction of exact exchange (EXX), hybrid DFT provides a more accurate and reliable description of electronic structure than traditional semi-local DFT (density functional theory). However, the conventional reciprocal-space EXX evaluation is cubic scaling and computationally demanding (typically 10x–100x more expensive than semi-local DFT), which limits the applicability of hybrid DFT. To overcome this bottleneck, we have developed an accurate linear-scaling approach that exploits the sparsity of the EXX interaction using a localized representation of the occupied space. For crystalline systems with general k-point sampling, further optimization can be made to exploit both locality and periodicity.

We develop algebraic geometry for coupled cluster theory using second quantization. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The truncated eigenstates parametrize well known varieties such as the Grassmannian, flag varieties and spinor varieties. We will offer a detailed study into the truncation varieties. Additionally in second quantization we work within the exterior algebra. There we can define interior and exterior operators called the creation and annihilation operators. They span a special Clifford algebra called the Fermi-Dirac algebra, which we will study as well.

To function, cells must move material internally. This intracellular transport is achieved by molecular motors, which transport vesicle-bound cargo along protein filaments. In vitro experiments have uncovered the mechanochemistry of how single, isolated motors turn chemical energy into mechanical work as they "walk" along a protein filament. In cells, however, multiple motors transport cargo. Some of these motors bind to the protein filament and contribute to cargo transport; others diffuse over the surface of the cargo, and the motors transition between the two roles. It is a challenge to experimentally determine the number of motors on a given cargo, let alone the time-varying number of these motors bound to the protein filament. To address this, we developed a method to estimate the number of motors and the number of filament-bound motors from position-time measurements of the cargo. The method uses our knowledge of the mechanics of the system and the mechanochemistry of the motors to infer the most likely number of all motors on the cargo and of filament-bound motors.

Graph convolutional networks (GCNs) are a powerful tool for graph representation learning. Due to the recursive neighborhood aggregations employed by GCNs, efficient training methods suffer from a lack of theoretical guarantees or are missing important practical elements from modern deep learning algorithms, such as adaptivity and momentum. We present several neighbor-sampling (NS) based Adam-type stochastic methods for solving a nonconvex GCN training problem. We utilize the control variate technique proposed by [Chen et al., 2017] to reduce the stochastic error caused by neighbor sampling. Under standard assumptions for Adam-type methods, we show that our methods enjoy the optimal convergence rate. In addition, we conduct extensive numerical experiments on node classification tasks with several benchmark datasets. The results demonstrate superior performance of our methods over classic NS-based SGD that also uses the control-variate technique, especially for large-scale graph datasets.

With the surge of AI models in everyday use, examining the trustworthiness of these models remains a crucial concern. Trustworthiness is defined by several key factors: vulnerabilities inherent in the model architecture that can be exploited by adversaries, leading to faulty model use; vulnerabilities in the training data that result in unfair demographic biases, along with mechanisms to mitigate these biases; and latent representations of models that can be used to recover sensitive training information. This presentation will explore these aspects in depth, discussing how adversarial attacks can compromise model integrity, the impact of biased training data on model fairness, and the risks associated with extracting sensitive information from model representations. Additionally, we will examine algorithmic strategies designed to enhance the robustness and fairness of machine learning models, ensuring their reliability and ethical deployment in real-world applications.