Optimization Seminar

Kyle Wright (Graduate Student, Applied Mathematics, University of California, Merced)

Title Talk: Structured Inverse Eigenvalue Problems with applications in quantum systems

Abstract:  Structured inverse eigenvalue problems (SIEPs) have applications in quantum systems.  We will explore SIEPs for symmetric matrices with parameterized diagonal components.  This includes methods for solving the general 2x2 case and other special cases.  We then discuss the applications of these methods in quantum systems.

 

Toby Isaac (Argonne National Laboratory)

Title Talk: Variable Metric Three Operator Splitting for Convex Optimization

Abstract:  We explore the convergence properties of a variable metric formulation of the three operator splitting iteration of Davis & Yin, trying to balance convergence rates with robust adaptability.

 

Shashwat Sharan (Graduate Student, Applied Mathematics, University of California, Merced)

Title Talk: Quantum Mechanical Fluid Flow in a Channel (Dam Break Problem)

 

Jacqueline Alvarez (Graduate Student, Applied Mathematics, University of California, Merced)

Title Talk:  Machine Learning for Classifying Images with Motion Blur

Abstract:  Motion blur in an image results from movements of objects within a scene or from the imaging system itself. In applications such as high-speed license plate recognition, motion blur introduces image artifacts leading to challenges in image classification. In this work, we investigate machine learning techniques for classifying images with motion blur using convolutional neural networks. In particular, we explore how different motion directions and lengths affect the predictive performance of our classification model. We used the MNIST dataset, which contains 70,000 images of handwritten digits, to generate training and testing datasets of images with motion blur using MATLAB. Specifically, we considered motion blurs at various angles and lengths to analyze the effects of different motion blurs on the classification accuracy of images of digits. We found that our model shows very high accuracy when training and testing on datasets with the same type of motion blur. In contrast, training and testing on datasets with different motion blurs result in lower accuracy. We describe how we can improve overall classification performance and offer insights on what additional information can be inferred from the MNIST dataset with motion blur.

Ruanui Nicholson (Lecturer, Department of Engineering Science Te Herenga Mātai Pūkaha | Faculty of Engineering Waipapa Taumata Rau | University of Auckland)

Title Talk:  A Bayesian approach to optimal experimental design of large scale inverse problems in the presence of model uncertainties

 

Stephanie Chaillat (Laboratoire POems, École Nationale Supérieure de Techniques Avancées Paris)

Abstract:  Assessing the impact of underwater explosions on submerged structures (submarines) is an important naval engineering problem. An underwater explosion mainly induces two distinct phenomena: a "shock wave" followed by an oscillating bubble of gas. Our goal is to create an efficient numerical method that accounts for the effects of both phenomena on submerged structures. Due to the unbounded nature of the ocean and the complex mechanical behavior of the submarine we want to take into account, it is natural to consider a Boundary Element Method/Finite Element Method (BEM/FEM) coupling. I will present how we can take advantage of fast BEMs in the frequency domain to model this time-domain problem, all the necessary improvements to be able to consider realistic configurations and the consequences on the convergence of the FEM/BEM coupling.

Irabiel Romero (Graduate Student, Applied Mathematics, University of California, Merced)

Title Talk:  Paper presentation: “Embedding Power Flow into Machine Learning for Parameter and State Estimation”, by Laurent Pagnier and Michael Chertkov

Abstract:  Modern state and parameter estimations in power systems consist of two stages: the outer problem of minimizing the mismatch between network observation and prediction over the network parameters, and the inner problem of predicting the system state for given values of the parameters. The standard solution of the combined problem is iterative: (a) set the parameters, e.g. to priors on the power line characteristics, (b) map input observation to the prediction of the output, (c) compute the mismatch between predicted and observed output, (d) make a gradient descent step in the space of parameters to minimize the mismatch, and loop back to (a). We show how modern Machine Learning (ML), and specifically training guided by automatic differentiation allows to resolve the iterative loop more efficiently. Moreover, we extend the scheme to the case of incomplete observations, where Phasor Measurement Units (reporting real and reactive powers, voltage, and phase) are available only at the generators (PV buses), while loads (PQ buses) report (via SCADA controls) only active and reactive powers. Considering it from the implementation perspective, our methodology of resolving the parameter and state estimation problem can be viewed as embedding of the Power Flow (PF) solver into the training loop of the Machine Learning framework (PyTorch, in this study). We argue that this embedding can help to resolve high-level optimization problems in power system operations and planning.

Roummel Marcia (Professor, Applied Mathematics, University of California, Merced)

Title Talk:  Interpretability of ReLU for Inversion

Abstract: In this talk, we focus on the mathematical interpretability of fully-connected neural networks, especially those that use a rectified linear unit (ReLU) activation function.  Our analysis elucidates the difficulty of approximating the reciprocal function. Notwithstanding, using the ReLU activation function halves the error compared with a linear model. In addition, one might have expected the errors to increase only towards a singular point, but both the linear and ReLU  errors are fairly oscillatory and increase near both edge points.

 

Jacqueline Alvarez (Graduate Student, Applied Mathematics, University of California, Merced)

Title Talk:  Synthetic aperture radar inverse scattering reconstruction using convolutional neural networks

Abstract: We address the reconstruction of synthetic aperture radar (SAR) images using machine learning. From previous work, we utilize a single, fully-connected layer to learn the sensing matrix of the forward scattering problem. We estimate the reflectivity of the SAR measurements by applying the conjugate transpose of the learned sensing matrix to the SAR measurements. We further improve the reconstructions of the reflectivity using convolutional layers. The model is trained to reconstruct images containing a single target but can be applied to data containing multiple targets without additional training. Resulting reconstructions are sharper images, where the background noise is significantly decreased.

 

Symeon Papadimitropoulos (Postdoctoral Researcher, Applied Mathematics, University of California, Merced)

Title Talk:  Imaging in waveguides using physically-informed data-driven techniques

Abstract: Inverse source problems are central to many applications in acoustics, geophysics, non-destructive testing, and more. Traditional imaging methods suffer from the resolution limit, preventing distinction of sources separated by less than the emitted wavelength. In this work we propose a method based on physically-informed neural-networks for solving the source refocusing problem, constructing a novel loss term which promotes super-resolving capabilities of the network and is based on the physics of wave propagation. We demonstrate the approach in the setup of imaging an a-priori unknown number of point sources in a two-dimensional rectangular waveguide from measurements of wavefield recordings along a vertical cross-section. The results show the ability of the method to approximate the locations of sources with high accuracy, even when placed close to each other.

 

Chrysoula Tsogka (Professor, Applied Mathematics, University of California, Merced)

Title Talk:   Phase and absorption contrast imaging using intensity measurements

Abstract:  We consider imaging absorbing as well as non-absorbing objects using intensity only measurements. Objects with high absorption contrast can be imaged effectively using multiple illuminations and/or masks as in ghost imaging. On the other hand, transparent objects with low absorption contrast are more challenging to be imaged when only intensities are measured, even when they significantly change the phase of the waves as they go through them. We present a computational imaging approach that allows quantitative imaging of both absorbing and transparent objects. This problem arises in various fields such as  X-ray crystallography,  electron microscopy, coherent diffractive imaging and astronomy. The proposed algorithm guarantees exact recovery if the image is sparse with respect to a given basis, and it can be used, without any modification, when the illumination is partially coherent. This is important for, for example,  phase-contrast X-ray imaging because fully coherent sources of X-rays are very hard to be obtained.

Ruanui Nicholson (Lecturer, The University of Auckland)

Abstract: In this talk we consider the use of surrogate (forward) models to efficiently solve Bayesian inverse problems. The problems considered are from a range of applications but are all high dimensional problems resulting from the discretisation of partial differential equations. We adopt the Bayesian approximation error approach to account for the model discrepancy, which is treated as an additional stochastic error term. We also prove a somewhat surprising result: under the assumption of a Gaussian prior and additive noise model the approximate posterior found by using a linear(-ised) surrogate is invariant to the choice of linear surrogate, so long as the model discrepancy is taken into account via the  Bayesian approximation error approach. This is ongoing joint work with Noemi Petra, Umberto Villa, and Jari P. Kaipio.

 

Ki-Tae Kim (Postdoctoral Researcher, Applied Mathematics, University of California, Merced)

 

Tucker Hartland (Graduate Student, Applied Mathematics, University of California, Merced)