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Graph Neural Networks and Applied Linear Algebra
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| Pengarang | : | Nicholas S. Moore, Eric C. Cyr, Peter Ohm, Christopher M. Siefert, Raymond S. Tuminaro |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 1) |
| Halaman | : | 141-175 |
| Abstrak | : | Sparse matrix computations are ubiquitous in scientific computing. Given the recent interest in scientific machine learning, it is natural to ask how sparse matrix computations can leverage neural networks (NNs). Unfortunately, multilayer perceptron (MLP) NNs are typically not natural for either graph or sparse matrix computations. The issue lies with the fact that MLPs require fixed-sized inputs, while scientific applications generally generate sparse matrices with arbitrary dimensions and a wide range of different nonzero patterns (or matrix graph vertex interconnections). While convolutional NNs could possibly address matrix graphs where all vertices have the same number of nearest neighbors, a more general approach is needed for arbitrary sparse matrices, e.g., those arising from discretized partial differential equations on unstructured meshes. Graph neural networks (GNNs) are one such approach suitable to sparse matrices. The key idea is to define aggregation functions (e.g., summations) that operate on variable-size input data to produce data of a fixed output size so that MLPs can be applied. The goal of this paper is to provide an introduction to GNNs for a numerical linear algebra audience. Concrete GNN examples are provided to illustrate how many common linear algebra tasks can be accomplished using GNNs. We focus on iterative and multigrid methods that employ computational kernels such as matrix-vector products, interpolation, relaxation methods, and strength-of-connection measures. Our GNN examples include cases where parameters are determined a priori as well as cases where parameters must be learned. The intent of this paper is to help computational scientists understand how GNNs can be used to adapt machine learning concepts to computational tasks associated with sparse matrices. It is hoped that this understanding will further stimulate data-driven extensions of classical sparse linear algebra tasks. |
Limits of Learning Dynamical Systems
| Pengarang | : | Tyrus Berry, Suddhasattwa Das |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 1) |
| Halaman | : | 107-137 |
| Abstrak | : | A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system and is the object of focus of many learning techniques. However, there are many secondary aspects of dynamical systems—invariant sets, the Koopman operator, and Markov approximations—that provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces—namely, interpolation spaces, compact Hausdorff sets, unitary operators, and Markov operators, respectively. Thus, learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is placed on methods of learning the primary feature—the dynamics law itself. The main question considered is the connection between learning this law and reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and they reveal how these properties determine the limits of forecasting techniques. |
The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems
| Pengarang | : | Nina M. Gottschling, Vegard Antun, Anders C. Hansen, Ben Adcock |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 1) |
| Halaman | : | 73-104 |
| Abstrak | : | Methods inspired by artificial intelligence (AI) are starting to fundamentally change computational science and engineering through breakthrough performance on challenging problems. However, the reliability and trustworthiness of such techniques is a major concern. In inverse problems in imaging, the focus of this paper, there is increasing empirical evidence that methods may suffer from hallucinations, i.e., false, but realistic-looking artifacts; instability, i.e., sensitivity to perturbations in the data; and unpredictable generalization, i.e., excellent performance on some images, but significant deterioration on others. This paper provides a theoretical foundation for these phenomena. We give mathematical explanations for how and when such effects arise in arbitrary reconstruction methods, with several of our results taking the form of “no free lunch” theorems. Specifically, we show that (i) methods that overperform on a single image can wrongly transfer details from one image to another, creating a hallucination; (ii) methods that overperform on two or more images can hallucinate or be unstable; (iii) optimizing the accuracy-stability tradeoff is generally difficult; (iv) hallucinations and instabilities, if they occur, are not rare events and may be encouraged by standard training; and (v) it may be impossible to construct optimal reconstruction maps for certain problems. Our results trace these effects to the kernel of the forward operator whenever it is nontrivial, but also apply to the case when the forward operator is ill-conditioned. Based on these insights, our work aims to spur research into new ways to develop robust and reliable AI-based methods for inverse problems in imaging. |
Risk-Adaptive Approaches to Stochastic Optimization: A Survey
| Pengarang | : | Johannes O. Royset |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 1) |
| Halaman | : | 3-70 |
| Abstrak | : | Uncertainty is prevalent in engineering design and data-driven problems and, more broadly, in decision making. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount their spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability. |
Uncertainty Analysis of a Simple River Quality Model Using Differential Inequalities
| Pengarang | : | Grace D Agostino, Hermann J. Eberl |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 2) |
| Halaman | : | 375-398 |
| Abstrak | : | We present and discuss the Streeter–Phelps equations, which were the first river quality model. If the parameters are constants, then the model in its linear formulation can be solved explicitly. This reveals, however, that depending on parameters and initial data, the model might predict negative oxygen concentrations, which marks a breakdown of the model. To address this shortcoming, we introduce a nonlinear modification which, in the case of constant parameters, we can study in the phase plane. In real-world applications, parameters are never constant and are usually known not exactly, but instead with some uncertainty. We show how we can use the solutions for the constant parameter case to obtain estimates for the unknown solutions from estimates of the model parameters, using differential inequalities. |
A Nonlocal-to-Local Approach to Aggregation-Diffusion Equations
| Pengarang | : | C. Falco, R. E. Baker, J. A. Carrillo |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 2) |
| Halaman | : | 353-372 |
| Abstrak | : | Over the past few decades, nonlocal models have been widely used to describe aggregation phenomena in biology, physics, engineering, and the social sciences. These are often derived as mean-field limits of attraction-repulsion agent-based models and consist of systems of nonlocal partial differential equations. Using differential adhesion between cells as a biological case study, we introduce a novel local model of aggregation-diffusion phenomena. This system of local aggregation-diffusion equations is fourth-order, resembling thin-film or Cahn–Hilliard type equations. In this framework, cell sorting phenomena are explained through relative surface tensions between distinct cell types. The local model emerges as a limiting case of short-range interactions, providing a significant simplification of earlier nonlocal models while preserving the same phenomenology. This simplification makes the model easier to implement numerically and more amenable to calibration to quantitative data. In addition, we discuss recent analytical results based on the gradient flow structure of the model, along with open problems and future research directions. |
Computerized Tomography and Reproducing Kernels
| Pengarang | : | Ho Yun, Victor M. Panaretos |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 2) |
| Halaman | : | 321-350 |
| Abstrak | : | In this review paper, we provide an overview of numerical methods used in the study of the Gross–Pitaevskii eigenvalue problem (GPEVP). The GPEVP is an important nonlinear Schrödinger equation that is used in quantum physics to describe the ground states of ultracold bosonic gases. The discretization of the GPEVP leads to a nonlinear eigenvalue problem with eigenvector nonlinearities. The rich variety of numerical techniques in the literature for tackling the GPEVP has ingredients from linear algebra, partial differential equations, and numerical optimization as well as gradient flows on Riemannian manifolds. We review this heterogeneous body of literature with a focus on a unified treatment of seemingly different approaches, algorithms, and method properties, and we point to open problems and future challenges in the field. |
The Gross Pitaevskii Equation and Eigenvector Nonlinearities: Numerical Methods and Algorithms
| Pengarang | : | Patrick Henning, Elias Jarlebring |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 2) |
| Halaman | : | 256-317 |
| Abstrak | : | In this review paper, we provide an overview of numerical methods used in the study of the Gross–Pitaevskii eigenvalue problem (GPEVP). The GPEVP is an important nonlinear Schrödinger equation that is used in quantum physics to describe the ground states of ultracold bosonic gases. The discretization of the GPEVP leads to a nonlinear eigenvalue problem with eigenvector nonlinearities. The rich variety of numerical techniques in the literature for tackling the GPEVP has ingredients from linear algebra, partial differential equations, and numerical optimization as well as gradient flows on Riemannian manifolds. We review this heterogeneous body of literature with a focus on a unified treatment of seemingly different approaches, algorithms, and method properties, and we point to open problems and future challenges in the field. |
Multiobjective Optimization Using the R2 Utility
| Pengarang | : | Ben Tu, Nikolas Kantas, Robert M. Lee, Behrang Shafei |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 2) |
| Halaman | : | 213-255 |
| Abstrak | : | The goal of multiobjective optimization is to identify a collection of points which describe the best possible trade-offs among the multiple objectives. In order to solve this vector-valued optimization problem, practitioners often appeal to the use of scalarization functions in order to transform the multiobjective problem into a collection of single-objective problems. This set of scalarized problems can then be solved using traditional single-objective optimization techniques. In this paper, we formalize this convention into a general mathematical framework. We show how this strategy effectively recasts the original multiobjective optimization problem into a single-objective optimization problem defined over sets. An appropriate class of objective functions for this new problem is that of the R2 utilities, which are utility functions that are defined as a weighted integral over the scalarized optimization problem. As part of our work, we show that these utilities are monotone and submodular set functions that can be optimized effectively using greedy optimization algorithms. We then analyze the performance of these greedy algorithms both theoretically and empirically. Our analysis largely focuses on Bayesian optimization, which is a popular probabilistic framework for black-box optimization. |
Vector-based signal processing and quantization for image and video compression
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Proceedings of the IEEE |
| Volume / Edisi | : | 83-2, FEBRUARY (No. 2) |
| Halaman | : | 317-335 |
| Abstrak | : | Image and video compression has become an increasingly important and active area. Many techniques have been developed in this area. Any compression technique can be modeled as a three-stage process. The first stage can be generally called a signal processing stage where an image or video signal is converted into a different domain. Usually, there is no or little loss of information in this stage. The second stage is quantization where loss of information occurs. The third stage is lossless coding that generates the compressed bit stream. The purpose of the signal processing stage is to convert an image or video signal into such a form that quantization can achieve better performance than without the signal processing stage. Because the quantization stage is the place where most of compression is achieved and loss of information occurs, it is naturally the central stage of any compression technique. Since scalar quantization or vector quantization may be used in the second stage, the operation in the first stage should be scalar-based or vector-based respectively in order to match the second stage so that the compression performance can be optimized. In this paper, we summarize the most recent research results on vector-based signal processing and quantization techniques that have shown high compression performance. |