Publications (82)
On the Cartan Decomposition for Classical Random Matrix Ensembles
Alan Edelman, Sungwoo Jeong
We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between sy…
Sample eigenvalue based detection of high dimensional signals in white noise using relatively few samples
N. Raj Rao, Alan Edelman
We present a mathematically justifiable, computationally simple, sample eigenvalue based procedure for estimating the number of high-dimensional signals in white noise using relati…
The Beta-Wishart Ensemble
Alexander Dubbs, Alan Edelman, Plamen Koev +1
This paper proves a matrix model for the Wishart Ensemble with general covariance and general dimension parameter beta. In so doing, we introduce a new and elegant definition of Ja…
Some New Results on the Maximum Growth Factor in Gaussian Elimination
Alan Edelman, John Urschel
This paper combines modern numerical computation with theoretical results to improve our understanding of the growth factor problem for Gaussian elimination. On the computational s…
The largest 5th pivot may be the root of a 61st degree polynomial
James Chen, Alan Edelman, John Urschel
This paper introduces a number of new techniques in the study of the famous question from numerical linear algebra: what is the largest possible growth factor when performing Gauss…
Eigenvalues of Hermite and Laguerre ensembles: Large Beta Asymptotics
Ioana Dumitriu, Alan Edelman
In this paper we examine the zero and first order eigenvalue fluctuations for the -Hermite and -Laguerre ensembles, using the matrix models we described in \cite{dumitriu02…
Condition Numbers of Indefinite Rank 2 Ghost Wishart Matrices
Ramis Movassagh, Alan Edelman
We define an indefinite Wishart matrix as a matrix of the form A=W^{T}WΣ, where Σis an indefinite diagonal matrix and W is a matrix of independent standard normals. We focus on t…
Isotropic Entanglement
Ramis Movassagh, Alan Edelman
The method of "Isotropic Entanglement" (IE), inspired by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue distribution of quantum many-body (spin) systems…
The Singular Values of the GUE (Less is More)
Alan Edelman, Michael La Croix
Some properties that nominally involve the eigenvalues of Gaussian Unitary Ensemble (GUE) can instead be phrased in terms of singular values. By discarding the signs of the eigenva…
On a perturbation analysis of Higham squared maximum Gaussian elimination growth matrices
Alan Edelman, John Urschel, Bowen Zhu
Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth…
Automated Translation and Accelerated Solving of Differential Equations on Multiple GPU Platforms
Utkarsh Utkarsh, Valentin Churavy, Yingbo Ma +8
We demonstrate a high-performance vendor-agnostic method for massively parallel solving of ensembles of ordinary differential equations (ODEs) and stochastic differential equations…
The Geometry of Algorithms with Orthogonality Constraints
Alan Edelman, T. A. Arias, Steven T. Smith
In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as…
The polynomial method for random matrices
N. Raj Rao, Alan Edelman
We define a class of "algebraic" random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., i…
Global spectrum fluctuations for the -Hermite and -Laguerre ensembles via matrix models
Ioana Dumitriu, Alan Edelman
We study the global spectrum fluctuations for -Hermite and -Laguerre ensembles via the tridiagonal matrix models introduced in \cite{dumitriu02}, and prove that the fluctua…
Multiscale Computation with Interpolating Wavelets
Ross A. Lippert, T. A. Arias, Alan Edelman
Multiresolution analyses based upon interpolets, interpolating scaling functions introduced by Deslauriers and Dubuc, are particularly well-suited to physical applications because…
Backpropagation through Back Substitution with a Backslash
Alan Edelman, Ekin Akyurek, Yuyang Wang
We present a linear algebra formulation of backpropagation which allows the calculation of gradients by using a generically written ``backslash'' or Gaussian elimination on triangu…
An Efficient Partitioning Oracle for Bounded-Treewidth Graphs
Alan Edelman, Avinatan Hassidim, Huy N. Nguyen +1
Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a generic tool for constant-time algorithms. For any epsilon > 0, a partitioning oracle provides query access…
Sampling Pfaffian point processes and the symplectic Arnoldi method
Alan Edelman, Sungwoo Jeong, Simeon Schaub
We present an exact sampling algorithm for Pfaffian point processes based on a skew-symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a…
Error analysis of free probability approximations to the density of states of disordered systems
Jiahao Chen, Eric Hontz, Jeremy Moix +5
Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approxim…
The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument
Plamen Koev, Alan Edelman
We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit…
ABM-UDE: Developing Surrogates for Epidemic Agent-Based Models via Scientific Machine Learning
Sharv Murgai, Utkarsh Utkarsh, Kyle C. Nguyen +3
Agent-based epidemic models (ABMs) encode behavioral and policy heterogeneity but are too slow for nightly hospital planning. We develop county-ready surrogates that learn directly…
Cross-Model Cross-Language AI Coding Agent Performance: Accuracy and Speed of Parallel CLRS Algorithms
Shiqi Cheng, Evelyne Ringoot, Rabab Alomairy +1
AI coding agents have quickly become omnipresent in software engineering. Their serial performance, both in terms of accuracy and speed, has been extensively covered. However, rece…
On the Limit of the Tridiagonal Model for -Dyson Brownian Motion
Alan Edelman, Sungwoo Jeong, Ron Nissim
In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a GE random matrix is provided by Edelman and Dumitriu. The resulting…
From Random Matrices to Stochastic Operators
Alan Edelman, Brian D. Sutton
We propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators. Three particular stochastic operators commonl…
Toward Portable GPU Performance: Julia Recursive Implementation of TRMM and TRSM
Vicki Carrica, Maxwell Onyango, Rabab Alomairy +3
This paper presents a performant and portable recursive implementation of triangular matrix-matrix multiplication (TRMM) and triangular solve (TRSM) in Julia for GPUs, two kernels…
Parallel Prefix Polymorphism Permits Parallelization, Presentation & Proof
Jiahao Chen, Alan Edelman
Polymorphism in programming languages enables code reuse. Here, we show that polymorphism has broad applicability far beyond computations for technical computing: parallelism in di…
Scientific Machine Learning of Chaotic Systems Learns Reduced-Order Equations for Neural Populations
Anthony G. Chesebro, David Hofmann, Vaibhav Dixit +6
Extracting interpretable mathematical models from complex dynamical systems is difficult, especially for chaotic dynamics observed with noisy experimental data. We present PEM-UDE,…
Array operators using multiple dispatch: a design methodology for array implementations in dynamic languages
Jeff Bezanson, Jiahao Chen, Stefan Karpinski +2
Arrays are such a rich and fundamental data type that they tend to be built into a language, either in the compiler or in a large low-level library. Defining this functionality at…
Topology-optimized distributed 3d anisotropic Raman emission
Ian M. Hammond, Pengning Chao, Henry O. Everitt +4
Topology optimization (TO) of 3D surface-enhanced Raman scattering (SERS) substrates faces challenges in managing field singularities and modeling orientation-averaged anisotropic…
Accelerating Bidiagonalization of Banded Matrices through Memory-Aware Bulge-Chasing on GPUs
Evelyne Ringoot, Rabab Alomairy, Alan Edelman
The reduction of a banded matrix to bidiagonal form is a critical step in the calculation of Singular Values, a cornerstone of scientific computing and AI. Although inherently para…
High-performance symbolic-numerics via multiple dispatch
Shashi Gowda, Yingbo Ma, Alessandro Cheli +4
As mathematical computing becomes more democratized in high-level languages, high-performance symbolic-numeric systems are necessary for domain scientists and engineers to get the…
Fifty Three Matrix Factorizations: A systematic approach
Alan Edelman, Sungwoo Jeong
The success of matrix factorizations such as the singular value decomposition (SVD) has motivated the search for even more factorizations. We catalog 53 matrix factorizations, most…
Oceananigans.jl: A Julia library that achieves breakthrough resolution, memory and energy efficiency in global ocean simulations
Simone Silvestri, Gregory L. Wagner, Christopher Hill +10
Climate models must simulate hundreds of future scenarios for hundreds of years at coarse resolutions, and a handful of high-resolution decadal simulations to resolve localized ext…
Infinite Random Matrix Theory, Tridiagonal Bordered Toeplitz Matrices, and the Moment Problem
Alexander Dubbs, Alan Edelman
The four major asymptotic level density laws of random matrix theory may all be showcased though their Jacobi parameter representation as having a bordered Toeplitz form. We compar…
Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints
Utkarsh Utkarsh, Pengfei Cai, Alan Edelman +2
Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs), offering scalable simulation and uncertainty-aware inferenc…
Stochastic Optimal Control via Local Occupation Measures
Flemming Holtorf, Alan Edelman, Christopher Rackauckas
Viewing stochastic processes through the lens of occupation measures has proved to be a powerful angle of attack for the theoretical and computational analysis of stochastic optima…
A Differentiable Programming System to Bridge Machine Learning and Scientific Computing
Mike Innes, Alan Edelman, Keno Fischer +4
Scientific computing is increasingly incorporating the advancements in machine learning and the ability to work with large amounts of data. At the same time, machine learning model…
How many zeros of a random polynomial are real?
Alan Edelman, Eric Kostlan
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distribute…
Locally Regularized Neural Differential Equations: Some Black Boxes Were Meant to Remain Closed!
Avik Pal, Alan Edelman, Chris Rackauckas
Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically…
Fast computation of the principal components of genotype matrices in Julia
Jiahao Chen, Andreas Noack, Alan Edelman
Finding the largest few principal components of a matrix of genetic data is a common task in genome-wide association studies (GWASs), both for dimensionality reduction and for iden…
Efficient Symbolic Computation via Hash Consing
Bowen Zhu, Aayush Sabharwal, Songchen Tan +3
Symbolic computation systems suffer from memory inefficiencies due to redundant storage of structurally identical subexpressions, commonly known as expression swell, which degrades…
Matrix Calculus (for Machine Learning and Beyond)
Paige Bright, Alan Edelman, Steven G. Johnson
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector…
Density of States of Quantum Spin Systems from Isotropic Entanglement
Ramis Movassagh, Alan Edelman
We propose a method which we call "Isotropic Entanglement" (IE), that predicts the eigenvalue distribution of quantum many body (spin) systems (QMBS) with generic interactions. We…
Julia: A Fast Dynamic Language for Technical Computing
Jeff Bezanson, Stefan Karpinski, Viral B. Shah +1
Dynamic languages have become popular for scientific computing. They are generally considered highly productive, but lacking in performance. This paper presents Julia, a new dynami…
Random Triangle Theory with Geometry and Applications
Alan Edelman, Gilbert Strang
What is the probability that a random triangle is acute? We explore this old question from a modern viewpoint, taking into account linear algebra, shape theory, numerical analysis,…
Accelerating Simulation of Stiff Nonlinear Systems using Continuous-Time Echo State Networks
Ranjan Anantharaman, Yingbo Ma, Shashi Gowda +4
Modern design, control, and optimization often requires simulation of highly nonlinear models, leading to prohibitive computational costs. These costs can be amortized by evaluatin…
AutoMat: Accelerated Computational Electrochemical systems Discovery
Emil Annevelink, Rachel Kurchin, Eric Muckley +17
Large-scale electrification is vital to addressing the climate crisis, but several scientific and technological challenges remain to fully electrify both the chemical industry and…
Continuous Deep Equilibrium Models: Training Neural ODEs faster by integrating them to Infinity
Avik Pal, Alan Edelman, Christopher Rackauckas
Implicit models separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and in…
Beyond universality in random matrix theory
Alan Edelman, A. Guionnet, S. Péché
In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the expansion of local eigenvalue statistics in both the bulk and at the…
The Beta-MANOVA Ensemble with General Covariance
Alexander Dubbs, Alan Edelman
We find the joint generalized singular value distribution and largest generalized singular value distributions of the -MANOVA ensemble with positive diagonal covariance, which…
Sample size cognizant detection of signals in white noise
N. Raj Rao, Alan Edelman
The detection and estimation of signals in noisy, limited data is a problem of interest to many scientific and engineering communities. We present a computationally simple, sample…
Statistical eigen-inference from large Wishart matrices
N. Raj Rao, James A. Mingo, Roland Speicher +1
We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues o…
Numerical Methods for Eigenvalue Distributions of Random Matrices
Alan Edelman, Per-Olof Persson
We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on ve…
Performant Unified GPU Kernels for Portable Singular Value Computation Across Hardware and Precision
Evelyne Ringoot, Rabab Alomairy, Valentin Churavy +1
This paper presents a portable, GPU-accelerated implementation of a QR-based singular value computation algorithm in Julia. The singular value ecomposition (SVD) is a fundamental n…
SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling
Alaina Kolli, Theodoros Xenakis, Utkarsh Utkarsh +4
Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, an…
Performance Bounds for Quantum Feedback Control
Flemming Holtorf, Frank Schäfer, Julian Arnold +2
The limits of quantum feedback control have immediate consequences for quantum information science at large, yet remain largely unexplored. Here, we combine quantum filtering theor…
Partial freeness of random matrices
Jiahao Chen, Troy Van Voorhis, Alan Edelman
We investigate the implications of free probability for random matrices. From rules for calculating all possible joint moments of two free random matrices, we develop a notion of p…
Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices
Alan Edelman, Timothy E. Holy
Positive-semidefinite matrices are most efficiently factored using the Cholesky decomposition. For indefinite matrices, the Cholesky factorization does not exist, and the alternati…
MOPS: Multivariate Orthogonal Polynomials (symbolically)
Ioana Dumitriu, Alan Edelman, Gene Shuman
In this paper we present a Maple library (MOPs) for computing Jack, Hermite, Laguerre, and Jacobi multivariate polynomials, as well as eigenvalue statistics for the Hermite, Laguer…
A New Upper Bound For the Growth Factor in Gaussian Elimination with Complete Pivoting
Ankit Bisain, Alan Edelman, John Urschel
The growth factor in Gaussian elimination measures how large the entries of an LU factorization can be relative to the entries of the original matrix. It is a key parameter in erro…
The GSVD: Where are the ellipses?, Matrix Trigonometry, and more
Alan Edelman, Yuyang Wang
This paper provides an advanced mathematical theory of the Generalized Singular Value Decomposition (GSVD) and its applications. We explore the geometry of the GSVD which provides…
Matrix Models for Beta Ensembles
Ioana Dumitriu, Alan Edelman
This paper constructs tridiagonal random matrix models for general () -Hermite (Gaussian) and -Laguerre (Wishart) ensembles. These generalize the well-known Gaussian…
Integral geometry for Markov chain Monte Carlo: overcoming the curse of search-subspace dimensionality
Oren Mangoubi, Alan Edelman
We introduce a method that uses the Cauchy-Crofton formula and a new curvature formula from integral geometry to reweight the sampling probabilities of Metropolis-within-Gibbs algo…
Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints
Avik Pal, Alan Edelman, Christopher Rackauckas
Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in…
Circuitscape in Julia: High Performance Connectivity Modelling to Support Conservation Decisions
Ranjan Anantharaman, Kimberly Hall, Viral Shah +1
Connectivity across landscapes influences a wide range of conservation-relevant ecological processes, including species movements, gene flow, and the spread of wildfire, pests, and…
Convex Network Flows
Theo Diamandis, Guillermo Angeris, Alan Edelman
We introduce a general framework for flow problems over hypergraphs. In our problem formulation, which we call the convex flow problem, we have a concave utility function for the n…
Mapping out phase diagrams with generative classifiers
Julian Arnold, Frank Schäfer, Alan Edelman +1
One of the central tasks in many-body physics is the determination of phase diagrams. However, mapping out a phase diagram generally requires a great deal of human intuition and un…
Composing Modeling and Simulation with Machine Learning in Julia
Chris Rackauckas, Ranjan Anantharaman, Alan Edelman +10
In this paper we introduce JuliaSim, a high-performance programming environment designed to blend traditional modeling and simulation with machine learning. JuliaSim can build acce…
Scalable higher-order nonlinear solvers via higher-order automatic differentiation
Songchen Tan, Keming Miao, Alan Edelman +1
This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives.…
The conditional DPP approach to random matrix distributions
Alan Edelman, Sungwoo Jeong
We present the conditional determinantal point process (DPP) approach to obtain new (mostly Fredholm determinantal) expressions for various eigenvalue statistics in random matrix t…
Hierarchical Recursive Precision for Accelerating Symmetric Linear Solves on MXUs
Vicki Carrica, Rabab Alomairy, Evelyne Ringoot +1
Symmetric positive-definite system solvers based on Cholesky factorization are fundamental to many scientific applications, such as climate modeling. We present a portable, nested…
On the structure of the solutions to the matrix equation
Alan Edelman, Sungwoo Jeong
We study the mathematical structure of the solution set (and its tangent space) to the matrix equation for a given square matrix . In the language of pure mathematics,…
Bridging HPC Communities through the Julia Programming Language
Valentin Churavy, William F Godoy, Carsten Bauer +9
The Julia programming language has evolved into a modern alternative to fill existing gaps in scientific computing and data science applications. Julia leverages a unified and coor…
TabulaROSA: Tabular Operating System Architecture for Massively Parallel Heterogeneous Compute Engines
Jeremy Kepner, Ron Brightwell, Alan Edelman +10
The rise in computing hardware choices is driving a reevaluation of operating systems. The traditional role of an operating system controlling the execution of its own hardware is…
Eigenvalue approximation of sums of Hermitian matrices from eigenvector localization/delocalization
Ramis Movassagh, Alan Edelman
We propose a technique for calculating and understanding the eigenvalue distribution of sums of random matrices from the known distribution of the summands. The exact problem is fo…
Julia Implementation of the Dynamic Distributed Dimensional Data Model
Alexander Chen, Alan Edelman, Jeremy Kepner +2
Julia is a new language for writing data analysis programs that are easy to implement and run at high performance. Similarly, the Dynamic Distributed Dimensional Data Model (D4M) a…
Accelerated Convolutions for Efficient Multi-Scale Time to Contact Computation in Julia
Alexander Amini, Berthold Horn, Alan Edelman
Convolutions have long been regarded as fundamental to applied mathematics, physics and engineering. Their mathematical elegance allows for common tasks such as numerical different…
Julia: A Fresh Approach to Numerical Computing
Jeff Bezanson, Alan Edelman, Stefan Karpinski +1
Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical…
Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards
Pengfei Cai, Utkarsh Utkarsh, Alan Edelman +2
Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert…
Signal Enhancement for Magnetic Navigation Challenge Problem
Albert R. Gnadt, Joseph Belarge, Aaron Canciani +10
Harnessing the magnetic field of the Earth for navigation has shown promise as a viable alternative to other navigation systems. A magnetic navigation system collects its own magne…
NonlinearSolve.jl: High-Performance and Robust Solvers for Systems of Nonlinear Equations in Julia
Avik Pal, Flemming Holtorf, Axel Larsson +6
Efficiently solving nonlinear equations underpins numerous scientific and engineering disciplines, yet scaling these solutions for challenging system models remains a challenge. Th…
Universal Differential Equations for Scientific Machine Learning
Christopher Rackauckas, Yingbo Ma, Julius Martensen +6
In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." In this manuscript we introduce the SciM…