The nonlocal cubic Gross-Pitaevskii equation, in comparison to the cubic Gross-Pitaevskii equation, incorporates a nonlocal diffusion operator and can capture a wider range of practical phenomena. However, this nonlocal formulation significantly increases the computational expenses in numerical simulations, necessitating the development of efficient and accurate time integration schemes. This paper

The paper is concerned with efficient time discretization methods based on exponential integrators for scalar hyperbolic conservation laws. The model problem is first discretized in space by the discontinuous Galerkin method, resulting in a system of nonlinear ordinary differential equations. To solve such a system, exponential time differencing of order 2 (ETDRK2) is employed with Jacobian linearization

This paper introduces an efficient high-order numerical method for solving the 1D stationary Schrödinger equation in the highly oscillatory regime. Building upon the ideas from the article (Arnold et al. SIAM J. Numer. Anal. 49, 1436–1460, 2011), we first analytically transform the given equation into a smoother (i.e., less oscillatory) equation. By developing sufficiently accurate quadratures for

The demagnetization field in micromagnetism is given as the gradient of a potential that solves a partial differential equation (PDE) posed in \(\mathbb {R}^d\). In its most general form, this PDE is supplied with continuity condition on the boundary of the magnetic domain, and the equation includes a discontinuity in the gradient of the potential over the boundary. Typical numerical algorithms to

In this paper, we investigate the optimal control problem governed by parabolic PDEs in random cylindrical domains, where the random domains are independent of time. We introduce a random mapping to transform the original problem in the random domain into the stochastic problem in the reference domain. The randomness of the transformed problem is reflected in the random coefficient matrix of the elliptic

SIAM Review, Volume 67, Issue 2, Page 213-255, May 2025. Abstract.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

We introduce a two-parameter Tikhonov regularization method to approximate an ill-posed problem with an unbounded matrix operator. The existence and uniqueness of regularized solutions to the problem are derived. With an a priori as well as an a posteriori parameter choice strategy, convergence analysis of the regularized solution is presented. As an application, we apply the regularization to a simultaneous

SIAM Review, Volume 67, Issue 2, Page 411-411, May 2025. A short discussion on stochastic calculus is given under the assumption that the fundamentals of probability theory are known to readers. Some related basic details on probability theory should have been included to make the book more self-contained. Further, analytic solutions of some stochastic differential equations (SDEs), which are used

SIAM Review, Volume 67, Issue 2, Page 408-411, May 2025. Optimal transport was originally invented by Gaspard Monge [“Mémoire sur la théorie des déblais et des remblais,” Mem. Math. Phys. Acad. Royale Sci., (1781), pp. 666–704] to model the problem of optimally mapping one distribution of mass onto another. This was later reformulated by Leonid Kantorovich as a well-posed linear program using the notion

SIAM Review, Volume 67, Issue 2, Page 406-408, May 2025. The 2024 Nobel Prize in Physics was awarded to John Hopfield and Geoffrey Hinton for their work on artificial intelligence and machine learning. The award has been somewhat controversial in the physics community and prompted some heated debates, since the only apparent use of physics is the Boltzmann distribution in the sampling function of the

SIAM Review, Volume 67, Issue 2, Page 405-406, May 2025. Big Data Analytics for Smart Transport and Healthcare Systems explores the praxis of data analysis for urban, human-focused datasets. Through a series of timely case studies, the authors demonstrate the need for interdisciplinary approaches to studying big data. This text, which covers topics ranging from flight status to the COVID-19 pandemic

SIAM Review, Volume 67, Issue 2, Page 404-405, May 2025. “Math is like a drag queen: marvelous, whimsical, at times even controversial, but never boring!” That it how the preface of Math in Drag begins. It is also an excellent description of the book. Math in Drag was authored by Kyne Santos, who often goes by Kyne. Kyne studied mathematics at the University of Waterloo and went viral teaching math

SIAM Review, Volume 67, Issue 2, Page 401-403, May 2025. It’s 7.30 am when my alarm wakes me up and I am greeted by my notifications. While eating breakfast, I watch videos YouTube recommends to me: sometimes news stories, sometimes my guilty pleasure of a new “Say Yes to the Dress” clip. On my way to campus, I play my daylist, a curated playlist from Spotify based on what I normally listen to on a

SIAM Review, Volume 67, Issue 2, Page 399-399, May 2025.

SIAM Review, Volume 67, Issue 2, Page 375-398, May 2025. Abstract.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

SIAM Review, Volume 67, Issue 2, Page 373-373, May 2025.

SIAM Review, Volume 67, Issue 2, Page 353-372, May 2025. Abstract.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

SIAM Review, Volume 67, Issue 2, Page 351-351, May 2025.

SIAM Review, Volume 67, Issue 2, Page 321-350, May 2025. Abstract.The X-ray transform is one of the most fundamental integral operators in image processing and reconstruction. In this paper, we revisit the formalism of the X-ray transform by considering it as an operator between reproducing kernel Hilbert spaces (RKHSs). Within this framework, the X-ray transform can be viewed as a natural analogue

SIAM Review, Volume 67, Issue 2, Page 319-319, May 2025.

SIAM Review, Volume 67, Issue 2, Page 256-317, May 2025. Abstract.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

SIAM Review, Volume 67, Issue 2, Page 211-211, May 2025.

Multiscale partial differential equations (PDEs), featuring heterogeneous coefficients oscillating across possibly non-separated scales, pose computational challenges for standard numerical techniques. Over the past two decades, a range of specialized methods has emerged that enables the efficient solution of such problems. Two prominent approaches are numerical multiscale methods with problem-adapted

Symbolic summation as an active research topic of symbolic computation provides efficient algorithmic tools for evaluating and simplifying different types of sums arising from mathematics, computer science, physics and other areas. Most of existing algorithms in symbolic summation are mainly applicable to the problem with univariate inputs. A long-term project in symbolic computation is to develop

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, \(\mathbb {Z}_2\)-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its

This paper develops fast and efficient algorithms for computing Tucker decomposition with a given multilinear rank. By combining random projection and the power scheme, we propose two efficient randomized versions for the truncated high-order singular value decomposition (T-HOSVD) and the sequentially T-HOSVD (ST-HOSVD), which are two common algorithms for approximating Tucker decomposition. To reduce

This paper deals with efficient numerical methods for computing the action of the matrix generating function of Bernoulli polynomials, say \(q(\tau ,A)\), on a vector when A is a large and sparse matrix. This problem occurs when solving some non-local boundary value problems. Methods based on the Fourier expansion of \(q(\tau ,w)\) have already been addressed in the scientific literature. The contribution

In this paper, we propose a discontinuous plane wave neural network (DPWNN) method with \(hp-\)refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with \(h-\)refinement and introduce new discretization sets spanned by element-wise neural network functions with a

Exponential integrators are an efficient alternative to implicit schemes for the time integration of stiff system of differential equations. In this paper, low-rank exponential integrators of orders one and two for stiff differential Riccati equations are proposed and investigated. The error estimates of the proposed schemes are established. The proposed approach allows to overcome the main difficulties

In this paper, we adopt a quasi-boundary-value method to solve the nonlinear space-fractional backward problem with perturbed both final value and variable diffusion coefficient in general dimensional space, which is a severely ill-posed problem. The existence, uniqueness and stability of the solution for the quasi-boundary-value problem are proved. Convergence estimates are presented under an a-priori

We introduce a simple initialization of the Maubach bisection routine for adaptive mesh refinement which applies to any conforming initial triangulation and terminates in linear time with respect to the number of initial vertices. We show that Maubach’s routine with this initialization always terminates and generates meshes that preserve shape regularity and satisfy the closure estimate needed for

We investigate numerical approximations for the stochastic Burgers equation driven by an additive cylindrical fractional Brownian motion with Hurst parameter \(H \in (\frac{1}{2}, 1)\). To discretize the continuous problem in space, a spectral Galerkin method is employed, followed by the presentation of a nonlinear-tamed accelerated exponential Euler method to yield a fully discrete scheme. By showing

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. 68, 337–404 1950), the product of positive semi-definite kernel functions is again positive semi-definite

The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in

In 1991, Moore (Nonlinearity 4:199–230, 1991) raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao (J Am Math Soc 29(3):601–674, 2016) asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in Cardona et al. (Proc Natl Acad Sci 118(19):e2026818118

We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of multi-dimensional non-linear stochastic differential equations, which relies upon discrete-time observations of the state. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker–Planck equation to such observations, yielding theoretical estimates

The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Its decay has extensively been studied in approximation theory and for the solution of partial differential equations (PDEs). Particular interest has occurred within model order reduction (MOR) of parameterized PDEs, e.g., by the reduced basis method (RBM). While it is known that the

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below \(H^1\). New harmonic analysis tools, including averaging approximations to the exponential phase functions and trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence

This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported on semialgebraic sets and subject to moment polynomial constraints is investigated. On the one hand, a positive solution to Hilbert’s 17th problem for pseudo-moments

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply

We count singular vector tuples of a system of tensors assigned to the edges of a directed hypergraph. To do so, we study the generalisation of quivers to directed hypergraphs. Assigning vector spaces to the nodes of a hypergraph and multilinear maps to its hyperedges gives a hyperquiver representation. Hyperquiver representations generalise quiver representations (where all hyperedges are edges) and

Recently, deep neural networks (DNNs) have become powerful tools for solving inverse scattering problems. However, the approximation and generalization rates of DNNs for solving these problems remain largely under-explored. In this work, we introduce two types of combined DNNs (uncompressed and compressed) to reconstruct two function-valued coefficients in the Helmholtz equation for inverse scattering

We introduce an algorithm to decompose matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a d-dimensional representation of \(S_n\) is shown to have a complexity of \({\mathcal {O}}(n^2 d^3)\) operations for determining which irreducible representations

A non-uniform implicit-explicit L1 mixed finite element method (IMEX-L1-MFEM) is investigated for a class of time-fractional partial integro-differential equations (PIDEs) with space-time-dependent coefficients and non-self-adjoint elliptic part. The proposed fully discrete method combines an IMEX-L1 method on a graded mesh in the temporal variable with a mixed finite element method in spatial variables

SIAM Review, Volume 67, Issue 1, Page 208-209, March 2025. The book Mathematical Pictures at a Data Science Exhibition aims to introduce the reader to the many mathematical ideas that congregate under the ever-expanding umbrella of data science. Given the meteoric rise of this field and the immense speed at which it often moves, this book acts as a welcome road map for graduate students and researchers

SIAM Review, Volume 67, Issue 1, Page 207-208, March 2025. Elegant Simulations covers various aspects of modeling and simulating mechanical systems described at the elementary level by many-interacting particles. The book presents the topics from an original and fresh point of view. The complex many-body dynamics is reproduced at the elementary level in terms of simple models that are easy to understand

SIAM Review, Volume 67, Issue 1, Page 206-207, March 2025. This is a bold book! Professor Zhu wants to provide the basic statistical knowledge needed by data scientists in a super-short volume. It reminds me a bit of Larry Wasserman’s All of Statistics (Springer, 2014), but is aimed at Masters students (often from fields other than statistics) or advanced undergraduates (also often from other fields)

SIAM Review, Volume 67, Issue 1, Page 205-206, March 2025. The first look at Probability Adventures brought back memories of a conference in Ubatuba, Brazil, in 2001, where as a young Master’s student I worried that true science had to be deadly serious. Fortunately, several inspiring teachers came to the rescue. Andrei Toom’s words resonated deeply with me when he began his lecture by saying, “Every

SIAM Review, Volume 67, Issue 1, Page 204-205, March 2025. Numerical Methods in Physics with Python by Alex Gezerlis is an excellent example of a textbook built on long and established teaching experience. The goals are clearly defined in the preface: Gezerlis aims to gently introduce undergraduate physics students to the branch of numerical methods and their concrete implementation in Python. To this

SIAM Review, Volume 67, Issue 1, Page 197-204, March 2025. The book under review was originally published under the auspices of the National Research Council in 1933 (the year John was born), and it was republished as a Dover edition in 1956 (three years before Rob was born). At 108 pages—including title page, preface, table of contents, and index—it’s very short. Even so, it contains a significant

SIAM Review, Volume 67, Issue 1, Page 195-196, March 2025.

SIAM Review, Volume 67, Issue 1, Page 176-193, March 2025. Abstract.This paper is intended to serve as a low-hurdle introduction to nonlocality for graduate students and researchers with an engineering mechanics or physics background who did not have a formal introduction to the underlying mathematical basis. We depart from simple examples motivated by structural mechanics to form a physical intuition

SIAM Review, Volume 67, Issue 1, Page 141-175, March 2025. Abstract.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

SIAM Review, Volume 67, Issue 1, Page 139-140, March 2025.

SIAM Review, Volume 67, Issue 1, Page 107-137, March 2025. Abstract.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

SIAM Review, Volume 67, Issue 1, Page 105-105, March 2025.

SIAM Review, Volume 67, Issue 1, Page 73-104, March 2025. Abstract.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

SIAM Review, Volume 67, Issue 1, Page 71-71, March 2025.

SIAM Review, Volume 67, Issue 1, Page 3-70, March 2025. Abstract.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

SIAM Review, Volume 67, Issue 1, Page 1-1, March 2025.