This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution

This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1, with emphasis on foundational issues. After a brief review of functional

We propose a geometric framework to describe and analyse a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion

Physics-informed neural networks (PINNs) and their variants have been very popular in recent years as algorithms for the numerical simulation of both forward and inverse problems for partial differential equations. This article aims to provide a comprehensive review of currently available results on the numerical analysis of PINNs and related models that constitute the backbone of physics-informed

Questions of ‘how best to acquire data’ are essential to modelling and prediction in the natural and social sciences, engineering applications, and beyond. Optimal experimental design (OED) formalizes these questions and creates computational methods to answer them. This article presents a systematic survey of modern OED, from its foundations in classical design theory to current research involving

The Moment-SOS hierarchy, first introduced in optimization in 2000, is based on the theory of the S-moment problem and its dual counterpart: polynomials that are positive on S. It turns out that this methodology can also be used to solve problems with positivity constraints ‘f(x) ≥ 0 for all $\mathbf{x}\in S$ ’ or linear constraints on Borel measures. Such problems can be viewed as specific instances

Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.Nonetheless, the paper is also addressed

Floating-point numbers have an intuitive meaning when it comes to physics-based numerical computations, and they have thus become the most common way of approximating real numbers in computers. The IEEE-754 Standard has played a large part in making floating-point arithmetic ubiquitous today, by specifying its semantics in a strict yet useful way as early as 1985. In particular, floating-point operations

This article surveys research on the application of compatible finite element methods to large-scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa’s C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces linked by the operators of differential calculus. The use

We discuss the modelling framework of port-Hamiltonian descriptor systems and their use in numerical simulation and control. The structure is ideal for automated network-based modelling since it is invariant under power-conserving interconnection, congruence transformations and Galerkin projection. Moreover, stability and passivity properties are easily shown. Condensed forms under orthogonal transformations

One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone. Cones that are homogeneous and self-dual are called symmetric. Conic optimization problems over symmetric cones have been extensively studied, particularly in the literature on interior-point algorithms, and as the foundation of modelling tools for convex optimization. In this paper we consider the

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing

The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at

Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved

Today’s floating-point arithmetic landscape is broader than ever. While scientific computing has traditionally used single precision and double precision floating-point arithmetics, half precision is increasingly available in hardware and quadruple precision is supported in software. Lower precision arithmetic brings increased speed and reduced communication and energy costs, but it produces results

We present the asymptotic transitions from microscopic to macroscopic physics, their computational challenges and the asymptotic-preserving (AP) strategies to compute multiscale physical problems efficiently. Specifically, we will first study the asymptotic transition from quantum to classical mechanics, from classical mechanics to kinetic theory, and then from kinetic theory to hydrodynamics. We then

Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture

In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete mathematical mosaic emerging from the efforts to understand the foundations of deep learning

We present an overviewof the basic theory, modern optimal transportation extensions and recent algorithmic advances. Selected modelling and numerical applications illustrate the impact of optimal transportation in numerical analysis.

Neural networks (NNs) are the method of choice for building learning algorithms. They are now being investigated for other numerical tasks such as solving high-dimensional partial differential equations. Their popularity stems from their empirical success on several challenging learning problems (computer chess/Go, autonomous navigation, face recognition). However, most scholars agree that a convincing

This article addresses the inference of physics models from data, from the perspectives of inverse problems and model reduction. These fields develop formulations that integrate data into physics-based models while exploiting the fact that many mathematical models of natural and engineered systems exhibit an intrinsically low-dimensional solution manifold. In inverse problems, we seek to infer uncertain

The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way through the lens of linear algebra and numerical linear algebra, elucidated with examples from computational and applied mathematics.

Liquid crystals are a type of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the past four decades, which is of great importance for fundamental scientific research and has widespread applications in industry. In this paper we review the mathematical models and their connections to liquid crystals, and

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations,nonlocal modelsthat account for interaction occurring

Phase retrieval,i.e.the problem of recovering a function from the squared magnitude of its Fourier transform, arises in many applications, such as X-ray crystallography, diffraction imaging, optics, quantum mechanics and astronomy. This problem has confounded engineers, physicists, and mathematicians for many decades. Recently, phase retrieval has seen a resurgence in research activity, ignited by

The semiclassically scaled time-dependent multi-particle Schrödinger equation describes, inter alia, quantum dynamics of nuclei in a molecule. It poses the combined computational challenges of high oscillations and high dimensions. This paper reviews and studies numerical approaches that are robust to the small semiclassical parameter. We present and analyse variationally evolving Gaussian wave packets

This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues.Topics include norm estimation, matrix approximation by sampling, structured and

We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities

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Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences

Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort

In many optimization problems arising from scientific, engineering and artificial intelligence applications, objective and constraint functions are available only as the output of a black-box or simulation oracle that does not provide derivative information. Such settings necessitate the use of methods for derivative-free, or zeroth-order, optimization. We provide a review and perspectives on developments

Kohn–Sham density functional theory (DFT) is the most widely used electronic structure theory. Despite significant progress in the past few decades, the numerical solution of Kohn–Sham DFT problems remains challenging, especially for large-scale systems. In this paper we review the basics as well as state-of-the-art numerical methods, and focus on the unique numerical challenges of DFT.

We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted $b$-matching, and minimization versions of weighted edge cover and $b$-edge cover. Exact

Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete- and continuous-time data assimilation, both in terms of mathematical

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of

This article is about numerical methods for the solution of nonlinear equations. We consider both the fixed-point form $\mathbf{x}=\mathbf{G}(\mathbf{x})$ and the equations form $\mathbf{F}(\mathbf{x})=0$ and explain why both versions are necessary to understand the solvers. We include the classical methods to make the presentation complete and discuss less familiar topics such as Anderson acceleration

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it

The use of computational models and simulations to predict events that take place in our physical universe, or to predict the behaviour of engineered systems, has significantly advanced the pace of scientific discovery and the creation of new technologies for the benefit of humankind over recent decades, at least up to a point. That ‘point’ in recent history occurred around the time that the scientific

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Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties

This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct

The structure from motion (SfM) problem in computer vision is to recover the three-dimensional (3D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional (2D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (i) extracting features in images (e.g. points of

Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function

This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract

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Many crucial scientific computing applications, ranging from national security to medical advances, rely on high-performance linear algebra algorithms and technologies, underscoring their importance and broad impact. Here we present the state-of-the-art design and implementation practices for the acceleration of the predominant linear algebra algorithms on large-scale accelerated multicore systems

A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and

In recent decades, condition numbers have joined forces with probabilistic analysis to give rise to a form of condition-based analysis of algorithms. In this paper we survey how this analysis is done via a number of examples. We precede this catalogue of examples with short primers on both condition numbers and probabilistic analyses.

Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical