This paper studies the stochastic Allen-Cahn equation involving random diffusion coefficient field and multiplicative force noise. A new time-stepping method based on auxiliary variable approach is proposed and analyzed. The proposed method is efficient thanks to its low computational complexity. Furthermore, it is unconditionally stable in the sense that a discrete energy is dissipative when the multiplicative

For the time-fractional neutron diffusion equation with a Caputo derivative of order \( \varvec{\alpha } \in ~(\varvec{0},\frac{\varvec{1}}{\varvec{2}}) \), we give the optimal error bounds of L1-type schemes under the spatial \( L^{\infty } \)-norm with lower regularity solution than typical \(| \varvec{\partial }_{\varvec{t}}^{\varvec{l}} \varvec{u}(\varvec{x},\varvec{t}) | \le \varvec{C} (\varv

We present a numerical study of solutions to the 2d cubic and quintic focusing nonlinear Schrödinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling towards the obstacle at different angles and with different velocities directions. We introduce

The author’s recent research papers, “Cumulative deviation of a subpopulation from the full population” and “A graphical method of cumulative differences between two subpopulations” (both published in volume 8 of Springer’s open-access Journal of Big Data during 2021), propose graphical methods and summary statistics, without extensively calibrating formal significance tests. The summary metrics and

We give a detailed analysis of the convergence in Sobolev norm of the method of lines for the classical heat and wave equations on \(\mathbb {R }^n \) using non-stationary radial basis function interpolation on regular grids \(h \mathbb {Z }^n \) (scaled cardinal interpolation), for basis functions whose native space is a Sobolev space of order \(\nu / 2 \) with \(\nu > n + 2\).

In earlier work, the second author showed that a closed subset of a polynomial functor can always be defined by finitely many polynomial equations. In follow-up work on \({\text {GL}}_\infty \)-varieties, Bik–Draisma–Eggermont–Snowden showed, among other things, that in characteristic zero every such closed subset is the image of a morphism whose domain is the product of a finite-dimensional affine

In this paper we study the a posteriori error estimates for the time dependent Navier-Stokes system coupled with the convection-diffusion-reaction equation. The problem is discretized in time using the implicit Euler method and in space using the finite element method. We establish a posteriori error estimates with two types of computable error indicators, the first one linked to the space discretization

SIAM Review, Volume 65, Issue 3, Page 905-915, August 2023. This collection of reviews encompasses a wide range of topics. We kick off with an insightful featured review by Chris Oats on the book Probabilistic Numerics, written by Philipp Hennig, Michael A. Osborne, and Hans P. Kersting. Oats expresses his own fascination with the topic and highly recommends delving into the substantial tome. Continuing

SIAM Review, Volume 65, Issue 3, Page 887-902, August 2023. Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system

SIAM Review, Volume 65, Issue 3, Page 869-886, August 2023. Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because

SIAM Review, Volume 65, Issue 3, Page 867-867, August 2023. In this issue, the Education section presents two contributions. “The One-Dimensional Version of Peixoto's Structural Stability Theorem: A Calculus-Based Proof,” by Aminur Rahman and D. Blackmore, proposes, in the one-dimensional setting, a novel proof of Peixoto's structural stability and density theorem, which is fundamental in dynamical

SIAM Review, Volume 65, Issue 3, Page 831-865, August 2023. Inverse problems describe the task of blending a mathematical model with observational data---a fundamental task in many scientific and engineering disciplines. The solvability of such a task is usually classified through its well-posedness. A problem is well-posed if it has a unique solution that depends continuously on input or data. Inverse

SIAM Review, Volume 65, Issue 3, Page 829-829, August 2023. The SIGEST article in this issue, which comes from the SIAM/ASA Journal on Uncertainty Quantification, is “Bayesian Inverse Problems Are Usually Well-Posed,” by Jonas Latz. The author investigates the well-posedness of Bayesian approaches to inverse problems, generalizing the framework of well-posedness introduced by Andrew Stuart to a set

SIAM Review, Volume 65, Issue 3, Page 806-828, August 2023. In $d$ dimensions, accurately approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim$$k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$ and in $d$ dimensions) suffers from the pollution effect if, as $k→∞$, the total number of degrees of freedom needed to

SIAM Review, Volume 65, Issue 3, Page 774-805, August 2023. The beauty and simplicity of compartment modeling makes it a useful approach for simulating dynamics in an amazingly wide range of applications, which are growing rapidly especially in global carbon cycling, hydrological network flows, and epidemiology and population dynamics. These contexts, however, often involve compartment-to-compartment

SIAM Review, Volume 65, Issue 3, Page 735-773, August 2023. We analyze neural ordinary differential equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of deep learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of

SIAM Review, Volume 65, Issue 3, Page 733-733, August 2023. The three articles in this issue's Research Spotlight section highlight the breadth of problems and approaches that have differential equations as a central component. In the first article, “Neural ODE Control for Classification, Approximation, and Transport,” authors Domènec Ruiz-Balet and Enrique Zuazua seek to expand understanding of some

SIAM Review, Volume 65, Issue 3, Page 732-732, August 2023. This erratum corrects an error in the coefficients of equation (6.23) in the original paper [H. Miao, X. Xia, A. S. Perelson, and H. Wu, SIAM Rev., 53 (2011), pp. 3--39].

SIAM Review, Volume 65, Issue 3, Page 686-731, August 2023. Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships

SIAM Review, Volume 65, Issue 3, Page 601-685, August 2023. Over the last 30 years a plethora of variational regularization models for image reconstruction have been proposed and thoroughly inspected by the applied mathematics community. Among them, the pioneering prototype often taught and learned in basic courses in mathematical image processing is the celebrated Rudin--Osher--Fatemi (ROF) model

SIAM Review, Volume 65, Issue 3, Page 599-599, August 2023. Apart from a short erratum, which concerns the correction of some coefficients in a differential equation in the original paper, this issue contains two Survey and Review articles. “On and Beyond Total Variation Regularization in Imaging: The Role of Space Variance,” authored by Monica Pragliola, Luca Calatroni, Alessandro Lanza, and Fiorella

Matrix skeletonizations like the interpolative and CUR decompositions provide a framework for low-rank approximation in which subsets of a given matrix’s columns and/or rows are selected to form approximate spanning sets for its column and/or row space. Such decompositions that rely on “natural” bases have several advantages over traditional low-rank decompositions with orthonormal bases, including

A splitting scheme is proposed for a class of backward doubly stochastic differential equations (BDSDEs). The main idea is to decompose the backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation, which are much easier to solve than the BDSDE itself. The two equations are then approximated by first-order finite difference

This paper deals with the numerical analysis for a family of nonlinear degenerate parabolic problems. The model is spatially discretized using a finite element method; an implicit Euler scheme is employed for time discretization. We deduce sufficient conditions to ensure that the fully discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Finally,

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution

Recently, physics-informed neural networks (PINNs) have offered a powerful new paradigm for solving problems relating to differential equations. Compared to classical numerical methods, PINNs have several advantages, for example their ability to provide mesh-free solutions of differential equations and their ability to carry out forward and inverse modelling within the same optimisation problem. Whilst

In this paper, we analyze discontinuous Galerkin methods based in the interior penalty method in order to approximate the eigenvalues and eigenfunctions of the Stokes eigenvalue problem. The considered methods in this work are based in discontinuous polynomials approximations for the velocity field and the pressure fluctuation in two and three dimensions. The methods under consideration are symmetric

In this paper, we study an online learning algorithm with a robust loss function \(\mathcal {L}_{\sigma }\) for regression over a reproducing kernel Hilbert space (RKHS). The loss function \(\mathcal {L}_{\sigma }\) involving a scaling parameter \(\sigma >0\) can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression

We present a family of high-order trapezoidal rule-based quadratures for a class of singular integrals, where the integrand has a point singularity. The singular part of the integrand is expanded in a Taylor series involving terms of increasing smoothness. The quadratures are based on the trapezoidal rule, with the quadrature weights for Cartesian nodes close to the singularity judiciously corrected

In this paper, we study an adaptive finite element approximation of optimal control problems with integral fractional Laplacian and pointwise control constraints. The state variable is approximated by piecewise linear polynomials, and the control variable is implicitly discretized. Upper and lower bounds of a posteriori error estimates for finite element approximation of the optimal control problem

Given pointwise samples of an unknown function belonging to a certain model set, one seeks in optimal recovery to recover this function in a way that minimizes the worst-case error of the recovery procedure. While it is often known that such an optimal recovery procedure can be chosen to be linear, e.g., when the model set is based on approximability by a subspace of continuous functions, a construction

In this paper, the discontinuous finite volume element method (DFVEM) is considered to solve the Allen-Cahn equation which contains strong nonlinearity. The method is based on the DFVEM in space and the backward Euler method in time. The energy stability and unique solvability of the proposed fully discrete scheme are derived. The error estimates for the semi-discrete and fully discrete scheme are

Phase retrieval in dynamical sampling is a novel research direction, where an unknown signal has to be recovered from the phaseless measurements with respect to a dynamical frame, i.e., a sequence of sampling vectors constructed by the repeated action of an operator. The loss of the phase here turns the well-posed dynamical sampling into a severe ill-posed inverse problem. In the existing literature

We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak–strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for

In this paper, a method for quartic minimization over the sphere is studied. It is based on an equivalent difference of convex (DC) reformulation of this problem in the matrix variable. This derivation also induces a global optimality certification for the quartic minimization over the sphere. An algorithm with the subproblem being solved by a semismooth Newton method is then proposed for solving the

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev

We establish a deep learning theory for distribution regression with deep convolutional neural networks (DCNNs). Deep learning based on structured deep neural networks has been powerful in practical applications. Generalization analysis for regression with DCNNs has been carried out very recently. However, for the distribution regression problem in which the input variables are probability measures

A new exponentially fitted version of the discrete variational derivative method for the efficient solution of oscillatory complex Hamiltonian partial differential equations is proposed. When applied to the nonlinear Schrödinger equation, this scheme has discrete conservation laws of charge and energy. The new method is compared with other conservative schemes from the literature on a benchmark problem

We present an error bound for a least squares version of the kernel based meshless finite difference method for elliptic differential equations on smooth compact manifolds of arbitrary dimension without boundary. In particular, we obtain sufficient conditions for the convergence of this method. Numerical examples are provided for the equation \(-\Delta _{\mathcal {M}} u + u = f\) on the 2- and 3-spheres

In this paper, we consider the question of finding an as small as possible family of operators \((T_j)_{j\in J}\) on \(L^2({\mathbb {R}})\) that does phase retrieval: every \(\varphi \) is uniquely determined (up to a constant phase factor) by the phaseless data \((|T_j\varphi |)_{j\in J}\). This problem arises in various fields of applied sciences where usually the operators obey further restrictions

We present a novel computational framework for the simulation of rigid spherical Janus particle suspensions in Stokes flow. For a wide array of Janus particle types, we show long-range interactions may be resolved using fast, spectrally accurate boundary integral methods. We incorporate this to our rigid body Stokes platform, which resolves hydrodynamic interactions and contact. Our approach features

Let A be an \(m\times n\) matrix with nonnegative real entries. The psd rank of A is the smallest k for which there exist two families \((P_1,\ldots ,P_m)\) and \((Q_1,\ldots ,Q_n)\) of positive semidefinite Hermitian \(k\times k\) matrices such that \(A(i|j)={\text {tr}}(P_i Q_j)\) for all i, j. Several questions on the algorithmic complexity of related matrix invariants were posed in recent literature:

In this paper, we investigate and analyze numerical solutions for the Volterra integrodifferential equations with tempered multi-term kernels. Firstly, we derive some regularity estimates of the exact solution. Then a temporal-discrete scheme is established by employing Crank-Nicolson technique and product integration (PI) rule for discretizations of the time derivative and tempered-type fractional

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees

Tensor completion aims to reconstruct a high-dimensional data from the partial element missing tensors under a low-rank constraint, which may be seen as a least-squares problem on manifold. In this paper, we propose a new Riemannian conjugate gradient method for the tensor completion which performs Riemannian optimization techniques on a fixed transformed multi-rank tensor manifold. More specifically

In this paper we present a construction of interpolatory Hermite multiwavelets for functions that take values in nonlinear geometries such as Riemannian manifolds or Lie groups. We rely on the strong connection between wavelets and subdivision schemes to define a prediction-correction approach based on Hermite subdivision schemes that operate on manifold-valued data. The main result concerns the decay

In this paper, linear multi-step methods are used to numerically solve gradient flow models, and the relations between different numerical stabilities (e.g., unconditional energy stability, A-stability and G-stability) of linear multi-step methods are discussed. First, we introduce the definition of the absolutely unconditional energy stability (AUES), in which the meaning of absoluteness is borrowed

A class of implicit Milstein type methods is introduced and analyzed in the present article for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters \(\theta , \eta \in [0, 1]\) into both the drift and diffusion parts, the new schemes are indeed a kind of drift-diffusion double implicit methods. Within a

In this paper, we give pointwise estimates of a Voronoï-based finite volume approximation of the Laplace-Beltrami operator on Voronoï-Delaunay decompositions of the sphere. These estimates are the basis for local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient. Here, we consider the Voronoï-based finite volume method as a perturbation of the

In Bayesian analysis, the so-called conjugate models allow obtaining the posterior distribution in exact form, in the sense that the posterior quantities can explicitly be written in a computable form. However, this class of models only involves a few structures, with some specific prior distribution for every data distribution. Although approximate methods such as numerical integration and MCMC are