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. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}<1/\sqrt{K+1}$, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

We develop a forward-reverse expectation-maximization (FREM) algorithm for estimating parameters of a discrete-time Markov chain evolving through a certain measurable state-space. For the construction of the FREM method, we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. We prove almost sure convergence of our algorithm for a Markov chain model with curved exponential family structure. On the numerical side, we carry out a complexity analysis of the forward-reverse algorithm by deriving its expected cost. Two application examples are discussed.

The hybrid variational model for restoration of texture images corrupted by blur and Gaussian noise we consider combines total variation regularisation and a fractional-order regularisation, and is solved by an alternating minimisation direction algorithm. Numerical experiments demonstrate the advantage of this model over the adaptive fractional-order variational model in image quality and computational time.

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

Image segmentation is a fundamental problem in both image processing and computer vision with numerous applications. In this paper, we propose a two-stage image segmentation scheme based on inexact alternating direction method. Specifically, we first solve the convex variant of the Mumford-Shah model to get the smooth solution, the segmentation are then obtained by apply the K-means clustering method to the solution. Some numerical comparisons are arranged to show the effectiveness of our proposed schemes by segmenting many kinds of images such as artificial images, natural images, and brain MRI images.

The Douglas-Rachford and Peaceman-Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments.

An inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

In this paper we develop a Kantorovich-like theory for Chebyshev’s method, a well-known iterative method for solving nonlinear equations in Banach spaces. We improve the results obtained previously by considering Chebyshev’s method as an element of a family of iterative processes.

We consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

Morozov’s discrepancy principle is one of the simplest and most widely used parameter choice strategies in the context of regularization of ill-posed operator equations. Although many authors have considered this principle under general source conditions for linear ill-posed problems, such study for nonlinear problems is restricted to only a few papers. The aim of this paper is to apply Morozov’s discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems under general source conditions.

Let be a unital Banach algebra. Assume that a has a generalized inverse a+. Then is said to be a stable perturbation of a if . In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation is closely related to the gap function . These results will be applied to error estimates for perturbations of the Moore-Penrose inverse in C*–algebras and the Drazin inverse in Banach algebras.

This paper is concerned with a characterization of the optimal order of convergence of Tikhonov regularization for first kind operator equations in terms of the “smoothness” of the data.