Hilbert Space Setting Research Articles

Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods exhibit strong convergence. There are presented convergence results, even for a combination of noisy right-hand side and imperfect forward operators, and for one of the approaches there are also convergence rates results. Specifically adapted discrepancy principles are used as a posteriori stopping rules of the established iterative regularization algorithms. For an application of the suggested new approaches, we consider a biosensor problem, which is modelled as a two dimensional linear Fredholm integral equation of the first kind. Several numerical examples, as well as a comparison with the projected Landweber method, are presented to show the accuracy and the acceleration effect of the novel methods. Case studies of a real data problem indicate that the developed methods can produce meaningful featured regularized solutions.

State dependent nonconvex sweeping processes in smooth Banach spaces

In the setting of 2-uniformly smooth and q -uniformly convex Banach spaces, we prove the existence of solutions of the following multivalued differential equation: -\frac{d}{dt} J(u(t)) \in N^C(C(t,u(t));u(t)) \text{ a.e. in } [0,T]. \:\:\: \mathrm{(SDNSP)} This inclusion is called State Dependent Nonconvex Sweeping Process (SDNSP). Here N^C(C(t,u(t)); u(t)) stands for the Clarke normal cone. The perturbed (SDNSPP) is also considered. Our results extend recent existing results from the setting of Hilbert spaces to the setting of Banach spaces. In our proofs we use some new results on V -uniformly generalized prox-regular sets in Banach spaces.

On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left- and right-hand sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.

The Split Various Variational Inequalities Problems for Three Hilbert Spaces

There are many methods for finding a common solution of a system of variational inequalities, a split equilibrium problem, and a hierarchical fixed-point problem in the setting of real Hilbert spaces. They proved the strong convergence theorem. Many split feasibility problems are generated in real Hillbert spaces. The open problem is proving a strong convergence theorem of three Hilbert spaces with different methods from the lasted method. In this research, a new split variational inequality in three Hilbert spaces is proposed. Important tools which are used to solve classical problems will be developed. The convergence theorem for finding a common element of the set of solution of such problems and the sets of fixed-points of discontinuous mappings has been proved.

Determination of convex functions via subgradients of minimal norm

We show, in Hilbert space setting, that any two convex proper lower semicontinuous functions bounded from below, for which the norm of their minimal subgradients coincide, they coincide up to a constant. Moreover, under classic boundary conditions, we provide the same results when the functions are continuous and defined over an open convex domain. These results show that for convex functions bounded from below, the slopes provide sufficient first-order information to determine the function up to a constant, giving a positive answer to the conjecture posed in Boulmezaoud et al. (SIAM J Optim 28(3):2049–2066, 2018) .

Regularization of ill-posed operator equations: an overview

We discuss and illustrate the stability issues associated with ill-posed linear operator equations and the need to have regularization methods for obtaining stable approximate solutions. As illustration, we describe two simple regularization methods, namely, the Lavrentiev regularization in the setting of a Banach spaces and Tikhonov regularization in the setting of Hilbert spaces, and discuss on the corresponding error estimates. We also indicate procedures to obtain error estimates under milder general source conditions and also to obtain better error estimates under modified methods. The discussed procedures include some of the work of the author as well.

Viscosity methods for inclusion and fixed-point problems

We are concerned with a variational inclusion problem with two monotone mappings, and fixed-point problems of asymptotically nonexpansive mappings and pseudocontractive mappings. A viscosity method is introduced and studied for the two problems. A convergence theorem in norm of common solutions is established in the setting of infinite dimensional real Hilbert spaces.

On the Derivation of Quasi-Newton Formulas for Optimization in Function Spaces

Newton’s method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton’s method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. It is known that quasi-Newton update formulas are solutions to certain variational problems over the space of symmetric matrices. In this paper, we formulate similar variational problems over the space of bounded symmetric operators in Hilbert spaces. By changing the constraints of the variational problem we obtain updates (for the Hessian and Hessian inverse) not only for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method but also for Davidon–Fletcher–Powell (DFP), Symmetric Rank One (SR1), and Powell-Symmetric-Broyden (PSB). In addition, for an inverse problem governed by a partial differential equation (PDE), we derive DFP and BFGS “structured” secant formulas that explicitly use the derivative of the regularization and only approximates the second derivative of the misfit term. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods.

Construction of a Preconditioner for General Elliptic Problems Using Riesz Map

The current work aspires to design and study the construction of an efficient preconditioner for linear symmetric systems in a Hilbert space setting. Compliantly to Josef Málek and Zdeněk Strakoš’s work [Preconditioning and the Conjugate Gradient Method in the Context of Solving[Formula: see text] PDEs, Vol. 1 (SIAM, USA).], we shed new light on the dependence of algebraic preconditioners with the resolution steps of partial differential equations (PDEs) and describe their impact on the final numerical solution. The numerical strength and efficiency of the proposed approach is demonstrated on a two-dimensional examples.

Von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.

A note on optimal H^1-error estimates for Crank-Nicolson approximations to the nonlinear Schr\xf6dinger equation

In this paper we consider a mass- and energy–conserving Crank-Nicolson time discretization for a general class of nonlinear Schrödinger equations. This scheme, which enjoys popularity in the physics community due to its conservation properties, was already subject to several analytical and numerical studies. However, a proof of optimal L^{infty }(H^1)-error estimates is still open, both in the semi-discrete Hilbert space setting, as well as in fully-discrete finite element settings. This paper aims at closing this gap in the literature. We also suggest a fixed point iteration to solve the arising nonlinear system of equations that makes the method easy to implement and efficient. This is illustrated by numerical experiments.

Fixing and extending some recent results on the ADMM algorithm

We investigate the techniques and ideas used in Shefi and Teboulle (SIAM J Optim 24(1), 269–297, 2014) in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration, we allow the use of variable metrics, while the investigations are carried out in the setting of infinite-dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties.

Convergence Rate of Inertial Proximal Algorithms with General Extrapolation and Proximal Coefficients

In a Hilbert space setting ${\mathcal{H}}$, in order to minimize by fast methods a general convex lower semicontinuous and proper function ${\Phi }: {\mathcal{H}} \rightarrow \mathbb {R} \cup \{+\infty \}$, we analyze the convergence rate of the inertial proximal algorithms. These algorithms involve both extrapolation coefficients (including Nesterov acceleration method) and proximal coefficients in a general form. They can be interpreted as the discrete time version of inertial continuous gradient systems with general damping and time scale coefficients. Based on the proper setting of these parameters, we show the fast convergence of values and the convergence of iterates. In doing so, we provide an overview of this class of algorithms. Our study complements the previous Attouch–Cabot paper (SIOPT, 2018) by introducing into the algorithm time scaling aspects, and sheds new light on the Guler seminal papers on the convergence rate of the accelerated proximal methods for convex optimization.

Stochastic Fubini Theorem for Jump Noises in Banach Spaces

We prove a general version of the stochastic Fubini theorem for stochastic integrals of Banach space valued processes with respect to compensated Poisson random measures under weak integrability assumptions, which extends this classical result from Hilbert space setting to Banach space setting.

Property (R) Under Compact Perturbations

This paper discusses the stability of a kind of spectral property called property (R) under compact perturbations in the setting of Hilbert space. Necessary and sufficient conditions are given for such a spectral property to be invariant under compact perturbations.

Stochastic variational formulation for a general random time-dependent economic equilibrium problem

In the paper, in a Hilbert space setting, a random time-dependent oligopolistic market equilibrium problem in presence of both production and demand excesses is studied and the random time-dependent Cournot–Nash equilibrium principle by means of a stochastic variational inequality is characterized. Then, some existence results to such problem are established and the stochastic continuity of the equilibrium solution is proved. Moreover a simple numerical example illustrates the theoretical results.

Convergence of proximal algorithms with stepsize controls for non-linear inverse problems and application to sparse non-negative matrix factorization

We consider a general ill-posed inverse problem in a Hilbert space setting by minimizing a misfit functional coupling with a multi-penalty regularization for stabilization. For solving this minimization problem, we investigate two proximal algorithms with stepsize controls: a proximal fixed point algorithm and an alternating proximal algorithm. We prove the decrease of the objective functional and the convergence of both update schemes to a stationary point under mild conditions on the stepsizes. These algorithms are then applied to the sparse and non-negative matrix factorization problems. Based on a priori information of non-negativity and sparsity of the exact solution, the problem is regularized by corresponding terms. In both cases, the implementation of our proposed algorithms is straight-forward since the evaluation of the proximal operators in these problems can be done explicitly. Finally, we test the proposed algorithms for the non-negative sparse matrix factorization problem with both simulated and real-world data and discuss reconstruction performance, convergence, as well as achieved sparsity.

Multi-step inertial proximal contraction algorithms for monotone variational inclusion problems

"In this article, we introduce the multi-step inertial proximal contraction algorithms (MiPCA) to approximate a zero of the sum of two monotone operators, with one of the two operators being monotone and Lipschitz continuous. The weak convergence of the MiPCA is shown under the summability condition formulated in terms of the iterative sequence in a Hilbert space setting. We also investigate the unconditional convergence of the one-step inertial proximal contraction algorithm. Finally, numerical experiments are given to illustrate the advantage of the multi-step inertial proximal contraction algorithms."

The split feasibility problem with multiple output sets in Hilbert spaces

We study the split feasibility problem with multiple output sets in Hilbert spaces. In order to solve this problem, we propose two new algorithms. We establish a weak convergence theorem for the first one and a strong convergence theorem for the second.

Common best proximity pairs via the concept of complete proximal normal structure

We introduce a concept of complete proximal normal structure and used to investigate the existence of a best proximity point for an arbitrary family of cyclic relatively nonexpansive mappings in the setting of strictly convex Banach spaces. We also prove that every bounded, closed and convex pair in uniformly convex Banach spaces as well as every compact and convex pairs in Banach spaces has complete proximal normal structure. Furthermore, we consider a class of cyclic relatively nonexpansive mappings in the sense of Suzuki and establish a new best proximity point theorem in the setting of Hilbert spaces.