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This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem’s coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.

We present global convergence rates for a line-search method which is based on random first-order models and directions whose quality is ensured only with certain probability. We show that in terms of the order of the accuracy, the evaluation complexity of such a method is the same as its counterparts that use deterministic accurate models; the use of probabilistic models only increases the complexity by a constant, which depends on the probability of the models being good. We particularize and improve these results in the convex and strongly convex case. We also analyze a probabilistic cubic regularization variant that allows approximate probabilistic second-order models and show improved complexity bounds compared to probabilistic first-order methods; again, as a function of the accuracy, the probabilistic cubic regularization bounds are of the same (optimal) order as for the deterministic case.

In this paper, we study the problem of identifying source term and initial value simultaneously for fractional pseudo-parabolic equation with involution. This is an ill-posed problem and the conditional stability is given. A hybrid regularization method, which combines the modified quasi-inverse and the modified quasi-initial value, is used to deal with the inverse problem. The error estimates of the exact solutions and the regularization solutions are given respectively under the priori and the posteriori regularization parameter choice rules. Several numerical examples are presented to certify the hybrid regularization method is effective.

This paper considers the inverse problem for identifying the initial value problem of a space-time fractional diffusion wave equation. In general, this problem is ill-posed and the Landweber iterative regularization method is used to solve this problem. The error estimates between the exact solution and the regularized solution are given under the a priori parameter choice rule and the a posteriori parameter choice rule, respectively. In order to verify the validity and stability of the used method, numerical examples of two different dimensional cases with experimental data are performed.

We investigate a backward problem in time of diffusion equation with Laplace and Riesz-Feller space fractional operators. Firstly, inspired by the classical Landweber iterative method, a modified iterative regularization method is proposed to restore the continuous dependence of solution on the measurement data, and then under the a-priori and a-posteriori selection rules of regularization parameter, we give and prove the convergence estimations of regularization solution. Finally, we verify the regularization effect of modified iterative method by doing some numerical experiments. Numerical results show that this method is stable and feasible when it is used to deal with the inverse problem in this article.

In this paper, we consider the regularization method for exact as well as for inexact proximal point algorithms for finding the singularities of maximal monotone set-valued vector fields. We prove that the sequences generated by these algorithms converge to an element of the set of singularities of a maximal monotone set-valued vector field. A numerical example is provided to illustrate the inexact proximal point algorithm with regularization. Applications of our results to minimization problems and saddle point problems are given in the setting of Hadamard manifolds.

Stability estimates of Hölder type for the problem of evaluating the Caputo fractional derivative are obtained. This ill-posed problem is regularized by a Tikhonov-type method, which guarantees error estimates of Hölder type. Numerical results are presented to confirm the theory.

We estimate the worst-case complexity of minimizing an unconstrained, nonconvex composite objective with a structured nonsmooth term by means of some first-order methods. We find that it is unaffected by the nonsmoothness of the objective in that a first-order trust-region or quadratic regularization method applied to it takes at most $\\mathcal{O}(\\epsilon^{-2})$ function evaluations to reduce the size of a first-order criticality measure below $\\epsilon$. Specializing this result to the case when the composite objective is an exact penalty function allows us to consider the objective- and constraint-evaluation worst-case complexity of nonconvex equality-constrained optimization when the solution is computed using a first-order exact penalty method. We obtain that in the reasonable case when the penalty parameters are bounded, the complexity of reaching within $\\epsilon$ of a KKT point is at most $\\mathcal{O}(\\epsilon^{-2})$ problem evaluations, which is the same in order as the function-evaluation complexity of steepest-descent methods applied to unconstrained, nonconvex smooth optimization. [PUBLICATION ABSTRACT]

We prove lower bounds on the complexity of finding ϵ-stationary points (points x such that ‖∇f(x)‖≤ϵ) of smooth, high-dimensional, and potentially non-convex functions f. We consider oracle-based complexity measures, where an algorithm is given access to the value and all derivatives of f at a query point x. We show that for any (potentially randomized) algorithm A, there exists a function f with Lipschitz pth order derivatives such that A requires at least ϵ-(p+1)/p queries to find an ϵ-stationary point. Our lower bounds are sharp to within constants, and they show that gradient descent, cubic-regularized Newton’s method, and generalized pth order regularization are worst-case optimal within their natural function classes.

We consider the problem of computed tomography (CT). This ill-posed inverse problem arises when one wishes to investigate the internal structure of an object with a non-invasive and non-destructive technique. This problem is severely ill-conditioned, meaning it has infinite solutions and is extremely sensitive to perturbations in the collected data. This sensitivity produces the well-known semi-convergence phenomenon if iterative methods are used to solve it. In this work, we propose a multigrid approach to mitigate this instability and produce fast, accurate, and stable algorithms starting from unstable ones. We consider, in particular, symmetric Krylov methods, like lsqr, as smoother, and a symmetric projection of the coarse grid operator. However, our approach can be extended to any iterative method. Several numerical examples show the performance of our proposal.