There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to
talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard Brent's zeroin method. The possibly new method is based on a cubic blend of tangent-line approximations for the inverse function. We rewrite this iteration for numerical stability as an average of two Newton steps and a secant step: only one new function evaluation and derivative evaluation is needed for each step. The total cost of the method is therefore only trivially more than Newton's method, and we will see that it has order $1+\sqrt{3} = 2.732...$, thus ensuring that to achieve a given accuracy it usually takes fewer steps than Newton's method while using essentially the same effort per step.
We present a perturbed subspace iteration algorithm to approximate the lowermost eigenvalue cluster of an elliptic eigenvalue problem. As a prototype, we consider the Laplace eigenvalue problem posed in a polygonal domain. The algorithm is motivated by the analysis of inexact (perturbed) inverse iteration algorithms in numerical linear algebra. We couple the perturbed inverse iteration approach with mesh refinement strategy based on residual estimators. We demonstrate our approach on model problems in two and three dimensions.
Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale. The associated inverse scale space flow can be obtained using the classical Bregman iteration applied to a convex, absolutely one-homogeneous regularizer. In order to extend these approaches to general energies with non-convex data term, we apply the Bregman iteration to a lifted version of the functional with sublabel-accurate discretization. We provide a condition for the subgradients of the regularizer under which this lifted iteration reduces to the standard Bregman ite…
In this paper, we propose a simple sparse approximate inverse for triangular matrices (SAIT). Using the Jacobi iteration method, we obtain an expression of the exact inverse of triangular matrix, which is a finite series. The SAIT is constructed based on this series. We apply the SAIT matrices to iterative methods with ILU preconditioners. The two triangular solvers in the ILU preconditioning procedure are replaced by two matrix-vector multiplications, which can be fine-grained parallelized. We test this method by solving some linear systems and eigenvalue problems with preconditioned iterativ…
Non-linear filtering approaches allow to obtain decompositions of images with respect to a non-classical notion of scale, induced by the choice of a convex, absolutely one-homogeneous regularizer. The associated inverse scale space flow can be obtained using the classical Bregman iteration with quadratic data term. We apply the Bregman iteration to lifted, i.e. higher-dimensional and convex, functionals in order to extend the scope of these approaches to functionals with arbitrary data term. We provide conditions for the subgradients of the regularizer -- in the continuous and discrete setting…
We give a self-contained randomized algorithm based on shifted inverse iteration which provably computes the eigenvalues of an arbitrary matrix $M\in\mathbb{C}^{n\times n}$ up to backward error $δ\|M\|$ in $O(n^4+n^3\log^2(n/δ)+\log(n/δ)^2\log\log(n/δ))$ floating point operations using $O(\log^2(n/δ))$ bits of precision. While the $O(n^4)$ complexity is prohibitive for large matrices, the algorithm is simple and may be useful for provably computing the eigenvalues of small matrices using controlled precision, in particular for computing Ritz values in shifted QR algorithms as in (Banks, Garza-…
We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical results are complemented by several numerical experiments for solving linear inverse problems for the Radon transform and a nonlinear inverse problem for Schlieren tomography.
In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not Gâteaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fréchet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt…
A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints
Purdue University（普渡大学）；University of Vienna（维也纳大学）
The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well-understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landw…
We investigate the iterative methods proposed by Maz'ya and Kozlov (see [3], [4]) for solving ill-posed reconstruction problems modeled by PDE's. We consider linear time dependent problems of elliptic, hyperbolic and parabolic types. Each iteration of the analyzed methods consists on the solution of a well posed boundary (or initial) value problem. The iterations are described as powers of affine operators, as in [4]. We give alternative convergence proofs for the algorithms, using spectral theory and some functional analytical results (see [5], [6]).
Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions
Universidade Federal de Minas Gerais（米纳斯吉拉斯联邦大学）；ETH Zürich（苏黎世联邦理工学院）；Argonne National Laboratory（阿贡国家实验室）；Universidade Federal de Ouro Preto（欧罗普雷图联邦大学）
Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems
School of Electrical Engineering, Tel Aviv University（特拉维夫大学电子工程学院）；Electrical Engineering Department, Technion - IIT（技术学院-理工学院电子工程系）；Computer Science Department, Technion - IIT（技术学院-理工学院计算机科学系）；Electrical and Computer Engineering Department, Duke University（杜克大学电气与计算机工程系）
