Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The $\ell_p$ norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of $p$ and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding $\ell_p$ regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional Generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing $\ell_1$ norm and edge-preserving total variation regularization.
Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding for large scale problems. We propose a new and efficient algorithm which simultaneously solves the Tikhonov problem and finds the corresponding regularization parameter such that the discrepancy…
The numerical solution of the algebraic Riccati equation is a challenging task especially for very large problem dimensions. In this paper we present a new algorithm that combines the very appealing computational features of projection methods with the convergence properties of the inexact Newton-Kleinman procedure equipped with a line search. In particular, the Newton scheme is completely merged in a projection framework with a single approximation space so that the Newton-Kleinman iteration is only implicitly performed. Moreover, the line search turns out to be exact in our setting, i.e., th…
We present PNKH-B, a projected Newton-Krylov method for iteratively solving large-scale optimization problems with bound constraints. PNKH-B is geared toward situations in which function and gradient evaluations are expensive, and the (approximate) Hessian is only available through matrix-vector products. This is commonly the case in large-scale parameter estimation, machine learning, and image processing. In each iteration, PNKH-B uses a low-rank approximation of the (approximate) Hessian to determine the search direction and construct the metric used in a projected line search. The key featu…
The non-convex $α\|\cdot\|_{\ell_1}-β\| \cdot\|_{\ell_2}$ $(α\geβ\geq0)$ regularization has attracted attention in the field of sparse recovery. One way to obtain a minimizer of this regularization is the ST-($α\ell_1-β\ell_2$) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to the classical $\ell_1$ sparsity regularization. In this paper, we present two accelerated alternatives to the ST-(…
We develop a computationally efficient algorithm for the automatic regularization of nonlinear inverse problems based on the discrepancy principle. We formulate the problem as an equality constrained optimization problem, where the constraint is given by a least squares data fidelity term and expresses the discrepancy principle. The objective function is a convex regularization function that incorporates some prior knowledge, such as the total variation regularization function. Using the Jacobian matrix of the nonlinear forward model, we consider a sequence of quadratically constrained optimiz…
In this paper, we consider the Newton-Schur method in Hilbert space and obtain quadratic convergence. For the symmetric elliptic eigenvalue problem discretized by the standard finite element method and non-overlapping domain decomposition method, we use the Steklov-Poincaré operator to reduce the eigenvalue problem on the domain $Ω$ into the nonlinear eigenvalue subproblem on $Γ$, which is the union of subdomain boundaries. We prove that the convergence rate for the Newton-Schur method is $ε_{N}\leq CH^{2}(1+\ln(H/h))^{2}ε^{2}$, where the constant $C$ is independent of the fine mesh size $h$ a…
In this contribution we device and analyze improved variants of the non-conforming dual approach for trust-region reduced basis (TR-RB) approximation of PDE-constrained parameter optimization that has recently been introduced in [Keil et al.. A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization. arXiv:2006.09297, 2020]. The proposed methods use model order reduction techniques for parametrized PDEs to significantly reduce the computational demand of parameter optimization with PDE constraints in the context of large-scale or multi…
Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions
University of Bucharest（布加勒斯特大学）；“Gheorghe Mihoc – Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院格奥尔基·米霍克-凯乌斯·雅各布数学统计与应用数学研究所）
We consider the quasilinear 1D inverse heat convection problem (IHCP) of determining the enthalpy-dependent heat fluxes from noisy internal enthalpy measurements. This problem arises in the Accelerated Cooling (ACC) process of producing thermomechanically controlled processed (TMCP) heavy plates made of steel. In order to adjust the complex microstructure of the underlying material, the Leidenfrost behavior of the hot surfaces with respect to the application of the cooling fluid has to be studied. Since the heat fluxes depend on the enthalpy and hence on the solution of the underlying initial …
Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known …
A modified projected walk on spheres method for elliptic equations on high-dimensional embedded manifolds: algorithm and error estimates
School of Mathematics, Shanghai University of Finance and Economics（上海财经大学数学学院）；School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
Regularity Analysis and Tensor Neural Network Methods for Quasiperiodic Elliptic Equations
Jilin University（吉林大学）；Peking University（北京大学）；Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学研究所）；University of Chinese Academy of Sciences（中国科学院大学）
The paper studies the well-posedness and optimal error estimates of spectral finite element approximations for the boundary value problems of semi-linear elliptic SPDEs driven by white or colored Gaussian noises. The noise term is approximated through the spectral projection of the covariance operator, which is not required to be commutative with the Laplacian operator. Through the convergence analysis of SPDEs with the noise terms replaced by the projected noises, the well-posedness of the SPDE is established under certain covariance operator-dependent conditions. These SPDEs with projected n…
Phase recovery from the bispectrum is a central problem in speckle interferometry which can be posed as an optimization problem minimizing a weighted nonlinear least-squares objective function. We look at two different formulations of the phase recovery problem from the literature, both of which can be minimized with respect to either the recovered phase or the recovered image. Previously, strategies for solving these formulations have been limited to first-order optimization methods such as gradient descent or quasi-Newton methods. This paper explores Gauss-Newton optimization schemes for the…
We introduce a domain decomposition-based nonlinear preconditioned iteration for solving nonlinear, nonsmooth elliptic optimal control problems, with a nonlinear reaction term, $L^1$ regularization and box constraints on the control function. The method is obtained by applying semismooth Newton to the fixed-point equation of the parallel optimized Schwarz iteration. As a proof of concept, numerical experiments are performed on two subdomains, as well as on a multi-subdomain test case. The results show that it is possible to obtain substantial improvements in robustness and efficiency with the …
Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {\mathcal L}^{-1} {\mathcal F}(\hat{u}) + {\mathcal L}^{-1} {\mathcal G}(w)$, where ${\mathcal L}$ is a linearized operator, ${\mathcal F}(\hat{u})$ is a residual, and ${\mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $\| {\mathcal L}^{-1} {\mathcal F}(\hat{u}) \|$ and $\| {\mathcal L…
