Multi-parameter Tikhonov Regularisation in Topological Spaces

We study the behaviour of Tikhonov regularisation on topological spaces with multiple regularisation terms. The main result of the paper shows that multi-parameter regularisation is well-posed in the sense that the results depend continuously on the data and converge to a true solution of the equation to be solved as the noise level decreases to zero. Moreover, we derive convergence rates in terms of a generalised Bregman distance using the method of variational inequalities. All the results in the paper, including the convergence rates, consider not only noise in the data, but also errors in the operator

Simulated identification on dynamic characteristics of large heavy-load bearing

It’s difficult to test repeatedly for large heavy-load bearings (LHLBs) with full-scale and real load due to complexity and costliness, so simulated identification on dynamic characteristics of 1750 MW nuclear generator bearing with diameter 800 mm and specific pressure 3.3 MPa is provided in this paper. The identification model of bearing dynamic characteristic is established, the calculating method of positive and negative dynamic problems is provided, and effects of signal disturbances on identification precision are analyzed. The results show that the LHLBs’ permitted displacement disturbance should not be over 5 μm and the permitted ratio of dynamic load and static load is about 1 %-2 %, which is different from common knowledge of 15 %-20 % for small light-load bearings. If identification error of the main stiffness and main damping coefficients is less than 5 %, the amplitude of periodical disturbance of the dynamic load and displacement signals should be less than 5 %. If identification error of the main damping coefficients is less than 10 %, the phase of these two signals should be less than 1°. The roundness error and rotation error of the large shaft should be eliminated

Incorporation of spatial- and connectivity-based cortical brain region information in regularized regression: Application to Human Connectome Project data

Studying the association of the brain’s structure and function with neurocognitive outcomes requires a comprehensive analysis that combines different sources of information from a number of brain-imaging modalities. Recently developed regularization methods provide a novel approach using information about brain structure to improve the estimation of coefficients in the linear regression models. Our proposed method, which is a special case of the Tikhonov regularization, incorporates structural connectivity derived with Diffusion Weighted Imaging and cortical distance information in the penalty term. Corresponding to previously developed methods that inform the estimation of the regression coefficients, we incorporate additional information via a Laplacian matrix based on the proximity measure on the cortical surface. Our contribution consists of constructing a principled formulation of the penalty term and testing the performance of the proposed approach via extensive simulation studies and a brain-imaging application. The penalty term is constructed as a weighted combination of structural connectivity and proximity between cortical areas. Simulation studies mimic the real brain-imaging settings. We apply our approach to the study of data collected in the Human Connectome Project, where the cortical properties of the left hemisphere are found to be associated with vocabulary comprehensio

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results

Constructing the Physical Parameters of a Damped Vibrating System From Eigendata

In this paper we consider the inverse problem for a discrete damped mass-spring system where the mass, damping, and sti®ness matrices are all symmetric tridiagonal. We ¯rst show that the model can be constructed from two real eigenvalues and three real eigenvectors or two complex conjugate eigenpairs and a real eigenvector. Then, we study the general under-determined and over-determined problems. In particular, we provide the su±cient and necessary conditions on the given two real or complex conjugate eigenpairs so that the under-determined problem has a physical solution. However, for large model order, the construction from these data may be sensitive to perturbations. To reduce the sensitivity, we propose the the minimum norm solution over the under-determined noisy data and the least squares solution to the over-determined measured data. We also discuss the physical realizability of the required model by the positivity-constrained regularization method for the ill-posed under-determined problem and the least squares optimization problems with positivity-constraints for the ill-posed over-determined problem. Finally, we give simple numerical examples to illustrate the e®ectiveness of our methods

Multi-parameter regularization techniques for ill-conditioned linear systems

none4When a system of linear equations is ill-conditioned, regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties: the ill-conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen. In this paper, we consider the case of several regularizations terms added simultaneously, thus overcoming the problem of the best choice of the regularization matrix. The error of this procedure is analyzed and numerical results prove its efficiency.mixedBREZINSKI C; REDIVO ZAGLIA M.; RODRIGUEZ G; SEATZU SBrezinski, C; REDIVO ZAGLIA, Michela; Rodriguez, G; Seatzu, S