In this work we consider the problem of finding optimal regularization parameters for general-form Tikhonov regularization using training data. We formulate the general-form Tikhonov solution as a spectral filtered solution using the generalized singular value decomposition of the matrix of the forward model and a given regularization matrix. Then, we find the optimal regularization parameter by minimizing the average of the errors between the filtered solutions and the true data. We extend the approach to the multi-parameter Tikhonov problem for the case where all the matrices involved are simultaneously diagonalizable. For problems where this is not the case, we describe an approach to compute optimal or near-optimal regularization parameters by using operator approximations for the original problem. Several tests are performed for 1D and 2D examples using different norms on the errors, showing the effectiveness of this approach.

OBJECTIVE The current diagnostic criterion for Chiari malformation Type I (CM-I), based on tonsillar herniation (TH), includes a diversity of patients with amygdalar descent that may be caused by a variety of factors. In contrast, patients presenting with an overcrowded posterior cranial fossa, a key characteristic of the disease, may remain misdiagnosed if they have little or no TH. The objective of the present study was to use machine-learning classification methods to identify morphometric measures that help discern patients with classic CM-I to improve diagnosis and treatment and provide insight into the etiology of the disease. METHODS Fifteen morphometric measurements of the posterior cranial fossa were performed on midsagittal T1-weighted MR images obtained in 195 adult patients diagnosed with CM. Seven different machine-learning classification methods were applied to images from 117 patients with classic CM-I and 50 controls matched by age and sex to identify the best classifiers discriminating the 2 cohorts with the minimum number of parameters. These classifiers were then tested using independent CM cohorts representing different entities of the disease. RESULTS Machine learning identified combinations of 2 and 3 morphometric measurements that were able to discern not only classic CM-I (with more than 5 mm TH) but also other entities such as classic CM-I with moderate TH and CM Type 1.5 (CM-1.5), with high accuracy (> 87%) and independent of the TH criterion. In contrast, lower accuracy was obtained in patients with CM Type 0. The distances from the lower aspect of the corpus callosum, pons, and fastigium to the foramen magnum and the basal and Wackenheim angles were identified as the most relevant morphometric traits to differentiate these patients. The stronger significance (p < 0.01) of the correlations with the clivus length, compared with the supraoccipital length, suggests that these 5 relevant traits would be affected more by the relative position of the basion than the opisthion. CONCLUSIONS Tonsillar herniation as a unique criterion is insufficient for radiographic diagnosis of CM-I, which can be improved by considering the basion position. The position of the basion was altered in different entities of CM, including classic CM-I, classic CM-I with moderate TH, and CM-1.5. The authors propose a predictive model based on 3 parameters, all related to the basion location, to discern classic CM-I with 90% accuracy and suggest considering the anterior alterations in the evaluation of surgical procedures and outcomes.

Abstract. We present a multilevel method for discrete ill-posed problems arising from the discretization of Fredholm integral equations of the first kind. In this method, we use the Haar wavelet transform to define restriction and prolongation operators within a multigrid-type iteration. The choice of the Haar wavelet operator has the advantage of preserving matrix structure, such as Toeplitz, between grids, which can be exploited to obtain faster solvers on each level where an edgepreserving Tikhonov regularization is applied. Finally, we present results that indicate the promise of this approach for restoration of signals and images with edges.

We present a Γ-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defectfree crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum-e.g., finite-elementapproximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with Γ-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a practical means of assessing the convergence of quasicontinuum approximations. We demonstrate the utility of the discrete patch test by means of selected examples of application. Introduction. The quasicontinuum method of Tadmor, Phillips, and Ortiz[59, 60] was originally conceived as an approximation scheme for zero-temperature molecular statics consisting of: (i) adaptive interpolation constraints on the motion of the atoms aimed at eliminating degrees of freedom in regions where the displacement field is nearly affine, and (ii) summation or quadrature rules for purposes of avoiding full lattice sums. The initial development of the method was application-driven, with primary emphasis given to probing multiscale phenomena straddling the atomistic and continuum scales. Examples of such applications include: dislocations and plasticity [44,50,61]; nanoindentation [58,31,32]; nanovoid growth [40,41]; fracture [43,45,44]; grain boundaries [56]; and others. Extensions to finite-temperature, be it at equilibrium [22,55,42,62], or with heat conduction accounted for [33,3,6], greatly extend the range of applicability of the method.The mathematical analysis of the quasicontinuum method is comparatively more

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.