Norm Estimates Research Articles

Plug-and-Play-Based Algorithm for Mixed Noise Removal with the Logarithm Norm Approximation Model

During imaging and transmission, images are easily affected by several factors, including sensors, camera motion, and transmission channels. In practice, images are commonly corrupted by a mixture of Gaussian and impulse noises, further complicating the denoising problem. Therefore, in this work, we propose a novel mixed noise removal model that combines a deterministic low-rankness prior and an implicit regularization scheme. In the optimization model, we apply the matrix logarithm norm approximation model to characterize the global low-rankness of the original image. We further adopt the plug-and-play (PnP) scheme to formulate an implicit regularizer by plugging an image denoiser, which is used to preserve image details. The above two building blocks are complementary to each other. The mixed noise removal algorithm is thus established. Within the framework of the PnP scheme, we address the proposed optimization model via the alternating directional method of multipliers (ADMM). Finally, we perform extensive experiments to demonstrate the effectiveness of the proposed algorithm. Correspondingly, the simulation results show that our algorithm can recover the global structure and detailed information of images well and achieves superior performance over competing methods in terms of quantitative evaluation and visual inspection.

Computability of magnetic Schrödinger and Hartree equations on unbounded domains

AbstractWe study the computability of global solutions to linear Schrödinger equations with magnetic fields and the Hartree equation on . We show that the solution can always be globally computed with error control on the entire space if there exist a priori decay estimates in generalized Sobolev norms on the initial state. Using weighted Sobolev norm estimates, we show that the solution can be computed with uniform computational runtime with respect to initial states and potentials. We finally study applications in optimal control theory and provide numerical examples.

Assessing HD-EEG functional connectivity states using a human brain computational model

Objective. Electro/Magnetoencephalography (EEG/MEG) source-space network analysis is increasingly recognized as a powerful tool for tracking fast electrophysiological brain dynamics. However, an objective and quantitative evaluation of pipeline steps is challenging due to the lack of realistic ‘controlled’ data. Here, our aim is two-folded: (a) provide a quantitative assessment of the advantages and limitations of the analyzed techniques and (b) introduce (and share) a complete framework that can be used to optimize the entire pipeline of EEG/MEG source connectivity. Approach. We used a human brain computational model containing both physiologically based cellular GABAergic and Glutamatergic circuits coupled through Diffusion Tensor Imaging, to generate high-density EEG recordings. We designed a scenario of successive gamma-band oscillations in distinct cortical areas to emulate a virtual picture-naming task. We identified fast time-varying network states and quantified the performance of the key steps involved in the pipeline: (a) inverse models to reconstruct cortical-level sources, (b) functional connectivity measures to compute statistical interdependency between regional signals, and (c) dimensionality reduction methods to derive dominant brain network states (BNS). Main results. Using a systematic evaluation of the different decomposition techniques, results show significant variability among tested algorithms in terms of spatial and temporal accuracy. We outlined the spatial precision, the temporal sensitivity, and the global accuracy of the extracted BNS relative to each method. Our findings suggest a good performance of weighted minimum norm estimate/ Phase Locking Value combination to elucidate the appropriate functional networks and ICA techniques to derive relevant dynamic BNS. Significance. We suggest using such brain models to go further in the evaluation of the different steps and parameters involved in the EEG/MEG source-space network analysis. This can reduce the empirical selection of inverse model, connectivity measure, and dimensionality reduction method as some of the methods can have a considerable impact on the results and interpretation.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space

We consider a singularly perturbed time-dependent problem with a shift term in space. On appropriately defined layer adapted meshes of Durán- and S-type we derive a-priori error estimates for the stationary problem. Using a discontinuous Galerkin method in time we obtain error estimates for the full discretisation. Introduction of a weighted scalar products and norms allows us to estimate the error of the time-dependent problem in energy and balanced norm. So far it was open to prove such a result. Error estimates in the energy norm is for the standard finite element discretization in space, and for the error estimate in the balanced norm the computation of the numerical solution is changed by using a different bilinear form. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

Baskakov-Kantorovich operators reproducing affine functions: inverse results

In a previous paper the author presented a Kantorovich modification of Baskakov operators which reproduce affine functions and he provided an upper estimate for the rate of convergence in polynomial weighted spaces. In this paper, for the same family of operators, a strong inverse inequality is given for the case of approximation in norm.

On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators

This paper deals with convergence theorems of the Galerkin finite element approximation for the second-order elliptic boundary value problems. Under some quite general settings, we show not only the pointwise convergence but also prove that the norm of approximate operator converges to the corresponding norm for the inverse of a linear elliptic operator. Since the approximate norm estimates of linearized inverse operator play an essential role in the numerical verification method of solutions for non-linear elliptic problems, our result is also important in terms of guaranteeing its validity. Furthermore, the present method can also be applied to more general elliptic problems, e.g., biharmonic problems and so on.

SMOOTHNESS OF SOLUTIONS (SEPARABILITY) OF THE NONLINEAR STATIONARY SCHR¨ODINGER EQUATION

The equation of motion of a microparticle in various force fields is the Schr¨odinger wave equation.Many questions of quantum mechanics, in particular the thermal radiation of electromagneticwaves, lead to the problem of separability of singular differential operators. One such operator isthe above Schr?dinger operator. In this paper, the named operator is studied by the methods offunctional analysis. Found sufficient conditions for the existence of a solution and the separabilityof an operator in a Hilbert space. All theorems were originally proved for the model Sturm-Liouvilleequation and extended to a more general case.In §1-2, for the nonlinear Sturm-Liouville equation, sufficient conditions are found that ensurethe existence of an estimate for coercivity, and estimates of weight norms are obtained for thefirst derivative of the solution. In Sections 3-4 the results of Sections 1-2 are generalized for theSchr¨odinger equation in the case m = 3.

The Kato-Ponce inequality with polynomial weights

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights \(\langle x\rangle ^a = (1+|x|^2)^{a/2}\) for \(a\ge 0\). We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential \(J^s = (1- \Delta )^{{s}/{2}}\) holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree s. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponce inequality. We also consider a few other variants such as commutator and mixed norm estimates, and analogous conclusions are derived. Our results contain strong-type inequalities for both \(L^1\) and \(L^\infty \) endpoints, which extend several existing results.

A new primal-dual weak Galerkin method for elliptic interface problems with low regularity assumptions

This article introduces a new primal-dual weak Galerkin (PDWG) finite element method for second order elliptic interface problems with ultra-low regularity assumptions on the exact solution and the interface and boundary data. It is proved that the PDWG method is stable and accurate with optimal order of error estimates in discrete and Sobolev norms. In particular, the error estimates are derived under the low regularity assumption of u∈Hδ(Ω) for δ>12 for the exact solution u. Extensive numerical experiments are conducted to provide numerical solutions that verify the efficiency and accuracy of the new PDWG method.

Alternating Minimization-Based Sparse Least-Squares Classifier for Accuracy and Interpretability Improvement of Credit Risk Assessment

When dealing with complex and redundant data classification problems, many classifiers cannot provide high predictive accuracy and interpretability. We also find that the least-squares support vector classifiers (LSSVCs) hardly identify important instances and features from data, so they cannot give an interpretable prediction. Although the LSSVC has the properties of low bias and high robustness, its high variance often gives a poor predictive performance. In this paper, we propose an alternating minimization-based sparse least-squares classifier (AMSLC) approach in the framework of LSSVCs to address the aforementioned problems. Based on the reconstructed row- and column-wise kernel matrices, the sparsity-induced [Formula: see text]-norm approximation function is introduced to the LSSVC model. By alternately solving two unconstrained quadratic optimization problems or two systems of linear equations, AMSLC can predict the class labels of given instances and extract the least number of important instances and features to obtain the interpretable classification. Compared with SVC, LSSVC, [Formula: see text]-norm SVC (L1SVC), [Formula: see text]-norm SVC (L0SVC), the least absolute shrinkage and selection operator classifier (LASSOC), and multiple kernel learning SVC (MKLSVC) on four real credit datasets, the experimental results show that the proposed AMSLC method generally obtains the best predictive accuracy and the interpretable classification with the minimum number of important instances and features.

Discontinuous Galerkin method for the diffusive-viscous wave equation

This paper develops and analyzes the discontinuous Galerkin method for the diffusive-viscous wave equation. The proposed numerical scheme is established by using the discontinuous Galerkin discretization for the spatial variable and a tailored finite difference scheme to approximate the first and second order temporal derivative terms. A second order temporal convergence rate and the optimal order spatial error estimate in the DG norm are derived. We also provide the optimal spatial error estimate in the L2(Ω)-norm under the extra parameter assumption. Numerical tests are presented to illustrate the predicted convergence behaviours and the versatility of the proposed method.

Non-radial weights and polynomial approximation in spaces of analytic functions

We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights.

Compact difference of composition operators on the Hardy spaces

Answering to a long-standing question raised by Shapiro and Sundberg in 1990, Choe et al. have recently obtained a characterization for compact differences of composition operators acting on the Hilbert-Hardy space over the unit disk. Their characterization is described in terms of certain Bergman-Carleson measures involving derivatives of the inducing maps. In this paper, based on such results, we take one step further to obtain a completely new characterization, which is more intuitive and much simpler. In particular, our new characterization does not involve derivatives of the inducing maps and includes the Reproducing Kernel Thesis characterization. Moreover, our proofs are constructive enough to yield optimal estimates for the essential norms.

Weak Galerkin coupled with conforming finite element method for hybrid-dimensional fracture model

In this paper, we propose the weak Galerkin coupled with conforming finite element method to solve the hybrid-dimensional fracture problem, which couples n-dimensional Darcy flow in the porous matrix with (n−1)-dimensional flow in the fracture. The generalized Robin type interface conditions present in the coupling scheme. We derive the optimal error estimates for L2 and energy norms with arbitrary degree polynomials. The error rate obtained by numerical examples agrees with the analysis results on both polygonal mesh and nonmatching mesh. Results of numerical examples also show that the proposed scheme is robust to the parameters in the reduced fracture model.

Source localization and functional network analysis in emotion cognitive reappraisal with EEG-fMRI integration.

BackgroundThe neural activity and functional networks of emotion-based cognitive reappraisal have been widely investigated using electroencephalography (EEG) and functional magnetic resonance imaging (fMRI). However, single-mode neuroimaging techniques are limited in exploring the regulation process with high temporal and spatial resolution.ObjectivesWe proposed a source localization method with multimodal integration of EEG and fMRI and tested it in the source-level functional network analysis of emotion cognitive reappraisal.MethodsEEG and fMRI data were simultaneously recorded when 15 subjects were performing the emotional cognitive reappraisal task. Fused priori weighted minimum norm estimation (FWMNE) with sliding windows was proposed to trace the dynamics of EEG source activities, and the phase lag index (PLI) was used to construct the functional brain network associated with the process of downregulating negative affect using the reappraisal strategy.ResultsThe functional networks were constructed with the measure of PLI, in which the important regions were indicated. In the gamma band source-level network analysis, the cuneus, the lateral orbitofrontal cortex, the superior parietal cortex, the postcentral gyrus, and the pars opercularis were identified as important regions in reappraisal with high betweenness centrality.ConclusionThe proposed multimodal integration method for source localization identified the key cortices involved in emotion regulation, and the network analysis demonstrated the important brain regions involved in the cognitive control of reappraisal. It shows promise in the utility in the clinical setting for affective disorders.

Subordinations and Norm Estimates for Functions Associated with Ma-Minda Subclasses

For a function p analytic in the open unit disc and satisfying p(0)=1, we prove certain subordination implications of the first order differential subordination 1+zp′(z)≺1+Mz, which provides sufficient conditions for a function to belong to various subclasses of Ma-Minda starlike functions. The pre-Schwarzian norm estimate and inclusion criteria for certain subclasses of analytic function are also obtained. Additionally, using Gronwall’s inequality we give a sufficient condition for a normalized function to belong to a class of functions with bounded arguments that extends the class of strongly α-Bazilevič functions of order γ studied by Gao in 1996.

Approximation of criminal sanctions in the European Union: A wild goose chase?

The era when criminal law was solely a matter of national legislation is long gone. In the European Union, the approximation of substantive criminal norms and procedural rights has not escaped the attention of the European legislator. Inevitably, the move towards supranational criminal law has affected the field of sentencing and criminal sanctions. This paper seeks to discuss the changes that the development of EU criminal law brings on sentencing. The focus lies on the approximation of criminal sanctions within the EU, in an attempt to understand the scope and the added value of the EU’s approach in the field. First, the legal basis of article 83 TFEU is presented and analyzed, in order to define the limits of the Union’s competence in the field of sentencing. Subsequently, the approach of the European legislator when adopting minimum rules on the approximation of criminal sanctions is unravelled, and the relationship between functional approximation and mutual recognition is discussed. Finally, this paper concludes with some critical reflections upon the approximation of criminal sanctions in the EU, with an eye towards the future of Europe’s Area of Freedom, Security and Justice (AFSJ).

Alterations in the default mode network in rolandic epilepsy with mild spike-wave index in non-rapid eye movement sleep.

PurposeRolandic epilepsy (RE) is one of the most common epilepsy syndromes during childhood. The aim of this study was to investigate the alterations in the default mode network (DMN) of RE patients whose spike-wave index (SWI) was within the 50–85% range during non-rapid eye movement (NREM) during sleep, as well as to detect early neuroimaging markers.MethodsResting-state data was recorded for each subject using magnetoencephalography (MEG). DMN-related brain regions were chosen as regions of interest. The spectral power and functional connectivity (FC) of the DMN were estimated through the use of minimum norm estimation (MNE) combined with Welch technique and corrected amplitude envelope correlation (AEC-c).ResultsThe patient group included 20 patients with NREM phase 50% ≤ SWI < 85% (mild SWI group), and 18 typical RE patients (SWI < 50% group). At the regional level, the mild SWI group exhibited enhanced spectral power in the delta band of the bilateral posterial cingulate cortex and attenuated the spectral power in the alpha band of the bilateral posterial cingulate cortex. Enhanced spectral power in the bilateral precuneus (PCu) in the delta band and attenuated spectral power in the right lateral temporal cortex (LTC) in the alpha band were common across all RE patients. At the FC level, patients in the mild SWI group indicated increased AEC-c values between the bilateral posterial cingulate cortex in the delta band and between the left medial frontal cortex (MFC) and bilateral posterial cingulate cortex in the alpha band. Increased AEC-c values between the right PCu and left MFC in the delta band, and between the left PCu and right MFC in the theta band, were common across all RE patients. Moreover, the spectral power in the bilateral posterial cingulate cortex in the alpha band and the AEC-c value between the bilateral posterial cingulate cortex in the delta band demonstrated good discrimination ability.ConclusionThe spectral power of the bilateral posterior cingulate cortex (PCC) in the alpha band and the AEC-c value between the bilateral PCC in the delta band may be promising indicators of early differentiation between mild SWI and typical RE.

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

This paper is a comprehensive study of Lp estimates for time fractional wave equations of order α∈(1,2) in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.

Equilibrium validation for triblock copolymers via inverse norm bounds for fourth-order elliptic operators

Block copolymers play an important role in materials sciences and have found widespread use in many applications. From a mathematical perspective, they are governed by a nonlinear fourth-order partial differential equation which is a suitable gradient of the Ohta–Kawasaki energy. While the equilibrium states associated with this equation are of central importance for the description of the dynamics of block copolymers, their mathematical study remains challenging. In the current paper, we develop computer-assisted proof methods which can be used to study equilibrium solutions in block copolymers consisting of more than two monomer chains, with a focus on triblock copolymers. This is achieved by establishing a computer-assisted proof technique for bounding the norm of the inverses of certain fourth-order elliptic operators, in combination with an application of a constructive version of the implicit function theorem. While these results are only applied to the triblock copolymer case, the obtained norm estimates can also be directly used in other contexts such as the rigorous verification of bifurcation points, or pseudo-arclength continuation in fourth-order parabolic problems.