Search

We use a sticky particle method to show global existence of (energy) conservative sticky N -peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of conservative N -peakon solutions. The dispersion limit avoids the collision between peakons, and numerical results show that the dispersion limit is exactly the sticky peakons. At last, when the splitting of peakons is allowed, we give an example to show the non-uniqueness of conservative solutions. [ABSTRACT FROM AUTHOR]

Estimation of high-dimensional sparse covariance matrix is one of the fundamental and important problems in multivariate analysis and has a wide range of applications in many fields. This paper presents a novel method for sparse covariance matrix estimation via solving a non-convex regularization optimization problem. We establish the asymptotic properties of the proposed estimator and develop a multi-stage convex relaxation method to find an effective estimator. The multi-stage convex relaxation method guarantees any accumulation point of the sequence generated is a first-order stationary point of the non-convex optimization. Moreover, the error bounds of the first two stage estimators of the multi-stage convex relaxation method are derived under some regularity conditions. The numerical results show that our estimator outperforms the state-of-the-art estimators and has a high degree of sparsity on the premise of its effectiveness. [ABSTRACT FROM AUTHOR]

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA. • A nonconvex TRPCA (N-TRPCA) model is proposed for visual data recovery, which can preserve the important information by shrinking tensor singular values differently. • The nonlocal prior is incorporated into N-TRPCA, resulting in a nonconvex and nonlocal TRPCA (NN-TRPCA) model. • An optimization algorithm based on ADMM is presented for solving NN-TRPCA. • Extensive experimental results confirm the superiority of the proposed methods. [ABSTRACT FROM AUTHOR]

Fault impulses induced by localized failure is critical for fault diagnosis and degradation prognostics of rotating machinery, however, weak transient fault impulses are always submerged in heavy noise and unrelated components. To overcome this bottleneck issue, a novel approach named smoothing sparse low-rank matrix (SSLRM) associated with asymmetric and singular value decomposition (SVD) penalty regularizers is proposed, for the first time. To be specific, the asymmetric and SVD penalty regularizers are utilized to alleviate the issues of underestimation deficiency of traditional sparse regularization, where the Parseval theorem in wavelet framework and smoothing operation are conducted, so as to enhance the sparsity of the estimated transient impulses while restraining the unrelated interference. Meanwhile, the strict convexity of the cost function is proved theatrically, and the effective optimization algorithm with fast convergence speed based upon the alternating direction method of multipliers (ADMM) is proposed through splitting the optimization function into two simple sub-parts. The simulated case and two experimental cases regarding bearings failure in corn thresher and gear-root crack failure in reducer are investigated for theoretical verification of the proposed approach, the results indicate that diagnosis accuracy and fault impulses amplitude of the algorithm are superior to the state-of-the-art benchmarks. [ABSTRACT FROM AUTHOR]

For many algorithms, parameter tuning remains a challenging task, which becomes tedious in a multi-parameter setting. Multi-penalty regularization, successfully used for solving undetermined sparse regression problems of unmixing type, is one of such examples. We propose a novel algorithmic framework for an adaptive parameter choice in multi-penalty regularization with focus on correct support recovery. By extending ideas on regularization paths, we provide an efficient procedure for the construction of regions containing structurally similar solutions, i.e., solutions with the same sparsity and sign pattern, over the range of parameters. Combined with a model selection criterion, regularization parameters are chosen in a data-adaptive manner. Another advantage of our algorithm is that it provides an overview on the solution stability over the parameter range. We provide a numerical analysis of our method and compare it to the state-of-the-art algorithms for compressed sensing problems to demonstrate the robustness and power of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study the Nyström type subsampling for large-scale kernel methods to reduce the computational complexities of big data. We discuss the multi-penalty regularization scheme based on Nyström type subsampling which is motivated from well-studied manifold regularization schemes. We develop a theoretical analysis of the multi-penalty least-square regularization scheme under the general source condition in vector-valued function setting, therefore the results can also be applied to multi-task learning problems. We achieve the optimal minimax convergence rates of the multi-penalty regularization using the concept of effective dimension for the appropriate subsampling size. We discuss an aggregation approach based on the linear function strategy to combine various Nyström approximants. Finally, we demonstrate the performance of the multi-penalty regularization based on Nyström type subsampling on the Caltech-101 dataset for multi-class image classification and NSL-KDD benchmark dataset for intrusion detection problem. [ABSTRACT FROM AUTHOR]

This paper discusses isotropic sparse regularization for a random field on the unit sphere S 2 in R 3 , where the field is expanded in terms of a spherical harmonic basis. A key feature is that the norm used in the regularization term, a hybrid of the ℓ 1 and ℓ 2 -norms, is chosen so that the regularization preserves isotropy, in the sense that if the observed random field is strongly isotropic then so too is the regularized field. The Pareto efficient frontier is used to display the trade-off between the sparsity-inducing norm and the data discrepancy term, in order to help in the choice of a suitable regularization parameter. A numerical example using Cosmic Microwave Background (CMB) data is considered in detail. In particular, the numerical results explore the trade-off between regularization and discrepancy, and show that substantial sparsity can be achieved along with small L 2 error. [ABSTRACT FROM AUTHOR]

Boosting as gradient descent algorithms is one popular method in machine learning. In this paper a novel Boosting-type algorithm is proposed based on restricted gradient descent with structural sparsity control whose underlying dynamics are governed by differential inclusions. In particular, we present an iterative regularization path with structural sparsity where the parameter is sparse under some linear transforms, based on variable splitting and the Linearized Bregman Iteration. Hence it is called Split LBI. Despite its simplicity, Split LBI outperforms the popular generalized Lasso in both theory and experiments. A theory of path consistency is presented that equipped with a proper early stopping, Split LBI may achieve model selection consistency under a family of Irrepresentable Conditions which can be weaker than the necessary and sufficient condition for generalized Lasso. Furthermore, some ℓ 2 error bounds are also given at the minimax optimal rates. The utility and benefit of the algorithm are illustrated by several applications including image denoising, partial order ranking of sport teams, and world university grouping with crowdsourced ranking data. [ABSTRACT FROM AUTHOR]

We discuss the problem of parameter choice in learning algorithms generated by a general regularization scheme. Such a scheme covers well-known algorithms as regularized least squares and gradient descent learning. It is known that in contrast to classical deterministic regularization methods, the performance of regularized learning algorithms is influenced not only by the smoothness of a target function, but also by the capacity of a space, where regularization is performed. In the infinite dimensional case the latter one is usually measured in terms of the effective dimension. In the context of supervised learning both the smoothness and effective dimension are intrinsically unknown a priori. Therefore we are interested in a posteriori regularization parameter choice, and we propose a new form of the balancing principle. An advantage of this strategy over the known rules such as cross-validation based adaptation is that it does not require any data splitting and allows the use of all available labeled data in the construction of regularized approximants. We provide the analysis of the proposed rule and demonstrate its advantage in simulations. • For smooth kernels, a power decay of effective dimensions may not be appropriate. • Learning rates under general smoothness assumptions on target functions. • Optimal learning rates in all involved norms based on a novel bound. • An adaptive parameter choice which uses the whole dataset without data splitting. [ABSTRACT FROM AUTHOR]

We provide a new estimator of integral operators with smooth kernels, obtained from a set of scattered and noisy impulse responses. The proposed approach relies on the formalism of smoothing in reproducing kernel Hilbert spaces and on the choice of an appropriate regularization term that takes the smoothness of the operator into account. It is numerically tractable in very large dimensions. We study the estimator's robustness to noise and analyze its approximation properties with respect to the size and the geometry of the dataset. In addition, we show minimax optimality of the proposed estimator. [ABSTRACT FROM AUTHOR]

The multiplicative ℓ q -regularization has been recently introduced in structural dynamics for solving force reconstruction problems. Practically, the resolution of this regularization strategy requires the implementation of an iterative procedure. To this end, an Iteratively Reweighted Least-Squares algorithm has been originally implemented. The core idea of this algorithm is to replace the direct resolution of the inverse problem by an equivalent iterative procedure having an explicit and unique solution at each iteration. However, the exploitation of this very general idea allows defining other Iteratively Reweighted schemes. The present paper aims at comparing the overall performances of three particular Iteratively Reweighted algorithms for solving force reconstruction problems derived from a more general and original iterative procedure. The numerical applications proposed in this contribution highlight the ability of the considered algorithms in providing consistent regularized solutions with respect to various parameters such as the measurement noise level or the tolerance chosen to stop the iterative process. • A general Iteratively Reweighted ℓ p algorithm is presented. • General recommendations are given to properly applied the proposed strategy. • The inverse problem is expressed as a multiplicative regularization problem. • The proposed method can be seen as a generalization of the IRLS algorithm. • IR ℓ p algorithm for p ≤ 1 is the better option for sparse excitation fields. [ABSTRACT FROM AUTHOR]

• This paper proposes an improved Jaya algorithm for damage identification. • Objective function is improved based on sparse regularization and Bayesian inference. • Benchmark studies are conducted to demonstrate the improvement of this approach. • Numerical and experimental investigations are conducted to demonstrate the performance. • Good identification results are obtained with a limited quantity of modal data. Structural damage identification can be considered as an optimization problem, by defining an appropriate objective function relevant to structural parameters to be identified with optimization techniques. This paper proposes a new heuristic algorithm, named improved Jaya (I-Jaya) algorithm, for structural damage identification with the modified objective function based on sparse regularization and Bayesian inference. To improve the global optimization capacity and robustness of the original Jaya algorithm, a clustering strategy is employed to replace solutions with low-quality objective values and a new updated equation is used for the best-so-far solution. The objective function that is sensitive and robust for effective and reliable damage identification is developed through sparse regularization and Bayesian inference and used for optimization analysis with the proposed I-Jaya algorithm. Benchmark tests are conducted to verify the improvement in the developed algorithm. Numerical studies on a truss structure and experimental validations on an experimental reinforced concrete bridge model are performed to verify the developed approach. A limited quantity of modal data, which is distinctively less than the number of unknown system parameters, are used for structural damage identification. Significant measurement noise effect and modelling errors are considered. Damage identification results demonstrate that the proposed method based on the I-Jaya algorithm and the modified objective function based on sparse regularization and Bayesian inference can provide accurate and reliable damage identification, indicating the proposed method is a promising approach for structural damage detection using data with significant uncertainties and limited measurement information. [ABSTRACT FROM AUTHOR]

• Fractional-order elastic net regularization is proposed for force identification. • Method retains traits of fractional Tikhonov and fractional l 1 -norm regularization. • A strategy for determining weights of different regularization parameters is proposed. • High accuracy and applicability for identifying various types of forces is achieved. Force identification is an important and fundamental problem in various engineering fields, and several regularization techniques have been introduced to improve the ill-posedness of this problem. While these regularization techniques can achieve satisfactory identified accuracy, they are best suited for dealing with specific forms of external forces. The external force, in reality, is always unknown and needs to be identified, making it difficult to select an appropriate regularization technique in advance. To overcome this disadvantage, a novel regularization technique, i.e. fractional-order elastic net regularization is proposed for force identification in this study. Firstly, the relationship between acceleration response and coefficient vector of external force is established. Then, a weighted matrix is introduced to define the difference between measured and calculated acceleration responses. Meanwhile, the external force and coefficient vector are constrained by their respective norm penalties with the help of regularization parameters. As a result, an optimization problem is defined for force identification, and the corresponding geometric significance is analyzed to propose a selection way for the regularization parameters. Finally, both numerical simulations and experimental verifications are conducted to evaluate the effectiveness of the proposed method. Comparison with existing methods reveals that the proposed method has great identification accuracy and applicability for various types of external forces. [ABSTRACT FROM AUTHOR]

We recently proposed a non-convex sparse regularization method in the ADMM framework to reconstruct and localize unknown impact forces. This method surpasses the convex ℓ 1 regularization not only in inducing sparsity but also in avoiding the high-amplitude underestimation of solutions. In this work, we aim to improve the identification performance and enable the monitoring of unknown impact forces with fewer sensors by incorporating prior information in the form of natural grouping of the solution components. To achieve this, we extend our previous work on non-convex sparse regularization by incorporating group information into the method. The resulting group sparsity problem is challenging to solve due to the mixed structure and possible grouping irregularity. To address this, we develop an efficient ADMM solver in a grouped manner, featuring a novel shrinkage operator. We validate our approach both numerically and experimentally on aircraft-like composite laminated plates. Our case studies demonstrate that the proposed method achieves high accuracy and strong robustness in impact localization and time–history reconstruction from single-sensor-based measurements. • A novel impact force identification method is proposed by considering non-convex group sparse regularization. • A versatile ADMM solver is developed to handle the non-convex group sparsity regularization problem. • A novel group shrinkage operator(G2IT) is derived. • The proposed method outperforms the existing methods including the group Lasso and non-convex ϕ exp regularization. [ABSTRACT FROM AUTHOR]

• A new compact-bandwidth regularization approach is proposed within the signal processing framework for operational modal analysis. • The key idea behind the proposed approach is that an arbitrary mode is compact in the frequency domain. • The spectrum-peak-based rule is proposed for choice of initial modal parameters. • The L-curve method along with a derived bound is used for regularization parameter estimation. • Results shown the effectiveness of the proposed approach for various types of excitations and the case with closely spaced modes. Operational modal analysis (OMA) has received tremendous interest from engineering fields in recent years. This paper develops a compact-bandwidth regularization approach for OMA within the signal processing framework. The key ingredient lies in the fact that a structural mode is always compact in the frequency domain and this results in the compact-bandwidth constraint for structural modes. To implicitly enforce the compact-bandwidth constraint, the compact-bandwidth regularization is introduced to conventional blind modal separation. Then, the alternating minimization algorithm is used to iteratively get the solution in the immediately-at-once or one-by-one manner. In proceeding so, the spectrum-peak-based rule is adopted for choice of initial modal parameters and the L-curve method in conjunction with a theoretically derived bound is applied to estimate a proper regularization parameter. Numerical examples and an experimental test case are studied along with comparison to some usual OMA approaches to see the performance and advantages of the proposed approach in modal identification. [ABSTRACT FROM AUTHOR]

Fast reconstruction of a bandlimited function from its finite oversampling data has been a fundamental problem in sampling theory. As the number of sample data increases to infinity, exponentially-decaying reconstruction errors can be achieved by many methods in the literature. In fact, it is generally conjectured that when the optimal method is used, the dominant term in the error of reconstructing a function bandlimited to [ − δ , δ ] (δ < π) from its data sampled at the integer points on [ − n , n ] is exp (− λ (π − δ) n). By far, the best estimate for the constant λ among regularization methods is 1 ∕ 2 and is achieved by the highly efficient Gaussian regularized Whittaker–Kotelnikov–Shannon sampling series. We prove in this paper that the exponential constant 1 ∕ 2 is optimal for this method. Moreover, the optimal variance of the Gaussian regularizer is provided. [ABSTRACT FROM AUTHOR]

We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [8]. We introduce a new path-by-path scaling property which is shown to be sufficient to imply regularizing effects. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for solving Facial Expression Recognition (FER) task which integrates Manifold Regularization Dictionary Pair Learning and Intra-class Variation Reduced Feature (MRDPL-IVRF) model and Iterative Optimization Classification Strategy (IOCS). In the proposed framework, we firstly divide the facial image into overlapping patches. Weighted Patch-based Local Binary Patterns (WPLBP) is proposed for feature extraction by cascading the weighted LBP features from each patch, which highlights the informative facial regions. MRDPL-IVRF model is then adopted for robust sparse representation of facial features, where IVRFs are generated from the difference between query images and their corresponding approximation of each expression class, which can reduce the intra-class variations resulted by illumination and identity. Finally, IOCS is presented for expression classification which can explore the most unlikely class according to the maximum reconstruction residual. Different from traditional sparse coding classification scheme, our IOCS method iteratively removes irrelevant samples to obtain a more precise representation model, thus can further improve the recognition performance. The feasibility of our framework has been successfully tested on CK, CK+, SEFW, CMU-PIE and Multi-PIE databases. Furthermore, the experiments combining deep features shows that our model can be integrated with existing deep models and obtain better FER results. • We present a novel Weighted Patch-based Local Binary Patterns (WPLBP) descriptor. • A novel Iterative Optimization Classification Strategy (IOCS) is proposed. • Great experiment results are obtained on four famous databases. [ABSTRACT FROM AUTHOR]

• A density filtering regularization method is proposed to smooth out strong local stiffness variations. • The density filter controls the length scale of the damaged zone by selecting a proper filter radius. • The method is compared with classical Tikhonov regularization in a numerical case study. • Compared with Tikhonov regularization, density filtering is found to be more successful in identifying the damaged zone. Finite element (FE) model updating is often used as a non-destructive method to detect structural damage. Stiffness parameters of an FE model of the structure are calibrated based on experimental vibration data. If the desired spatial resolution is high, the problem is likely to be ill-conditioned and requires regularization. Tikhonov regularization is frequently used for FE model updating problems, but the selection of a proper regularization parameter and a good initial estimate of the stiffness parameters is difficult. This paper proposes an alternative, density-filtering-based method where the filter radius acts as regularization parameter. Since the filter radius controls the minimal length scale of the identifiable damaged zones, it has a clear physical meaning. Furthermore, an initial estimate of the stiffness parameters is only required as a starting point for the optimization algorithm whereas in the case of Tikhonov regularization it also guides the optimization. Both regularization methods are compared in a numerical case study. The density filtering regularization method is found to be more successful in identifying the damaged zone, while Tikhonov regularization sometimes fails to do so. [ABSTRACT FROM AUTHOR]

• A preconditioned least square QR-factorization approach is developed for moving force identification. • The regularization matrix L is introduced to improve the ill-posed problems. • The number of iterations j is introduced to avoid noise disturbance and ensure the robustness. • Preconditioned least square QR-factorization approach is validated through numerical simulation. This paper develops a novel method for moving force identification (MFI) called preconditioned least square QR-factorization (PLSQR) method. The algorithm seeks to reduce the impact of identification errors caused by unknown noise. The biaxial moving forces travel on a simply supported bridge at three different speeds is used to generate numerical simulations to assess the effectiveness and applicability of the algorithm. Results indicate that the method is more robust towards ill-posed problem and has higher identification precision than the conventional time domain method (TDM). In addition, the robustness and ill-posed immunity of PLSQR are directly affected by two kinds of regularization parameters, namely, number of iterations j and regularization matrix L. Compared with the standard form of least square QR-factorization (LSQR), i.e., the regularization matrix L being the identity matrix I n , the PLSQR with the optimal number of iterations j and regularization matrix L has many advantages on MFI and it is more suitable for field trials due to better adaptability with type of sensors and number of sensors. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the following singular Monge–Ampère equation (0.1) { det ⁡ D 2 u = 1 (H u) n + k + 2 u ⁎ k in Ω ⊂ ⊂ R n , u = 0 , on ∂ Ω where k ≥ 0 , H < 0 are constants and u ⁎ = x ⋅ ∇ u (x) − u (x) is the Legendre transformation of u. Equation (0.1) is related to proper affine hyperspheres. We will show the existence of solutions of (0.1) u ∈ C ∞ (Ω) ∩ C (Ω ¯) via regularization method. Using the technique in [10,12] , we also obtain the optimal graph regularity of the solution of (0.1). [ABSTRACT FROM AUTHOR]

Highlights • An enhanced sparse regularization method is developed for improving impact force identification. • A weighted l 1 -norm convex optimization model for impact force identification is developed. • Iteratively reweighted l 1 -norm minimization algorithm is proposed for solving impact force identification. • Compared with existing regularizations, the enhanced sparse regularization has much sparser and more accurate result. Abstract The standard sparse regularization method based on l 1 -norm minimization for impact force identification has already proved to be an interesting alternative to the classical regularization method based on l 2 -norm minimization. However, choosing the l 1 -norm as a convex relaxation of the l 0 -norm, the corresponding sparse regularization model generally offers a sparse but underestimated solution. In this paper, considering the sparsity of impact force, an enhanced sparse regularization method based on reweighted l 1 -norm minimization is developed for reducing the peak force error and improving the identification accuracy of impact force. First, a weighted l 1 -norm convex optimization model is presented to overcome the ill-posed nature of the inverse problem of impact force identification. Second, to solve such a regularized model efficiently, an iteratively reweighted l 1 -norm minimization algorithm is introduced, where the weights are adaptively updated from the previous solution. The application of the iteratively reweighted scheme is to overcome the mismatch between l 1 -norm minimization and l 0 -norm minimization, while keeping the enhanced sparse regularization problem solvable and convex. Finally, numerical simulation and experimental verification including the single and double impact force identification on a plate structure are presented to illustrate the superior performance of the enhanced sparse regularization method compared to classical regularization approaches. Effects of reweighting iteration number, tuning parameters, initial conditions and response locations are successfully investigated in detail. Results demonstrate that compared with the standard l 1 -norm regularization method and the classical l 2 -norm regularization method, the enhanced sparse regularization method based on reweighted l 1 -norm minimization whose solution is much sparser, can greatly improve the identification accuracy of impact force. Moreover, the proposed method is much more robust to the choice of tuning parameters and noisy measurements. [ABSTRACT FROM AUTHOR]

Highlights • An optimal Bayesian regularization is presented. • Generalized Gaussian distributions are used to describe prior knowledge of the sources. • The approach estimates the most probable parameters given a measured vibration field. • An adapted resolution algorithm is proposed. • The pertinence of the method is evaluated from numerical and experimental validations. Abstract In a paper, recently published in Mechanical Systems and Signal Processing , we have proposed a full Bayesian inference for reconstructing mechanical sources acting on a linear and time invariant structure. The main interest of this approach is to propose an estimation of all the parameters of the model and quantify the posterior uncertainty associated to each parameter. Since all the necessary information about the problem is available, statistical measures, such as the mean, the median and the mode of the solution, can be easily estimated. In many practical situations, however, one only wants to determine the most probable parameters given the available data. Consequently, it is not relevant to implement a full Bayesian inference to only extract a point estimate. To overcome this potential issue, this paper introduces an optimal Bayesian regularization aiming at computing the Maximum a Posteriori estimate of the Bayesian formulation previously introduced by the authors. In doing so, the most probable parameters are obtained without heavy computations. The validity of the proposed method is assessed numerically and experimentally. In particular, obtained results highlight the ability of the proposed regularization strategy in computing solutions with a minimal amount of prior information on the sources to identify. [ABSTRACT FROM AUTHOR]

Abstract We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The singular Stokeslet and stresslet kernels are regularized and, for the nearly singular case, corrections are added to reduce the regularization error. These corrections are derived analytically for both the Stokeslet and the stresslet using local asymptotic analysis. For the case of evaluating the integrals on the surface, as needed when solving integral equations, we design high order regularizations for both kernels that do not require corrections. This approach is direct in that it does not require grid refinement or special quadrature near the singularity, and therefore does not increase the computational complexity of the overall algorithm. Numerical tests demonstrate the uniform convergence rates for several surfaces in both the singular and near singular cases, as well as the importance of corrections when two surfaces are close to each other. Highlights • A highly accurate numerical method for 3D Stokeslet and stresslet integrals. • Regularization and analytically derived corrections used for the nearly singular case. • Enhanced accuracy when evaluating on the surface, without requiring corrections. • Straightforward quadrature for surface integrals. • No grid refinement or special quadrature required when evaluating near a surface. [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove a weak version of Lusin's theorem for the space of Sobolev- (1 , p) volume preserving homeomorphisms on closed and connected n -dimensional manifolds, n ≥ 3 , for p < n − 1. We also prove that if p > n this result is not true. More precisely, we obtain the density of Sobolev- (1 , p) homeomorphisms in the space of volume preserving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball centered at the identity can be done in a Sobolev- (1 , p) ball with the same radius centered at the identity. [ABSTRACT FROM AUTHOR]

We present an approach on training classifiers or regressors using the latent embedding of variational auto-encoders (VAE), an unsupervised deep learning method, as features. Usually VAEs are trained using unlabeled data and independently from the classifier, whereas we investigate and analyze the performance of a classifier or regressor that is trained jointly with the variational deep network. We found that models trained this way can improve the embedding s.t. to increase classification performance, and also can be used for semi-supervised learning, building up the information extracting latent representation in an incremental fashion. The model was tested on two widely known computer vision benchmarks, and its generalization power was evaluated on an independent dataset. Additionally, generally applicable statistical methods are presented for evaluating similarly performing classifiers, and used to quantify the performance increase. The general applicability and ease-of-use of deep learning approaches allows for a wide applicability of the method. [ABSTRACT FROM AUTHOR]

Abstract We propose a novel infimal convolution type functional based on total generalized variation (TGV) and shearlet transform, which can be easily incorporated into image restoration problems. We employ the TGV functional to represent the piecewise smooth cartoon part of an image and the L 1 norm of shearlet transform for the edge-like texture part. The proposed model recovers both fine details and edge features of images satisfactorily. The existence of solutions to our proposed model is proved. We also design a general algorithm using the classical first-order Primal–Dual method for solving our imaging problems. Numerical experiments are carried out to illustrate the ability of the new regularizer in preserving edge-like textures better than some existing variational and sparsity-based methods. Highlights • The new infimal convolution model separates an image to cartoon and texture parts. • The new regularizer can deal with images with edge-like textures. • We illustrate the existence of solutions of our general model. • We design a practical algorithm and also estimate the convergence condition. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Mathematical aspects of regularization are presented in a compact and unified way. • The noise level provides an optimal stopping criterion for Bregman iterations. • Osher's Bregman-iterative regularization can outperform Tikhonov regularization. • Features of the distribution are better preserved by Bregman-iterative regularization. • The regularization parameter selection is uncritical for Bregman iterations. Abstract Tikhonov regularization is the standard processing technique for the inversion of double electron-electron resonance (DEER) data to distance distributions without assuming a parametrized model. In other fields it has been surpassed by modern regularization methods. We analyze such alternative regularization methods based on the Tikhonov, total variation (TV) and Huber penalties with and without the use of Bregman iterations. For this, we provide a general mathematical framework and its open-source software implementation. We extend an earlier approach by Edwards and Stoll for the selection of an optimal regularization parameter to all of these penalties and use their big test data set of noisy DEER traces with known ground truth for assessment. The results indicate that regularization methods based on Bregman iterations provide an improvement upon Tikhonov regularization in recognizing features and recovering distribution width at moderate signal-to-noise ratio, provided that noise variance is known. Bregman-iterative methods are robust with respect to the method used in the choice of regularization parameter. [ABSTRACT FROM AUTHOR]

Abstract We study a class of 2D solutions of a Bloch–Torrey regularization of the Rosensweig system in the whole space, which arise when the initial data and the external magnetic field are 2D. We prove that such solutions are globally defined if the initial data is in H k (R 2) , k ⩾ 1. [ABSTRACT FROM AUTHOR]

Abstract The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study semilinear elliptic systems with critical nonlinearity of the form (0.1) Δ u = Q (x , u , ∇ u) , for u : R n → R K , Q has quadratic growth in ∇ u. Our work is motivated by elliptic systems for harmonic map and biharmonic map. When n = 2 , such a system does not have smooth regularity in general for W 1 , 2 weak solutions, by a well-known example of J. Frehse. Classical results of harmonic map, proved by F. Hélein (for n = 2) and F. Béthuel (for n ≥ 3), assert that a W 1 , n weak solution of harmonic map is always smooth. We extend Béthuel's result to general system (0.1) , that a W 1 , n weak solution of the system is smooth for n ≥ 3. For a fourth order semilinear elliptic system with critical nonlinearity which extends biharmonic map, we prove a similar result, that a W 2 , n / 2 weak solution of such system is always smooth, for n ≥ 5. We also construct various examples, and these examples show that our regularity results are optimal in various sense. [ABSTRACT FROM AUTHOR]

Abstract Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are defined on a sphere. Due to the localization property in the frequency domain, we prove that the regularization parameter of the kernel ridge regression associated with the needlet kernel can decrease arbitrarily fast. A natural consequence is that the regularization term for the kernel ridge regression is not necessary in the sense of rate optimality. Based on the excellent localization property in the spacial domain further, we prove that all l q (0 < q ≤ 2) kernel regularization estimates associated with the needlet kernel, including the kernel lasso estimate and kernel bridge estimate, possess almost same generalization capability for a large range of regularization parameters in the sense of rate optimality. This finding tentatively reveals that, if the needlet kernel is utilized, then the choice of q might not have a strong impact on the generalization capability in some modeling contexts. The above two properties reveal the theoretical advantages of the needlet kernel in kernel methods for spherical nonparametric regression problems. [ABSTRACT FROM AUTHOR]

Highlights • Accurate and efficient numerical simulation of granular flow. • Two regularized models of the μ (I) -rheology. • Application of the PFEM to frictional material simulation. • Validation against 3D experimental tests. Abstract This paper proposes two regularized models of the μ (I) -rheology and shows their application to the numerical simulation of 3D dense granular flows. The proposed regularizations are inspired by the Papanastasiou and Bercovier–Engleman methods, typically used to approximate the Bingham law. The key idea is to keep limited the value of the apparent viscosity for low shear rates without introducing a fixed cutoff. The proposed techniques are introduced into the Particle Finite Element Method (PFEM) framework to deal with the large deformations expected in free-surface granular flows. After showing the numerical drawbacks associated to the standard μ (I) -rheology, the two regularization strategies are derived and discussed. The regularized μ (I) -rheology is then applied to the simulation of the collapse of 2D and 3D granular columns. The numerical results show that the regularization techniques improve substantially the conditioning of the linear system without affecting the solution accuracy. A good agreement with the experimental tests and other numerical methods is obtained in all the analyzed problems. [ABSTRACT FROM AUTHOR]

Abstract We study thin obstacle problems involving the energy functional with p (x) -growth. We prove higher integrability and Hölder regularity for the gradient of minimizers of the thin obstacle problems under the assumption that the variable exponent p (x) is Hölder continuous. [ABSTRACT FROM AUTHOR]

Highlights • The regularization parameter and the flagellum discretization are decoupled. • The filament with regularized forces behaves like a slender body. • The method support Stokeslets, rotlets, dipoles. • Examples include flagella near a plane boundary. Abstract We present a variation of the method of regularized Stokeslet (MRS) specialized for the case of forces and torques distributed over filaments in three dimensions. The new formulation is based on the exact solution of Stokes equation generated by a linear continuous distribution of regularized forces along a line segment. Therefore, a straight filament with linearly varying forces does not require discretization. A general filament is approximated by a piecewise linear curve in three dimensions where the length of each line segment is chosen only based on the variation of the force field and the desired accuracy of its piecewise linear approximation. The most significant advantage of this formulation is that the values of the regularization parameter ϵ and the length of the segments h are decoupled as long as ϵ < h so that ϵ can be selected as a proxy for the radius of the filament and h is chosen to discretize the forces and torques. We analyze the performance on test problems and present biological applications of sperm motility based on existing models of swimming flagella in open space and near a plane wall. The results show, for example, that because the forces along the flagellum vary mildly, a flagellum can be approximated with as few as 11 segments of length h while fixing the regularization parameter to ϵ = h / 30 , overcoming the need for hundreds of discretization nodes required by the MRS when ϵ is small. The filament behaves like a slender cylindrical tube of radius ≈ 0.97 ϵ so that the value of ϵ influences the flagellum's swimming speed. For fixed regularization, doubling the number of line segments does not affect the results significantly as long as the force field is resolved. Examples that require rotlets and potential dipoles along the filament are also presented. [ABSTRACT FROM AUTHOR]

Abstract Quantum simulations would be highly desirable in order to investigate the finite density physics of QCD. (1 + 1) -d C P (N − 1) quantum field theories are toy models that share many important features of QCD: they are asymptotically free, have a non-perturbatively generated massgap, as well as θ -vacua. S U (N) quantum spin ladders provide an unconventional regularization of C P (N − 1) models that is well-suited for quantum simulation with ultracold alkaline-earth atoms in an optical lattice. In order to validate future quantum simulation experiments of C P (2) models at finite density, here we use quantum Monte Carlo simulations on classical computers to investigate S U (3) quantum spin ladders at non-zero chemical potential. This reveals a rich phase structure, with single- or double-species Bose–Einstein "condensates", with or without ferromagnetic order. [ABSTRACT FROM AUTHOR]

We prove the density of smooth functions in the modular topology in Musielak–Orlicz–Sobolev spaces essentially extending the results of Gossez [17] obtained in the Orlicz–Sobolev setting. We impose new systematic regularity assumption on M which allows to study the problem of density unifying and improving the known results in Orlicz–Sobolev spaces, as well as variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of W 0 1 , p ( Ω ) functions by smooth functions in the double-phase space governed by the modular function H ( x , s ) = s p + a ( x ) s q with a ∈ C 0 , α ( Ω ) excluding the Lavrentiev phenomenon within the sharp range q / p ≤ 1 + α / N . See [11, Theorem 4.1] for the sharpness of the result. [ABSTRACT FROM AUTHOR]

The global existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions is studied in this paper. We show the global existence if the initial velocity u 0 is small in H η with an arbitrary η ∈ ( 0 , 1 2 ) and the perturbation of ( ρ 0 , F 0 ) around the constant state ( 1 , I ) are small in L 2 ∩ B ˙ p , 1 2 p with p ∈ ( − 1 + 1 + 16 η 2 η , 4 ) . One of the main ingredients is that the velocity and the “effective viscous flux” G i are sufficiently regular for positive time. The regularity of G i helps to obtain the L ∞ estimate of density and deformation gradient, and hence eliminates the possible concentration and oscillation issues. [ABSTRACT FROM AUTHOR]

Spherical equivalent source method (S-ESM) with rigid spherical arrays is able to achieve good sound field reconstruction and acoustic source identification in three-dimensional free-field spaces. However, the S-ESM solved by the standard Tikhonov regularization is restricted to the low-frequency reconstruction and source identification at small measurement distances. To make S-ESM achieve good reconstruction and source identification at high frequencies and large hologram distances, this study proposes a sparsity-promoting approach denoted as wideband holography based S-ESM (WBH-based S-ESM), which applies a steepest descent method to iteratively solve S-ESM. Firstly, the framework of WBH-based S-ESM is established. Subsequently, to examine its validity, the performance of reconstruction and source identification is compared with Tikhonov regularization. Finally, a focus is concerned with the adaptability to large hologram distances. Several meaningful results have emerged from simulations and experiments: (1) WBH-based S-ESM can make good sound field reconstruction and acoustic source identification at medium-high frequencies. It extends the upper frequency limit of S-ESM. (2) The maximum hologram distance of WBH-based S-ESM at high frequencies is greater than that of Tikhonov regularization. It enlarges the measurement distance of S-ESM. This study will demonstrate the potential of WBH-based S-ESM as a useful tool for reconstruction and source identification. [ABSTRACT FROM AUTHOR]

We study distributed regression learning with coefficient regularization scheme in a reproducing kernel Hilbert space (RKHS). The algorithm randomly partitions the sample set { z i } i = 1 N into m disjoint sample subsets of equal size, applies the coefficient regularization scheme to each sample subset to produce an output function, and averages the individual output functions to get the final global estimator. We deduce the error bound in expectation in the L 2 -metric and prove the asymptotic convergence for this distributed coefficient regularization learning. Satisfactory learning rates are then derived under a standard regularity condition on the regression function, which reveals an interesting phenomenon that when m ≤ N s and s is small enough, this distributed learning has the same convergence rate compared with the algorithm processing the whole data in one single machine. [ABSTRACT FROM AUTHOR]

A new dipole field inversion method for whole head quantitative susceptibility mapping (QSM) is proposed. Instead of performing background field removal and local field inversion sequentially, the proposed method performs dipole field inversion directly on the total field map in a single step. To aid this under-determined and ill-posed inversion process and obtain robust QSM images, Tikhonov regularization is implemented to seek the local susceptibility solution with the least-norm (LN) using the L-curve criterion. The proposed LN-QSM does not require brain edge erosion, thereby preserving the cerebral cortex in the final images. This should improve its applicability for QSM-based cortical grey matter measurement, functional imaging and venography of full brain. Furthermore, LN-QSM also enables susceptibility mapping of the entire head without the need for brain extraction, which makes QSM reconstruction more automated and less dependent on intermediate pre-processing methods and their associated parameters. It is shown that the proposed LN-QSM method reduced errors in a numerical phantom simulation, improved accuracy in a gadolinium phantom experiment, and suppressed artefacts in nine subjects, as compared to two-step and other single-step QSM methods. Measurements of deep grey matter and skull susceptibilities from LN-QSM are consistent with established reconstruction methods. [ABSTRACT FROM AUTHOR]

A computation technique, known as inverse perturbation-fractional norm regularization (IP-FNR), is proposed in this wok for a sparse signal recovery problem. The objective function of this method is derived using a general ℓ p norm, when p is a positive fractional number. Numerical examples are conducted for both noiseless and noisy cases. Performance of the proposed approach in terms of root-mean-square relative error (RMSRE), mean normalized squared error, standard deviation mean, occupied memory during the computation, and computational time is compared to several previous methods. It is found that in the noiseless case, the IP-FNR method significantly outperforms the former fixed-point algorithms for a certain range of the norm exponent p , provided that the perturbation parameter and the regularization multiplier are properly chosen. In the noisy case, at the expense of computational time, the IP-FNR approach provides noticeably lower RMSRE when the signal-to-noise ratio or the sparsity ratio is high and the compression ratio is quite low. [Display omitted] • We derive a closed-form solution of the fractional norm regularization. • Simulation results demonstrate that the new proposed algorithm can provide lower error than the former approaches. • The numerical simulation also takes into account the displacement time response of the top floor of the undamped structure without an impact damper. [ABSTRACT FROM AUTHOR]

Mathematical modelling allows to predict the state of a physical system based on a set of parameters. The corresponding inverse problem, i.e., when parameter values are inferred based on constraints imposed on the system state, is commonly ill-posed. Here, we consider consistent underdetermined problems, where infinitely many combinations of possible parameter values achieve a state that accurately matches all constraints. For such problems a unique regularization strategy can be used, where the ambiguity of the solution space is reduced by sequentially incorporating additional constraints. As this regularization approach always yields a consistent problem by construction, we now show that the Gauss-Newton method is the prime choice to achieve fast convergence. Moreover, using the adjoint method allows to efficiently compute the required Jacobian matrix, which makes the overall solution approach ideally suited for large test cases with many constraints and unknown parameters. We present results for several illustrative examples related to network flow, for which we visualize the solution manifolds of the regularized inverse problem. This provides an intuitive explanation of the hierarchical approach to reduce the ambiguity of the solution. Furthermore, we confirm that combining our regularization strategy with a Gauss-Newton method results in an order of magnitude lower computational cost compared to a gradient descent algorithm. This highlights the potential of our regularization strategy in combination with the Gauss-Newton method, which likely is beneficial for many comparable inverse problems, especially with large parameter spaces. • We present a fast convergence strategy for inverse problems with ambiguous solutions. • A hierarchical regularization approach is used to reduce the solution ambiguity. • We show that the Gauss-Newton method is optimally suited for such inverse problems. • We intuitively explain the solution strategy by discussing illustrative examples. • Fast convergence rates are achieved for inverse problems with large parameter spaces. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the generic fold-fold singularity of Filippov systems on the plane, its unfoldings and its Sotomayor–Teixeira regularization. We work with general Filippov systems and provide the bifurcation diagrams of the fold-fold singularity and their unfoldings, proving that, under some generic conditions, is a codimension one embedded submanifold of the set of all Filippov systems. The regularization of this singularity is studied and its bifurcation diagram is shown. In the visible–invisible case, the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding, specially the appearance and disappearance of periodic orbits that are not present in the Filippov vector field. In the case of a linear regularization, we prove that the regularized system is equivalent to a general slow-fast system studied by Krupa and Szmolyan [10] . [ABSTRACT FROM AUTHOR]

We consider the problem of analyzing spatially distributed data characterized by spatial anisotropy. Following a functional data analysis approach, we propose a method based on regression with partial differential regularization, where the differential operator in the regularizing term is anisotropic and is derived from data. We show that the method correctly identifies the direction and intensity of anisotropy and returns an accurate estimate of the spatial field. The method compares favorably to both isotropic and anisotropic kriging, as tested in simulation studies under various scenarios. The method is then applied to the analysis of Switzerland rainfall data. [ABSTRACT FROM AUTHOR]

Traditional force reconstruction techniques require prior knowledge on the force nature to determine the regularization term. When such information is unavailable, the inappropriate term is easily chosen and the reconstruction result becomes unsatisfactory. In this paper, we propose a novel method to automatically determine the appropriate q as in ℓ q regularization and reconstruct the force history. The method incorporates all to-be-determined variables such as the force history, precision parameters and q into a hierarchical Bayesian formulation. The posterior distributions of variables are evaluated by a Metropolis-within-Gibbs sampler. The point estimates of variables and their uncertainties are given. Simulations of a cantilever beam and a space truss under various loading conditions validate the proposed method in providing adaptive determination of q and better reconstruction performance than existing Bayesian methods. [ABSTRACT FROM AUTHOR]

In this paper we consider a degenerate pseudoparabolic equation for the wetting saturation of an unsaturated two-phase flow in porous media with dynamic capillary pressure-saturation relationship where the relaxation parameter depends on the saturation. Following the approach given in [13] the existence of a weak solution is proved using Galerkin approximation and regularization techniques. A priori estimates needed for passing to the limit when the regularization parameter goes to zero are obtained by using appropriate test-functions, motivated by the fact that considered PDE allows a natural generalization of the classical Kullback entropy. Finally, a special care was given in obtaining an estimate of the mixed-derivative term by combining the information from the capillary pressure with the obtained a priori estimates on the saturation. [ABSTRACT FROM AUTHOR]

Dynamic forces reconstruction from vibration data is an ill-posed inverse problem. A standard approach to stabilize the reconstruction consists in using some prior information on the quantities to identify. This is generally done by including in the formulation of the inverse problem a regularization term as an additive or a multiplicative constraint. In the present article, a space-frequency multiplicative regularization is developed to identify mechanical forces acting on a structure. The proposed regularization strategy takes advantage of one’s prior knowledge of the nature and the location of excitation sources, as well as that of their spectral contents. Furthermore, it has the merit to be free from the preliminary definition of any regularization parameter. The validity of the proposed regularization procedure is assessed numerically and experimentally. It is more particularly pointed out that properly exploiting the space-frequency characteristics of the excitation field to identify can improve the quality of the force reconstruction. [ABSTRACT FROM AUTHOR]

Considering the 2D regularized Boussinesq equations with fractional dissipations ( Λ α u , Λ β θ ) and convection terms ( Λ − γ u ⋅ ∇ u , Λ − γ u ⋅ ∇ θ ) , where Λ = − Δ and γ ≥ 0 , we prove the global existence and uniqueness of the solution in two critical cases. The first case has fractional dissipations ( Λ α u , Λ β θ ) , where α + β = 1 − γ , β > 0 , and the second one has particular dissipation ( Λ 1 − γ u , 0 ) . In particular, for the case γ = 0 , we give some decay estimates for ( θ , u ) and the uniform estimate for G independent of time, where G = ∂ 1 u 2 − ∂ 2 u 1 − ∂ 1 Λ − α θ . [ABSTRACT FROM AUTHOR]