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This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ ( 0 , 1 ) {L^{2}(0,1)} , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.

A novel iterative algorithm is introduced for solving the split common fixed point problem with demicontractive operators. Weak and strong convergence of the algorithm is discussed. The step-sizes ...

We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results., 24 pages; Accepted by Inverse Problems and Imaging

The split common fixed point problem (SCFPP) for the demicontractive operators is studied. An iterative viscosity approximation method (VAM) for SCFPP is introduced. It is shown that the sequence g...

Based on sparse information recovery, we develop a new method for locating multiple multiscale acoustic scatterers. Firstly, with the prior information of the scatterers’ shape, we reformulate the location identification problem into a sparse information recovery model which brought the power of sparse recovery method into this type of inverse scattering problems. Specifically, the new model can advance the judgment of the existence of alternative scatterers and, in the meantime, conclude the number and locating of each existing scatterers. Secondly, as well known, the core model (l0-minimization) in sparse information recovery is an NP-hard problem. According to the characteristics of the proposed sparse model, we present a new substitute method and give a detailed theoretical analysis of the new substitute model. Relying on the properties of the new model, we construct a basic algorithm and an improved one. Finally, we verify the validity of the proposed method through two numerical experiments.

We found some errors in this above titled paper (Cui et al., 2018) which should be corrected in this Corrigendum.

Many practical problems in the real world can be formulated as the non-negative l 0 -minimisation problems, which seek the sparsest non-negative signals to underdetermined linear equations. They have been widely applied in signal and image processing, machine learning, pattern recognition and computer vision. Unfortunately, this non-negative l 0 -minimisation problem is non-deterministic polynomial hard (NP-hard) because of the discrete and discontinuous nature of the l 0 -norm. Inspired by the good performances of the fraction function in the authors’ former work, in this paper, the authors replace the l 0 -norm with the non-convex fraction function and study the minimisation problem of the fraction function in recovering the sparse non-negative signal from an underdetermined linear equation. They discuss the equivalence between non-negative l 0 -minimisation problem and non-negative fraction function minimisation problem, and the equivalence between non-negative fraction function minimisation problem and regularised non-negative fraction function minimisation problem. It is proved that the optimal solution to the non-negative l 0 -minimisation problem could be approximately obtained by solving their regularised non-negative fraction function minimisation problem if some specific conditions are satisfied. Then, they propose a non-negative iterative thresholding algorithm to solve their regularised non-negative fraction function minimisation problem. At last, numerical experiments on some sparse non-negative signal recovery problems are reported.

This article investigates a space–time fractional diffusion equation with a nonlinear source and a space nonlocal operator. The well-posedness is demonstrated from using contraction mapping theorem...

Recently, the l p -norm regularization minimization problem ( P p λ ) has attracted great attention in compressed sensing. However, the l p -norm ‖x‖ p p in problem ( P p λ ) is nonconvex and non-Lipschitz for all p ∈ ( 0 , 1 ) , and there are not many optimization theories and methods proposed to solve this problem. In fact, it is NP-hard for all p ∈ ( 0 , 1 ) and λ > 0 . In this paper, we study one modified l p -norm regularization minimization problem to approximate the NP-hard problem ( P p λ ) . Inspired by the good performance of Half algorithm in some sparse signal recovery problems, an iterative thresholding algorithm is proposed to solve our modified l p -norm regularization minimization problem ( P p , 1 ∕ 2 , ϵ λ ) . Numerical results on some sparse signal recovery problems show that our algorithm performs effectively in finding the sparse signals compared with some state-of-art methods.

In this paper, we introduce a two-step inertial primal-dual algorithm (TSIPD) for solving the minimizations of the sum a smooth function with Lipschitzian gradient and two non-smooth convex functions with linear operators. This is a complete splitting approach, in the sense that non-smooth functions are treated separately by their proximity operators. In order to prove the convergence of the TSIPD, we transform the problem into a fixed point equation with good performance, and prove the convergence of the algorithm base on the fixed point theory. This work brings together and significantly extends several classical splitting schemes, like the primal-dual method (PD3O) proposed by Yan, and the recent three-operator splitting scheme proposed by Davis and Yin. The validity of the proposed method is demonstrated on an image denoising problem. Numerical results show that our iterative algorithm (TSIPD) has better performance than the original one (PD3O).

In this work, we introduced an iterative algorithm for finding a common element of the set of common fixed points of a one-parameter nonexpansive semigroup, the set of solutions to a variational inclusion and the set of solutions to a generalized equilibrium problem in a real Hilbert space. Furthermore, we proved the convergence theorem of the proposed iterative algorithm under some mild conditions on algorithm parameters. Finally, we gave some numerical examples which support our main theorem at last part. The results presented in this paper generalize the well-known theorems.

In this paper, we propose a new primal–dual fixed point algorithm with dynamic stepsize ( $$\hbox {PDFP}^{2}O_{DS_{n}}$$ ) for solving convex minimization problems involving the sum of a smooth function with Lipschitzian gradient and the composition of a nonsmooth convex function with a continuous linear operator. Based on modified Mann iteration and the firmly nonexpansive properties of the proximity operator, we achieve the convergence of the proposed algorithm. Moreover, we give the connection of the proposed algorithm with other existing $$\hbox {PDFP}^{2}O$$ (Chen et al. in Inverse Probl 29:025011–025033, 2013). Finally, we illustrate the efficiency of $$\hbox {PDFP}^{2}O_{DS_{n}}$$ through some numerical examples on the CT image reconstruction problem. Numerical results show that our iterative algorithm ( $$\hbox {PDFP}^{2}O_{DS}$$ ) performs better than the original one ( $$\hbox {PDFP}^{2}O$$ ).

In this paper, we first present a modified Chambolle–Pock primal-dual method (MCPPDM) to solve a convex composite optimization problem which minimizes the sum of two convex functions with one composed by a linear operator. It is well known that the Chambolle–Pock primal-dual method (CPPDM) with the combination parameter being 1 is an application of the proximal point algorithm and thus is convergent, however, when the combination parameter is not 1, the method may be not convergent. To choose flexibly the combination parameter, we develop a slightly modified version with little additional computation cost. In CPPDM, one variable is updated twice but another variable is updated only once at each iteration. However, in the modified version, two variables are respectively updated twice at each iteration. Another main task of this paper is that we reformulate some well-known sparse recovery problems as special cases of the convex composite optimization problem and then apply MCPPDM to address these sparse recovery problems. A large number of numerical experiments have demonstrated that the efficiency of the proposed method is generally comparable or superior to that of existing well-known methods such as the linearized alternating direction method of multipliers and the graph projection splitting algorithm in terms of solution quality and run time.

When the finite difference (FD) method is employed to simulate the wave propagation , high-order FD method is preferred in order to achieve better accuracy. However, if the order of FD scheme is high enough, the coefficient matrix of the formula for calculating finite difference coefficients is close to be singular. In this case, when the FD coefficients are computed by matrix inverse operator of MATLAB, inaccuracy can be produced. In order to overcome this problem, we have suggested an algorithm based on Vandermonde matrix in this paper. After specified mathematical transformation, the coefficient matrix is transformed into a Vandermonde matrix. Then the FD coefficients of high-order FD method can be computed by the algorithm of Vandermonde matrix, which prevents the inverse of the singular matrix. The dispersion analysis and numerical results of a homogeneous elastic model and a geophysical model of oil and gas reservoir demonstrate that the algorithm based on Vandermonde matrix has better accuracy compared with matrix inverse operator of MATLAB.

This paper studies algorithms for solving quadratically constrained l 1 minimization and Dantzig selector which have recently been widely used to tackle sparse recovery problems in compressive sensing. The two optimization models can be reformulated via two indicator functions as special cases of a general convex composite model which minimizes the sum of two convex functions with one composed with a matrix operator. The general model can be transformed into a fixed-point problem for a nonlinear operator which is composed of a proximity operator and an expansive matrix operator, and then a new iterative scheme based on the expansive matrix splitting is proposed to find fixed-points of the nonlinear operator. We also give some mild conditions to guarantee that the iterative sequence generated by the scheme converges to a fixed-point of the nonlinear operator. Further, two specific proximal fixed-point algorithms based on the scheme are developed and then applied to quadratically constrained l 1 minimization and Dantzig selector. Numerical results have demonstrated that the proposed algorithms are comparable to the state-of-the-art algorithms for recovering sparse signals with different sizes and dynamic ranges in terms of both accuracy and speed. In addition, we also extend the proposed algorithms to solve two harder constrained total-variation minimization problems.

The problem of recovering a low-rank matrix from partial entries, known as low-rank matrix completion, has been extensively investigated in recent years. It can be viewed as a special case of the affine constrained rank minimization problem which is NP-hard in general and is computationally hard to solve in practice. One widely studied approach is to replace the matrix rank function by its nuclear-norm, which leads to the convex nuclear-norm minimization problem solved efficiently by many popular convex optimization algorithms. In this paper, we propose a new nonconvex approach to better approximate the rank function. The new approximation function is actually the Moreau envelope of the rank function (MER) which has an explicit expression. The new approximation problem of low-rank matrix completion based on MER can be converted to an optimization problem with two variables. We then adapt the proximal alternating minimization algorithm to solve it. The convergence (rate) of the proposed algorithm is proved and its accelerated version is also developed. Numerical experiments on completion of low-rank random matrices and standard image inpainting problems have shown that our algorithms have better performance than some state-of-art methods.

In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.

The joint sparse recovery problem is a generalization of the single measurement vector problem widely studied in compressed sensing. It aims to recover a set of jointly sparse vectors, i.e., those that have nonzero entries concentrated at a common location. Meanwhile $l_{p}$ -minimization subject to matrixes is widely used in a large number of algorithms designed for this problem, i.e., $l_{2,p}$ -minimization $$\begin{aligned} \min_{X \in\mathbb {R}^{n\times r}} \Vert X \Vert _{2,p}\quad \text{s.t. }AX=B. \end{aligned}$$ Therefore the main contribution in this paper is two theoretical results about this technique. The first one is proving that in every multiple system of linear equations there exists a constant $p^{\ast}$ such that the original unique sparse solution also can be recovered from a minimization in $l_{p}$ quasi-norm subject to matrixes whenever $0< p

The success of orthogonal matching pursuit (OMP) in the sparse signal recovery heavily depends on its ability for correct support recovery. Based on a support recovery guarantee for OMP expressed in terms of the mutual coherence, and a result about the concentration of the extreme singular values of a Gaussian random matrix, this paper proposes a preconditioning method for increasing the recovery rate of OMP from random and noisy measurements. Compared to several existing preconditionings, the proposed method can reduce the mutual coherence with a proven high probability. Simultaneously, the proposed preconditioning can also succeed with a high probability in providing slight signal-to-noise ratio reduction, which is empirically shown to be less severe than that caused by a recently suggested technique for the noisy case. The simulations show the advantages of the proposed preconditioning over other currently relevant ones in terms of both the performance improvement for OMP, and computation time.

Let P be a convex polygon with n vertices. We consider a variation of the K-center problem called the connected disk covering problem (CDCP), i.e., finding K congruent disks centered in P whose union covers P with the smallest possible radius, while a connected graph is generated by the centers of the K disks whose edge length can not exceed the radius. We give a 2.81-approximation algorithm in O(Kn) time.

This paper studies a new version of the location problem called the mixed center location problem. Let P be a set of n points in the plane. We first consider the mixed 2-center problem, where one of the centers must be in P, and we solve it in $$O(n^2\log n)$$ time. Second, we consider the mixed k-center problem, where m of the centers are in P, and we solve it in $$O(n^{m+O(\sqrt{k-m})})$$ time. Motivated by two practical constraints, we propose two variations of the problem. Third, we present a 2-approximation algorithm and three heuristics solving the mixed k-center problem ( $$k>2$$ ).

Sparse recovery from indirectly under-sampled or possibly noisy data is a burgeoning topic drawing the attention of many researchers. Since sparse recovery problems can be cast as a class of the constrained convex optimization models which minimize a nonsmooth convex objection function in a convex closed set, fast and efficient methods for solving the constrained optimization models are highly needed. By introducing the indicator functions related to constrained sets in sparse recovery models, we reformulate these models as two general unconstrained optimization problems. To develop fast first-order methods, two smoothing approaches are proposed based on the Moreau envelope: smoothing the related indicator functions or the objection functions. By using the first smoothing approach, we obtain more proper unconstrained models for sparse recovery from noisy data. Fast iterative shrinkage-thresholding algorithm (FISTA) is applied to solve the smoothed models. When smoothing the objection functions, we propose an efficient first-order method based on FISTA to establish the rate of convergence of order O ( log k k ) for the iterative sequence of values of the original objective functions. Numerical experiments have demonstrated that the two proposed smoothing methods are comparable to the state-of-the-art first-order methods with respect to accuracy and speed when applied to the sparse recovery problems such as compressed sensing, matrix completion, and robust and stable principal component analysis.

In this paper, we introduce an iteration scheme for viscosity approximation of zeros of accretive operators and solutions to fixed point problems in q-uniformly smooth Banach spaces with a weakly continuous duality mapping. A new iterative algorithm is introduced and strong convergence of the sequences generated by this algorithm is proved. Our results improve and extend corresponding ones announced by many others.

In this paper, we focus on a space–time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.

Compressive sensing has achieved great success in many scientific research fields. It has revealed that sparse signals can be stably recovered from a small number of noisy measurements by solving the constrained convex $${\ell }_1$$ -minimization problem. In practice, a faster algorithm for solving this optimization problem is the key to compressive sensing. The Douglas–Rachford splitting method is a well-known operator splitting method that has been widely applied for solving a certain class of convex composite problems. In particular, its dual application results in the popular alternating direction method of multipliers (ADMM). In this paper, we reformulate the constrained convex $${\ell }_1$$ -minimization problem as a convex composite problem with a special structure and then apply the primal Douglas–Rachford splitting method to solve it. The computational cost of the developed algorithm in each iteration is dominated by the projection onto the constraint set. A fast and efficient method of computing the projection is proposed. Numerical results show that the developed algorithm performs better than the popular NESTA and LADMM (inexact ADMM) in terms of accuracy and run time for large-scale sparse signal recovery.

In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a, u, B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a, u, B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\|(u, B)\|_{L^{1}(\mathbb{R}^{+};\dot{B}_{2,1}^{5/2})} < C$, which is different to Navier-Stokes equations., arXiv admin note: text overlap with arXiv:1206.6144 by other authors

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a Levy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that positive solutions will blow up in finite time in mean L p -norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrate the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by Levy noise has a global solution.

In this work, we introduce a composite iterative scheme with Meir–Keeler contractions for nonexpansive semigroups in a q-uniformly smooth and strictly convex Banach space which admits a weakly sequentially continuous duality mapping. We also prove strong convergence theorems for the composite iterative schemes. Our results improve and extend several recent results.

In this paper, we are concerned with the optimal time convergence rate of the global strong solution to some constant equilibrium states for the compressible viscoelastic fluids in the whole space. Green’s matrix method and energy estimate method are used to obtain the optimal time decay rate under the critical Besov space framework. Our result implies the optimal -time decay rate and only need the initial datum to be small in some critical Besov space which have very low regularity compared with the classical Sobolev space.

This article discusses the influence of preconditioning on the mutual coherence and the restricted isometry property of Gaussian or Bernoulli measurement matrices. The mutual coherence can be reduced by preconditioning, although it is fairly small due to the probability estimate of the event that it is less than any given number in (0, 1). This can be extended to a set that contains either of the two types of matrices with a high probability but a subset with Lebesgue measure zero. The numerical results illustrate the reduction in the mutual coherence of Gaussian or Bernoulli measurement matrices. However, the first property can be true after preconditioning for a large type of measurement matrices having the property of s-order restricted isometry and being full row rank. This leads to a better estimate of the condition number of the corresponding submatrices and a more accurate error estimate of the conjugate gradient methods for the least squares problems typically used in greedy-like recovery algorithms.

The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell _{0}$\end{document}ℓ0-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho_{a}$\end{document}ρa in this quasi-linear compressed sensing. We propose an iterative fraction thresholding algorithm to solve the regularization problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(QP_{a}^{\lambda})$\end{document}(QPaλ) for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}a>0. With the change of parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a>0$\end{document}a>0, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods.

In signal processing theory, $l_{0}$ -minimization is an important mathematical model. Unfortunately, $l_{0}$ -minimization is actually NP-hard. The most widely studied approach to this NP-hard problem is based on solving $l_{p}$ -minimization ( $0< p\leq1$ ). In this paper, we present an analytic expression of $p^{\ast}(A,b)$ , which is formulated by the dimension of the matrix $A\in\mathbb{R}^{m\times n}$ , the eigenvalue of the matrix $A^{T}A$ , and the vector $b\in\mathbb{R}^{m}$ , such that every k-sparse vector $x\in\mathbb{R}^{n}$ can be exactly recovered via $l_{p}$ -minimization whenever $0< p< p^{\ast}(A,b)$ , that is, $l_{p}$ -minimization is equivalent to $l_{0}$ -minimization whenever $0< p< p^{\ast}(A,b)$ . The superiority of our results is that the analytic expression and each its part can be easily calculated. Finally, we give two examples to confirm the validity of our conclusions.

This paper is concerned with the well-posedness of a problem for age-dependent delayed population equations describing the birth and death process with account of death caused by pregnancy. The proof of well-posedness is based on the authors’ recent paper [Proceedings of the American Mathematical Society, 138(12)(2010) 4455-4468].

This paper is concerned with developing an operator theory for Riemann-Liouville fractional Cauchy problem. Two notions, named Riemann-Liouville fractional resolvent and solution operator, are developed. Some of their properties are deduced. Moreover, the obtained results are applied to Riemann-Liouville fractional Cauchy problem.

In this paper, we consider the global solutions to a generalized 2D Boussinesq equation \begin{align*} \left \{\begin{aligned} & \partial_{t} \omega + u\cdot \nabla \omega + \nu \Lambda^{\alpha} \omega = \theta_{x_{1}} , \quad \\ & u = \nabla^{\bot} \psi = (-\partial_{x_{2}} , \partial_{x_{1}}) \psi , \quad \Delta \psi = \Lambda^{\sigma} (\log (I-\Delta))^{\gamma} \omega , \quad \\ & \partial_{t} \theta + u\cdot \nabla \theta + \kappa \Lambda^{\beta} \theta = 0, \quad \\ & \omega(x,0) = \omega_{0}(x) , \quad \theta(x,0) = \theta_{0}(x), \end{aligned}\right. \end{align*} with $\sigma \geq 0$, $\gamma \geq 0$, $\nu >0$, $\kappa>0$, $\alpha < 1$ and $\beta < 1$. When $\sigma = 0$, $\gamma \geq 0$, $\alpha \in [0.95,1)$ and $\beta \in (1-\alpha,g(\alpha))$, where $g(\alpha), Comment: Submitted to NODEA about one year ago. arXiv admin note: text overlap with arXiv:0910.0311, arXiv:1111.2082 by other authors

The iterative projection methods for solving the multiple-sets split feasibility problem have been paid much attention in recent years. In this paper, we introduce a cyclic and simultaneous iterative sequence with self-adaptive step size for solving this problem. The advantage of the self-adaptive step size is that it does not need to know the Lipschitz constant of the gradient operator in advance. Furthermore, we propose a relaxed cyclic and simultaneous iterative sequence with self-adaptive step size, respectively. The theoretical convergence results are established in an infinite-dimensional Hilbert spaces setting. Preliminary numerical experiments show that these iteration methods are practical and easy to implement.

In this paper, we are concerned with a class of abstract fractional relaxation equations. We develop a new notion, named fractional resolvent and derive some of its properties. By virtue of the obtained properties and the properties of general Mittag-Leffler function, we present some sufficient conditions to guarantee that the classical solutions of homogeneous and inhomogeneous fractional relaxation equations exist. An illustrative example is presented.

In this paper, we construct local solution with highly oscillating initial velocity and then get the global strong solution in the L p based Besov space which improves the result of J. Qian, Z. Zhang (2010) [25] and X. Hu, D. Wang (2011) [14] . The local existence and uniqueness lies on the Lagrange coordinate transform and the contraction mapping theorem. The global result lies on a decomposition of the system and some commutator estimates. In the last part, we prove a time-decay in the critical Besov space framework which seems to have little investigation. The proof is based on the properties of the Green's matrix and various interpolations between Besov type spaces.

This paper is concerned with a class of dynamic boundary systems with boundary feedback. The well-posedness of the considered systems is proved under some regularity conditions. Moreover, some spectral properties are derived. As an application, the well-posedness and the asymptotic behavior of population dynamical systems with unbounded birth process ‘ ’ are solved. Such population dynamical systems were pointed out in [S. Piazzera, Math. Methods Appl. Sci., 27 (2004), 427-439] to be a current research topic in semigroup theory and still an open problem. Copyright © 2014 John Wiley & Sons, Ltd.

The goal of affine matrix rank minimization problem is to reconstruct a low-rank or approximately low-rank matrix under linear constraints. In general, this problem is combinatorial and NP-hard. In this paper, a nonconvex fraction function is studied to approximate the rank of a matrix and translate this NP-hard problem into a transformed affine matrix rank minimization problem. The equivalence between these two problems is established, and we proved that the uniqueness of the global minimizer of transformed affine matrix rank minimization problem also solves affine matrix rank minimization problem if some conditions are satisfied. Moreover, we also proved that the optimal solution to the transformed affine matrix rank minimization problem can be approximately obtained by solving its regularization problem for some proper smaller $\lambda>0$. Lastly, the DC algorithm is utilized to solve the regularization transformed affine matrix rank minimization problem and the numerical experiments on image inpainting problems show that our method performs effectively in recovering low-rank images compared with some state-of-art algorithms., Comment: arXiv admin note: text overlap with arXiv:1611.07777

Seismic waves in earth materials are subject to attenuation and dispersion in a broad range of frequencies. The commonly accepted mechanism of intrinsic attenuation and dispersion is the presence of fluids in the pore space of rocks. The diffusive-viscous model was proposed to explain low-frequency seismic anomalies related to hydrocarbon reservoirs. But, the model is only a description of compressional wave. In this work, we firstly discuss the extended elastic diffusive-viscous model. Then, we extend reflectivity method to the diffusive-viscous medium. Finally, we present two numerical models to simulate the attenuation of diffusive-viscous wave in horizontal and dip multi-layered media compared with the results of viscoelastic wave. The modeling results show that the diffusive-viscous wave has strong amplitude attenuation and phase shift when it propagates across absorptive layers.

This paper is concerned with the modeling errors appeared in the numerical methods of inverse medium scattering problems (IMSP). Optimization based iterative methods are wildly employed to solve IMSP, which are computationally intensive due to a series of Helmholtz equations need to be solved numerically. Hence, rough approximations of Helmholtz equations can significantly speed up the iterative procedure. However, rough approximations will lead to instability and inaccurate estimations. Using the Bayesian inverse methods, we incorporate the modelling errors brought by the rough approximations. Modelling errors are assumed to be some complex Gaussian mixture (CGM) random variables, and in addition, well-posedness of IMSP in the statistical sense has been established by extending the general theory to involve CGM noise. Then, we generalize the real valued expectation-maximization (EM) algorithm used in the machine learning community to our complex valued case to learn parameters in the CGM distribution. Based on these preparations, we generalize the recursive linearization method (RLM) to a new iterative method named as Gaussian mixture recursive linearization method (GMRLM) which takes modelling errors into account. Finally, we provide two numerical examples to illustrate the effectiveness of the proposed method., Comment: 26 pages

Kernel methods have been popular in machine learning. The Gaussian kernel is a good choice as a basis function. In this paper, a new kernel, the fractional Gaussian kernel, is presented and its positive definiteness is verified.