Search

In this paper, we study an unconstrained l2,p minimization and its associated iteratively reweighted least squares algorithm (UBIRLS) for recovering block sparse signals. Wang et al. [Y. Wang, J. Wang and Z. Xu, On recovery of block-sparse signals via mixed l2/lq (0 < q ≤ 1) norm minimization, EURASIP J. Adv. Signal Process. 2013(76) (2013) 76] have used numerical experiments to show the remarkable performance of UBIRLS algorithm for recovering a block sparse signal, but no theoretical analysis such as convergence and convergence rate analysis of UBIRLS algorithm was given. We focus on providing convergence and convergence rate analysis of UBIRLS algorithm for block sparse recovery problem. First, the convergence of UBIRLS is proved strictly. Second, based on the block restricted isometry property (block RIP) of linear measurement matrix A, we give the error bound analysis of the UBIRLS algorithm. Lastly, we also characterize the local convergence behavior of the UBIRLS algorithm. The simplicity of UBIRLS algorithm, along with the theoretical guarantees provided in this paper, will make a compelling case for its adoption as a standard tool for block sparse recovery. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global existence of small Gevrey-2 solutions to the 3D axially symmetric Prandtl equations. The index 2 is the optimal index for well-posedness result in smooth Gevrey function spaces for data without monotonic assumptions. The novelty of our paper lies in two aspects: one is the tangentially weighted energy construction to match the r weight in the incompressibility and the other is introducing of the new linearly good unknowns to obtain the fast decay of the lower order Gevrey-2 norms of the solutions and auxiliary functions. [ABSTRACT FROM AUTHOR]

In this paper, we present a sharper version of the results in the paper [H. N. Mhaskar, Dimension independent bounds for general shallow networks, Neural Netw. 123 (2020) 142–152]. Let and be compact metric spaces. We consider approximation of functions of the form x ↦ ∫ G (x , y) d τ (y) , x ∈ , by G -networks of the form x ↦ ∑ k = 1 n a k G (x , y k) , y 1 , ... , y n ∈ , a 1 , ... , a n ∈ ℝ. Defining the dimensions of and in terms of covering numbers, we obtain dimension independent bounds on the degree of approximation in terms of n , where also the constants involved are all dependent at most polynomially on the dimensions. Applications include approximation by power rectified linear unit networks, zonal function networks, certain radial basis function networks as well as the important problem of function extension to higher-dimensional spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the following biharmonic elliptic problem involving slightly subcritical non-power nonlinearity {Δ2u = |u|2∗−2u [ln(e+|u|)]휖inΩ,Δu = u = 0 on∂Ω, where Ω ⊂ ℝn is a bounded smooth domain, 2∗ = 2n n−4, n ≥ 5, 휖 > 0 is a small parameter. By Lyapunov–Schmidt procedure, under suitable assumptions on the Robin function related to Ω, we construct the multi-peak solutions which blow-up and concentrate in different points of Ω as 휖 goes to 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with a linear mapping. Besides, we are interested in its application to the graph Fourier basis problem. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many favorable properties. Thanks to this and the scheme for updating the smoothing parameter, we show that any accumulation point of the solution sequence generated by the proposed algorithms is a stationary point of the original optimization problem. Numerical experiments on building graph Fourier basis are conducted to demonstrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we show that the radius of analyticity σ (t) of solutions to the one-dimensional nonlinear Schrödinger (NLS) equation i ∂ t u + ∂ x 2 u = | u | p − 1 u is bounded from below by c | t | − 2 3 when p > 3 and by c | t | − 4 5 when p = 3 as | t | → + ∞ , given initial data that is analytic with fixed radius. This improves results obtained by Tesfahun [On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations 263(11) (2017) 7496–7512] for p = 3 and Ahn et al. [On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 40(1) (2020) 423–439] for any odd integers p > 3 , where they obtained a decay rate σ (t) ≥ c | t | − 1 for larger t. The proof of our main theorems is based on a modified Gevrey space introduced in [T. T. Dufera, S. Mebrate and A. Tesfahun, On the persistence of spatial analyticity for the beam equation, J. Math. Anal. Appl. 509(2) (2022) 126001], the local smoothing effect, maximal function estimate of the Schrödinger propagator, a method of almost conservation law, Schrödinger admissibility and one-dimensional Sobolev embedding. [ABSTRACT FROM AUTHOR]

This paper deals with a study on an initial-boundary value problem for incompressible non-Newtonian fluids of degree two, in a bounded and simply connected open set Ω of ℝ3. It will be shown that the global-in-time weak solution is uniformly stable (i.e. the behavior of the solution changes continuously with the data), and consequently this solution is unique. Furthermore, an exponential decay rate for the energy of the weak solution will also be established. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in Hk(k ≥ 2) in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a determinantal point process on the interval (−s,s), associated with the confluent hypergeometric kernel. Let 풦s(α,β) denote the trace class integral operator acting on L2(−s,s) with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant det(I − γ풦s(α,β)) as s → +∞, while simultaneously γ → 1− in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues λk(α,β)(s) of the integral operator 풦s(α,β) as s → +∞. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift–Zhou nonlinear steepest descent method to analyze the related Riemann–Hilbert problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the real interpolation theory for variable Lorentz–Karamata spaces as well as for the corresponding martingale Hardy spaces. As a consequence, the generalization of Doob’s maximal inequality will be proved. Moreover, the atomic decompositions of the variable martingale Hardy–Lorentz–Karamata spaces are also presented. The results obtained here are new even for martingale Hardy–Lorentz–Karamata spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors establish a general (two-weight) boundedness criterion for a pair of functions, (F,f), on ℝn in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz–)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón–Zygmund operators, Littlewood–Paley g-functions, Lusin area functions, Littlewood–Paley gλ∗-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) boundedness criterion, the authors further obtain the (two-weight) boundedness of Riesz transforms, Littlewood–Paley g-functions, and fractional integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on ℝn in the aforementioned function spaces. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study the well-posedness of a diffuse interface model of non-Newtonian two-phase flows in ℝ 3 . First, we prove the local existence of a unique strong solution provided that the initial velocity and initial concentration of two phases are sufficiently regular. Then, by using the energy estimates and standard continuity argument, we show that there exists a unique global strong solution provided that the initial velocity and initial concentration are sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × ℝ. We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that u (x , t) must be constant with respect to x. Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution u (t) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

Distributed learning has attracted considerable attention in recent years due to its power to deal with big data in various science and engineering problems. Based on a divide-and-conquer strategy, this paper studies the distributed robust regression algorithm associated with correntropy losses and coefficient regularization in the scheme of kernel networks, where the kernel functions are not required to be symmetric or positive semi-definite. We establish explicit convergence results of such distributed algorithm depending on the number of data partitions, robustness and regularization parameters. We show that with suitable parameter choices the distributed robust algorithm can obtain the optimal convergence rate in the minimax sense, and simultaneously reduce the computational complexity and memory requirement in the standard (non-distributed) algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of phase retrieval with short-time linear canonical transform (STLCT). The relation between signal, window and their STLCT is provided through Fourier transform. Based on this theorem, a uniqueness result is established for all square integrable functions. For nonseparable real continuous signal, we prove the uniqueness theorems under some weaker conditions. In complex bandlimited and cardinal B-spline spaces, uniqueness results are provided with magnitude-only STLCT. [ABSTRACT FROM AUTHOR]

Deep learning based on deep neural networks has been very successful in many practical applications, but it lacks enough theoretical understanding due to the network architectures and structures. In this paper, we establish some analysis for linear feature extraction by a deep multi-channel convolutional neural networks (CNNs), which demonstrates the power of deep learning over traditional linear transformations, like Fourier, wavelets, redundant dictionary coding methods. Moreover, we give an exact construction presenting how linear features extraction can be conducted efficiently with multi-channel CNNs. It can be applied to lower the essential dimension for approximating a high-dimensional function. Rates of function approximation by such deep networks implemented with channels and followed by fully connected layers are investigated as well. Harmonic analysis for factorizing linear features into multi-resolution convolutions plays an essential role in our work. Nevertheless, a dedicate vectorization of matrices is constructed, which bridges 1D CNN and 2D CNN and allows us to have corresponding 2D analysis. [ABSTRACT FROM AUTHOR]

This paper proposes a novel prediction approach for a semi-functional linear model comprising a functional and a nonparametric component. The study establishes the minimax optimal rates of convergence for this model, revealing that the functional component can be learned with the same minimax rate as if the nonparametric component were known and vice versa. This result can be achieved by using a double-penalized least squares method to estimate both the functional and nonparametric components within the framework of reproducing kernel Hilbert spaces. Thanks to the representer theorem, the approach also offers other desirable features, including the algorithm efficiency requiring no iterations. We also provide numerical studies to demonstrate the effectiveness of the method and validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of learning variational models in the context of supervised learning via risk minimization. Our goal is to provide a deeper understanding of the two approaches of learning of variational models via bilevel optimization and via algorithm unrolling. The former considers the variational model as a lower level optimization problem below the risk minimization problem, while the latter replaces the lower level optimization problem by an algorithm that solves said problem approximately. Both approaches are used in practice, but unrolling is much simpler from a computational point of view. To analyze and compare the two approaches, we consider a simple toy model, and compute all risks and the respective estimators explicitly. We show that unrolling can be better than the bilevel optimization approach, but also that the performance of unrolling can depend significantly on further parameters, sometimes in unexpected ways: While the stepsize of the unrolled algorithm matters a lot (and learning the stepsize gives a significant improvement), the number of unrolled iterations plays a minor role. [ABSTRACT FROM AUTHOR]

Kernel-based learning algorithms have been extensively studied over the past two decades for their successful applications in scientific research and industrial problem-solving. In classical kernel methods, such as kernel ridge regression and support vector machines, an unregularized offset term naturally appears. While its importance can be defended in some situations, it is arguable in others. However, it is commonly agreed that the offset term introduces essential challenges to the optimization and theoretical analysis of the algorithms. In this paper, we demonstrate that Kernel Ridge Regression (KRR) with an offset is closely connected to regularization schemes involving centered reproducing kernels. With the aid of this connection and the theory of centered reproducing kernels, we will establish generalization error bounds for KRR with an offset. These bounds indicate that the algorithm can achieve minimax optimal rates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the various aspects of normal hyperbolic mean curvature flow by LeFloch and Smoczyk. It is remarkable that the equation admits the null condition in three-dimensional case and only satisfies the first null condition when n = 2. Based on the interesting findings, we can obtain the results of global existence of smooth solutions, as well as the stability of hyperplanes under this flow when 2 ≤ n ≤ 7 , which relates to the famous Bernstein theorem. Some explicit solutions for this flow have been also derived. It should be emphasized that the null structures of this hyperbolic mean curvature flow have not been found before. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence and uniqueness of mild solutions for stochastic nonlocal evolution equations with Lévy diffusion operator and nonlocal initial conditions. Based on the continuity of the Lévy semigroup and the technique of the measure of noncompactness, we establish the local existence of mild solutions in C 0 , T ; L q (Ω ; L q (ℝ N)) under some weaker growth conditions. Moreover, we obtain the existence of mild solutions on any finite interval by using the general growth conditions on the nonlinear. Finally, the global existence and uniqueness of mild solutions follow from the additional Lipschitz conditions on nonlinear terms. [ABSTRACT FROM AUTHOR]

The paper is concerned with the change of probability measures μ along non-random probability measure-valued trajectories ν t , t ∈ [ − 1 , 1 ]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map ν ≡ ν 0 ↦ ν t does not exit the state space, for t ∈ [ − 1 , 0 ] or for t ∈ [ 0 , 1 ] , the Radon–Nikodym derivative d μ ∘ ν t / d μ is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness and optimal control problem of a reaction–diffusion system for an epidemic susceptible–exposed–infected–recovered–susceptible mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates u e and u i of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time T of the controlled evolution of the system. More precisely, we search for the optimal u i and u e such that the number of infected plus exposed does not exceed at the final time a threshold value Λ , fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Zakharov–Kuznetsov (ZK) equation in the n -dimensional setting with n ≥ 2 and investigate its smoothness properties. We extend the well-known regularity propagation phenomenon observed in the 2D and 3D cases, where the regularity of the initial data on certain half-spaces propagates with infinite speed, to the case where the regularity of the initial data is measured on a fractional scale. To achieve this, we introduce new localization formulas that enable us to describe the regularity of the solution on a specific class of subsets in Euclidean space. This work provides insights into the regularity behavior of solutions of the ZK equation in higher dimensions and with more general initial data. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear Schrödinger system with fractional derivative and quadratic interaction. Using constrained variational methods, we prove that this problem has a prescribed solution. Our contribution is that we can deal with a case in the supercritical range. [ABSTRACT FROM AUTHOR]

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in H 1 norm for DRM using deep neural networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of the number of training samples. [ABSTRACT FROM AUTHOR]

In this paper, we establish L p -boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions. [ABSTRACT FROM AUTHOR]