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In this paper, we consider the generalized inverse iteration for computing ground states of the Gross–Pitaevskii eigenvector (GPE) problem. For that we prove explicit linear convergence rates that depend on the maximum eigenvalue in magnitude of a weighted linear eigenvalue problem. Furthermore, we show that this eigenvalue can be bounded by the first spectral gap of a linearized Gross–Pitaevskii operator, recovering the same rates as for linear eigenvector problems. With this we establish the first local convergence result for the basic inverse iteration for the GPE without damping. We also show how our findings directly generalize to extended inverse iterations, such as the Gradient Flow Discrete Normalized (GFDN) proposed in [W. Bao and Q. Du, Computing the ground state solution of Bose–Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput.25 (2004) 1674–1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross–Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal.58 (2020) 1744–1772]. Our analysis also reveals why the inverse iteration for the GPE does not react favorably to spectral shifts. This empirical observation can now be explained with a blow-up of a weighting function that crucially contributes to the convergence rates. Our findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

A long range observing systems can be sturdily affected by scintillations. These scintillations are caused by changes in atmospheric conditions. In recent years, various turbulence mitigation approaches for turbulence mitigation have been exhibiting a promising nature. In this paper, we propose an effectual method to alleviate the effects of atmospheric distortion on observed images and video sequences. These sequences are mainly affected through floating air turbulence which can severely degrade the image quality. The existing algorithms primarily focus on the removal of turbulence and provides a solution only for static scenes, where there is no moving entity (real motion). As in the traditional SGL algorithm, the updated frame is iteratively used to correct the turbulence. This approach reduces the turbulence effect. However, it imposes some artifacts on the real motion that blurs the object. The proposed method is an alteration of the existing Sobolev Gradient and Laplacian (SGL) algorithm to eliminate turbulence. It eliminates the ghost artifact formed on moving object in the existing approach. The proposed method alleviates turbulence without harming the moving objects in the scene. The method is demonstrated on significantly distorted sequences provided by OTIS and compared with the SGL technique. The information conveyed in the scene becomes clearly visible through the method on exclusion of turbulence. The proposed approach is evaluated using standard performance measures such as MSE, PSNR and SSIM. The evaluation results depict that the proposed method outperforms the existing state-of-the-art approaches for all three standard performance measures. [ABSTRACT FROM AUTHOR]

We investigate two types of characterizations for anisotropic Sobolev and BV spaces. In particular, we establish anisotropic versions of the Bourgain–Brezis–Mironescu formula, including the magnetic case both for Sobolev and BV functions. [ABSTRACT FROM AUTHOR]

We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradient variable. The approach is mainly based on classical techniques for uniformly parabolic quasilinear equations and on the Lipschitz estimates provided in [S. N. Armstrong and H. V. Tran, Viscosity solutions of general viscous Hamilton–Jacobi equations, Math. Ann.361 (2015) 647–687], as well as on viscosity solution arguments. [ABSTRACT FROM AUTHOR]

The skew mean curvature flow (SMCF), which origins from the study of fluid dynamics, describes the evolution of a codimension two submanifold along its binormal direction. We study the basic properties of the SMCF and prove the existence of a short-time solution to the initial value problem of the SMCF of compact surfaces in Euclidean space ℝ 4 . A Sobolev-type embedding theorem for the second fundamental forms of two-dimensional surfaces is also proved, which might be of independent interest. [ABSTRACT FROM AUTHOR]

We study an asymptotically regular problem of p (x) -Laplacian type with discontinuous nonlinearity in a nonsmooth bounded domain. A global Calderón–Zygmund estimate is established for such a nonlinear elliptic problem with nonstandard growth under the assumption that the associated nonlinearity has a more general kind of the asymptotic behavior near the infinity with respect to the gradient variable. We also address an optimal regularity requirement on the nonlinearity as well as a minimal geometric assumption on the boundary of the domain for the nonlinear Calderón–Zygmund theory in the setting of variable exponent Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev-Maz'ya equation with critical growth: ﹛-Δμ-λμ/|y|2=|u|2*(t)-2u/|y|t+μu in Ω, u=0 on ∂Ω, provided N > 6 + t, where 0 ≤ λ < (k-2)2/4, when k > 2, λ = 0 when k = 2, 2*(t)=2(N-t)/N-2, 0 ≤t <2, x = (y,z) ∈Rk x RN-k, 2 ≤ k 0 and Ω is an open bounded domain in RN, which contains some points x° = (0, z°). [ABSTRACT FROM AUTHOR]

A numerical procedure for solving a class of integro-differential equations of Volterra type using the Sobolev gradient method is presented. Results are compared with those from the variational iteration method (VIM) and Adomian decomposition method (ADM) (Batiha, B., Noorani, M. S. M. and Hashmi, I. [2008] "Numerical solutions of the nonlinear integro-differential equations," Int. J. Open Probl. Compt. Math. 1, 34-42). The capabilities of our codes are briefly described and test results from some examples are presented. [ABSTRACT FROM AUTHOR]

The aim of collaborative clustering is to reveal the common structure of data distributed on different sites. In this paper, we present a formalism of topological collaborative clustering using prototype-based clustering techniques; in particular we formulate our approach using Kohonen's Self-Organizing Maps. Maps representing different sites could collaborate without recourse to the original data, preserving their privacy. We present two different approaches of collaborative clustering: horizontal and vertical. The strength of collaboration (confidence exchange) between each pair of datasets is determined by a parameter, we call coefficient of collaboration, to be estimated iteratively during the collaboration phase using a gradient-based optimization, for both the approaches. The proposed approaches have been validated on several datasets and experimental results have shown very promising performance. [ABSTRACT FROM AUTHOR]

In this paper, we construct the solutions of semilinear parabolic PDEs with singular coefficients and establish the link to solutions of backward stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this contribution, we prove the global existence and uniqueness of a trajectory that globally converges to the minimizer of the Gross-Pitaevskii energy functional for a large class of external potentials. Using the method of Sobolev gradients, we can provide an explicit construction of this minimizing sequence. Based upon this theoretical framework we numerically apply these results to a specific realization of the external potential and illustrate the main benefits of the method of Sobolev gradients, which are high numerical stability and rapid convergence toward the minimizer. [ABSTRACT FROM AUTHOR]

We consider Fokker–Planck type equations on the abstract Wiener space. Under the assumptions that the coefficients have a certain Sobolev regularity and they, together with their gradients and divergences, are exponentially integrable, we establish the existence of solutions to these equations, based on the estimates for solutions to Fokker–Planck equations in the finite-dimensional case. Moreover, the solution is unique if it belongs to the first-order Sobolev space. [ABSTRACT FROM AUTHOR]

Let f be a positive smooth function on a closed Riemann surface (M, g). The f-energy of a map u from M to a Riemannian manifold (N, h) is defined as $$ E_f(u)=\int_Mf|\nabla u|^2dV_g, $$ and its L2-gradient is: $$ \tau_f(u) = f\tau(u)+ \nabla f\cdot\nabla u. $$ We will study the blow-up properties of some approximate f-harmonic map sequences in this paper. For a sequence uk : M → N with |τf(uk)|L2 < C1 and Ef(uk) < C2, we will show that, if the sequence is not compact, then it must blow-up at some critical points of f or some concentrate points of |τf(uk)|2dVg. For a minimizing α-f-harmonic map sequence in some homotopy class of maps from M into N we show that, if the sequence is not compact, the blow-up points must be the minimal point of f and the energy identity holds true. [ABSTRACT FROM AUTHOR]

We study Milnor fibrations of real analytic maps $(\mathbb{R}^n, \underline 0) \stackrel{f}{\rightarrow} (\mathbb{R}^p, 0)$, n ≥ p, with an isolated critical value. We do so by looking at a pencil associated canonically to every such map, with axis V = f-1(0). The elements of this pencil are all analytic varieties with singular set contained in V. We introduce the concept of d-regularity, which means that away from the axis each element of the pencil is transverse to all sufficiently small spheres. We show that if V has dimension 0, or if f has the Thom af-property, then f is d-regular if and only if it has a Milnor fibration on every sufficiently small sphere, with projection map f/|f|. Our results include the case when f has an isolated critical point. Furthermore, we show that if f is d-regular, then its Milnor fibration on the sphere is equivalent to its fibration on a Milnor tube. To prove these fibration theorems we introduce the spherefication map, which is rather useful to study Milnor fibrations. It is defined away from V; one of its main properties is that it is a submersion if and only if f is d-regular. Here restricted to each sphere in ℝn the spherefication gives a fiber bundle equivalent to the Milnor fibration. [ABSTRACT FROM AUTHOR]

Denote points in ℝk × ℝN - k as pairs ξ = (x,y), and assume 2 ≤ k < N. In this paper, we study the problem $$ \left\{\begin{array}{@{}l@{\quad}l@{}} -\Delta v=\lambda|x|^{-2} ~ v+ |x|^{-b}~v^{p-1}\,\, {\rm in}\, \mathbb{R}^N,\, \enskip x\neq 0 \\ v > 0, \end{array}\right. $$ where $p > 2, b = N - p{N - 2\over 2}$ and $\lambda \le ({k-2\over2})^2$, the Hardy constant. Our results are the following: (i) Let $p < 2_{N-k+1}^*$. Then there exists at least an entire cylindrically symmetric solution. (ii) Let $p\le 2_N^*$ and λ ≥ 0. Then any solution v ∈ Lp(ℝN;|x|-bdξ) is cylindrically symmetric. (iii) Let $p < 2^*_N$ and $\lambda\le \left({k-2\over2}\right)^2-{k-1\over p-2}$. Then ground state solutions are not cylindrically symmetric, and therefore there exist at least two distinct entire solutions. We prove also similar results for the degenerate problem $$ \left\{\begin{array}{@{}l@{\quad}l@{}} -\rm div(|x|^{-2\alpha}\nabla u)=|x|^{-\beta p}u^{p-1},\,\, x\neq 0 \\ u > 0, \end{array}\right. $$ namely, for the Euler–Lagrange equations of the Maz'ya inequality with cylindrical weights. [ABSTRACT FROM AUTHOR]

The one-dimensional nonlinear Schrödinger equation (NLSE) has a wide range of applications in nonlinear optical and laser physics. The design of new optical systems requires analysis of the propagation characteristics of pulses in the fiber media. In this paper, the Sobolev gradient technique has been used for numerical solution of the NLSE. We show that gradients in a suitably chosen Sobolev space (Sobolev gradients) can be used in finite difference settings in a computationally efficient way to approximate the profile of a pulse given by the NLSE. [ABSTRACT FROM AUTHOR]

Conjugate gradient methods are efficient to minimize differentiable objective functions in large dimension spaces. Recently, Dai and Yuan introduced a tree-parameter family of nonlinear conjugate gradient methods and show their convergence. However, line search strategies usually bring computational burden. To overcome this problem, in this paper, we study the global convergence of a special case of three-parameter family(the CD-DY family) in which the line search procedures are replaced by fixed formulae of stepsize. [ABSTRACT FROM AUTHOR]

The role of a surface oxygen vacancy in the N2O dissociation over a Cu2O surface was studied using density functional theory — generalized gradient approximation periodic calculations. To perform this study, surface slab model — ${\rm Cu}_{2}{\rm O}(111)-(\sqrt{3} \times \sqrt{3}){\rm R}30^{\circ}$ surface cell was used. The results indicated that N2O adsorption properties were associated with the surface oxygen vacancies. The presence of a surface oxygen vacancy, prepared by removing one OSUF atom from $(\sqrt{3} \times \sqrt{3}){\rm R}30^{\circ}$ supercell, makes the N2O decomposition possible, which was believed to occur at a oxygen vacancy site. The N2O decomposition at the surface oxygen vacant site depends on whether the molecule is adsorbed with the O- or N-end on a vacancy. For the first case, the reaction proceeds spontaneously, which leads to an ejection of N2 from the surface and the rest of the oxygen atoms occupying the vacancy. In the second case, even though there is an elongation of both N–N and N–O bonds, the dissociation is unfeasible due to very small adsorption energy and high activation energy. The role of the defective surface with surface oxygen vacancy is to provide electrons to the adsorbed molecule. [ABSTRACT FROM AUTHOR]

The Cauchy problem for the scalar wave equation in the Kerr geometry is considered, with initial data which is smooth and compactly supported outside the event horizon. A time-independent energy estimate for the outgoing wave is obtained. As an application, the outgoing energy for wave-packet initial data is estimated, uniformly as the support of the initial data is shifted to infinity. The main mathematical tool is the previously derived integral representation of the wave propagator. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝn be a bounded open set. If a sequence fk : Ω → ℝN converges to f in L∞ in a certain "controlled" manner while bounded in W1,p (1 < p < + ∞) or BV, we show that f ∈ W1,p (respectively, f ∈ BV) and ∇fk → ∇f almost everywhere, where ∇fk and ∇f are the usual gradients if fk ∈ W1,p (respectively, the absolutely continuous part of the gradient measures if fk ∈ BV). Our main theorem generalizes results for Lipschitz mappings. We show by an example that when p = 1, the limit of a sequence of increasing functions may fail to be in W1,1 and can even be nowhere C1. [ABSTRACT FROM AUTHOR]

We consider the problem -Δu = &varbar;u[SUP2*] [SUP-2]u + γu in Σ u = 0 on δΣ, where Σ is an open regular subset of &Rdbl; (N ≥ 3), 2[SUP*] = 2N/N-2 is the critical Sobolev exponent and γ is a constant in ]0; γ[SUB1][ where γ[SUB1] is the first eigenvalue of -Δ. In this paper we show that, when N ≥ 4, the problem has at least N/2+ 1 pairs of solutions, improving a result 2 obtained in [4] for N ≥ 6. [ABSTRACT FROM AUTHOR]

Let *[This character cannot be converted to ASCII text] be a smoothly bounded pseudoconvex complex manifold with a family of almost complex structures *[This character cannot be converted to ASCII text], 0 ∈ I. Assume that there is a smooth function λ which is strictly plurisubharmonic with respect to the structure *[This character cannot be converted to ASCII text] in a neighborhood of bM. Then we prove the global regularity for *[This character cannot be converted to ASCII text]-Neumann problem in Sobolev spaces both in space and parameter variables. [ABSTRACT FROM AUTHOR]