Search

We prove that the energy dissipation property of gradient flows extends to semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the p-parabolic extension does not increase the p-norm of the gradient when p > 2 . We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients. These are the first regularity results for vertical maximal functions without convolution structure. [ABSTRACT FROM AUTHOR]

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein. [ABSTRACT FROM AUTHOR]

In this paper, we exploit necessary/sufficient optimality conditions for ε-quasi positively properly efficient solutions of the semi-infinite multiobjective optimization problems with data uncertainty. We also consider Wolfe type dual problems/Mond--Weir type dual problems under the assumptions of generalized convexity. Finally, several illustrative examples are also provided. [ABSTRACT FROM AUTHOR]

This paper concerns with convergence properties of an inertial proximal algorithm that contains a Tikhonov term regularization, time scale parameter, and a Hessian-driven damping in a Hilbert space. More precisely, we prove the strong convergence of the proximal algorithm obtained by temporal discretization of a continuous dynamic that we treated earlier in a previous work. We also obtain the convergence of the values to the global minimum of the objective function, and a strong convergence of the gradient and the velocity towards zero. Finally, we present a numerical example to illustrate our results. [ABSTRACT FROM AUTHOR]

We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross--Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross--Pitaevskii energy functional with respect to the H1 0 -metric and two other equivalent metrics on H1 0, including the iterate-independent a0-metric and the iterate-dependent au-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross--Pitaevskii energy for the discrete-time H1 and a0-gradient flow. We also prove local exponential convergence of all three schemes to the ground state. [ABSTRACT FROM AUTHOR]

In this article, we consider the Sobolev boundedness for the commutators of fractional new maximal operators both in the global and local settings. More precisely, in the global case, we prove that M b , φ , β and [ b , M φ , β ] are bounded on Sobolev space W 1 , q (R n) ; in the local case, the boundedness for M b , φ , β , Ω and [ b , M φ , β , Ω ] on Sobolev space W 1 , q (Ω) and Sobolev space with zero boundary values W 0 1 , q (Ω) is given. It is worth noting that the analysis and calculation for the commutators of fractional new maximal operators in the local case are more complex than those in the global case. Moreover, the pointwise gradient inequalities of the above commutators will also be showed. [ABSTRACT FROM AUTHOR]

We expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 gradient descent or other optimization methods. [ABSTRACT FROM AUTHOR]

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]

In this paper, we generalize a classical comparison result for solutions to Hamilton-- Jacobi equations with Dirichlet boundary conditions, to solutions to Hamilton--Jacobi equations with non-zero boundary trace. As a consequence, we prove the isoperimetric inequality for the torsional rigidity (with Robin boundary conditions) and for other functionals involving such boundary conditions. [ABSTRACT FROM AUTHOR]

The gradient based optimisation algorithms combined with the finite volume or element based adjoint approaches have been very successful in aerodynamic shape optimization (ASO). The meshfree least squares kinetic upwind method (LSKUM), which works on a cloud of points and its connectivity, has the advantage in terms of flexibility of generating a cloud of points and having an arbitrary number of points in the connectivity. The LSKUM based meshfree solvers have been successfully applied to problems involving multi-body configurations and moving boundaries. In the present work, the LSKUM based primal and discrete adjoint meshfree solvers for steady Euler flows have been used to perform aerodynamic shape optimisation. It is well-known that the raw shape sensitivities of a defined objective function are highly oscillatory. In this research, we have used the Sobolev gradient smoothing. An approach is developed to find a suitable choice for the smoothing parameter in the Sobolev gradient algorithm. Numerical results are presented for the shape optimisation of the NACA0012 airfoil at transonic and supersonic flows with aerodynamic and geometric constraints. The performance of the discrete adjoint LSKUM solver in accurate computation of shape sensitivities is demonstrated. The advantages of using LSKUM in aerodynamic shape optimisation have been shown. [ABSTRACT FROM AUTHOR]

26 F. Seuffert, An extension of the Bianchi-Egnell stability estimate to Bakry, Gentil, and Ledoux's generalization of the Sobolev inequality to continuous dimensions, J. Funct. Morrey's inequality, calculus of variations, extremal, Euler-Lagrange, p-Laplacian, functional inequality, p-Laplace, 49J40, 31C45, nonlinear potential theory, stability estimate, 35J62 Keywords: Morrey's inequality; nonlinear potential theory; calculus of variations; extremal; Euler-Lagrange; p-Laplacian; functional inequality; stability estimate; p-Laplace; 35J62; 49J40; 31C45 EN Morrey's inequality nonlinear potential theory calculus of variations extremal Euler-Lagrange p-Laplacian functional inequality stability estimate p-Laplace 35J62 49J40 31C45 863 877 15 10/03/22 20221001 NES 221001 1 Introduction xtremals, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4335-4347. [Extracted from the article]

Recently, we have studied the h-Matsumoto change and obtained Cartan connection coefficient for the transformed space1. In this paper, we have derived the necessary and sufficient condition for the h-Matsumoto change to be projective and also obtained that there is no non-trivial projective h-Matsumoto change such that the h-vector 𝑏𝑖 is gradient. [ABSTRACT FROM AUTHOR]

Based on density functional theory, the ground state for ultracold Fermi gases with dipole–dipole interaction is a functional minimization problem with orthonormality constraints. Many methods such as direct minimization, self-consistent iteration method and first-order gradient flow had been proposed to solve such problem. In this paper, we introduce the second-order Sobolev gradient flows, solve them to the steady state and get the ground state of ultracold Fermi gases numerically. Two contributions have been made: firstly we extend the usual first-order gradient flows to the second-order gradient flows; secondly, we extend the usual L 2 gradient to the Sobolev gradient and get the Sobolev gradient flows. We prove that the second-order Sobolev gradient flows have the properties of orthonormality preserving and 'modified' energy diminishing, which are desirable for the computation of the ground state solution. Several numerical techniques have been used to discretize the time-dependent second-order Sobolev gradient flows. These include explicit leap-frog method and stabilized semi-implicit method for temporal discretization and Fourier pseudo-spectral method for spatial variables. Extensive numerical examples for the ground state are reported to demonstrate the power of the numerical methods. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that the following form of local multilinear fractional new maximal operator M φ , β , Ω (f →) (x) = sup 0 < r < dist (x , Ω c) ⁡ r β (1 + r n) γ β n φ (| B (x , r) |) m | B (x , r) | m × ∏ i = 1 m ∫ B (x , r) | f i (y) | d y is bounded from W 1 , p 1 (Ω) × W 1 , p 2 (Ω) × ⋯ × W 1 , p m (Ω) into W 1 , q (Ω) by applying the pointwise gradient inequalities for M φ , β , Ω , and it is also bounded on Sobolev space W 1 , q (Ω) as f i ∈ L p i (Ω). Meanwhile, the pointwise gradient estimate and the Sobolev boundedness for the local multilinear new maximal operator M φ , Ω are given. Moreover, the authors also investigate the continuity of M φ , β , Ω and M φ , Ω on Sobolev space W 1 , q (Ω). As applications, the bounds for M φ , Ω and M φ , β , Ω on Sobolev space with zero boundary values W 0 1 , q (Ω) will be showed. [ABSTRACT FROM AUTHOR]

Wc study the Sobolcv regularity of p-harmonic functions. Wc show that |D-z|, -, Du belongs to the Sobolev space ITl-2, s > --1 -- for any p-harmonic function u. The proof is based on an elementary inequality. [ABSTRACT FROM AUTHOR]

Aiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the W 3 / 2 , 2 -inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on L 2 -gradients. [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]

In this paper we study the global well-posedness of the Allen-Cahn-Ohta-Nakazawa model with two fixed nonlinear volume constraints. Utilizing the gradient flow structure of its free energy, we prove the existence and uniqueness of the solution by following De Giorgi's minimizing movement scheme in a novel way. [ABSTRACT FROM AUTHOR]

We present a framework for real-time 3D reconstruction of non-rigidly moving surfaces captured with a single RGB-D camera. Based on the variational level set method, it warps a given truncated signed distance field (TSDF) to a target TSDF via gradient flow without explicit correspondence search. We optimize an energy that contains a data term which steers towards voxel-wise alignment. To ensure geometrically consistent reconstructions, we develop and compare different strategies, namely an approximately Killing vector field regularizer, gradient flow in Sobolev space and newly devised accelerated optimization. The underlying TSDF evolution makes our approach capable of capturing rapid motions, topological changes and interacting agents, but entails loss of data association. To recover correspondences, we propose to utilize the lowest-frequency Laplacian eigenfunctions of the TSDFs, which encode inherent deformation patterns. For moderate motions we are able to obtain implicit associations via a term that imposes voxel-wise eigenfunction alignment. This is not sufficient for larger motions, so we explicitly estimate voxel correspondences via signature matching of lower-dimensional eigenfunction embeddings. We carry out qualitative and quantitative evaluation of our geometric reconstruction fidelity and voxel correspondence accuracy, demonstrating advantages over related techniques in handling topological changes and fast motions. [ABSTRACT FROM AUTHOR]

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman's entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument. [ABSTRACT FROM AUTHOR]

Mahavier and Montgomery construct a Sobolev space for approximate solution of linear initial value problems in a finite difference setting in single-iteration sobolev descent for linear initial value problems, Mahavier, Montgomery, MJMS, 2013. Their Sobolev space is constructed so that gradient-descent converges to a solution in a single iteration, demonstrating the existence of a best Sobolev gradient for finite difference approximation of solutions of linear initial value problems. They then ask if there is a broader class of problems for which convergence in a single iteration in an appropriate Sobolev space occurs. We use their results to show the existence of single-step iteration to solution in a lower dimensional Sobolev space for their examples and then a class of problems for single-step convergence. [ABSTRACT FROM AUTHOR]

This work focuses on the application of a two-way approach, where Molecular Dynamics (MD) simulations and the Square Gradient Theory (SGT) have been used for describing the phase and interface behavior of binary and ternary Lennard-Jones (LJ) mixtures, along a condition of three-phase equilibrium. The unequivocal correspondence between MD and SGT has been achieved by using the global phase diagram of binary mixtures composed by equally sized Lennard-Jones molecules, from which representative molecular parameters for Type-I, Type-II, and Type-III systems have been determined. The so selected binaries have been used then to scale the behavior of a ternary mixture characterized by complex phase equilibrium patterns. For the case of the theoretical SGT approach applied to the Lennard-Jones equation of state was used for predicting phase equilibrium and interfacial properties. In addition the corresponding MD simulations of these macroscopic properties have been conducted for the LJ potential by using equivalent molecular parameters and conditions than in the theoretical approach. Excellent agreement has been observed between the predictions obtained from theory and simulations. Particularly, our results concerning the characterization of the three phase line of a binary Type-III mixture indicate that the bulk liquid (α) and the bulk gas (G) regions are sharply separated by a bulk liquid region (β) for all the explored temperature, pressure, and concentration conditions. The structural analysis of these bulk phases reveals that a secondary liquid phase (β) perfectly wets the liquid-gas interface (α-G), as previously found for Type-II mixture [A. Mejía and L. F. Vega, J. Chem. Phys. 124, 244505 (2006)]. The exploration along the threephase line for the ternary mixture shows good agreement between SGT and MD. Particularly, we observed the specific influence of a third component in the phase and interface behavior. From all the previous results, we conclude that the SGT applied to an EoS with appropriate mixing rules produces reliable predictions of the properties of ternary mixtures. [ABSTRACT FROM AUTHOR]

We derive and implement the analytic energy gradient for the symmetry Projected Hartree-Fock (PHF) method avoiding the solution of coupled-perturbed HF-like equations, as in the regular unprojected method. Our formalism therefore has mean-field computational scaling and cost, despite the elaborate multi-reference character of the PHF wave function. As benchmark examples, we here apply our gradient implementation to the ortho-, meta-, and para-benzyne biradicals, and discuss their equilibrium geometries and vibrational frequencies. [ABSTRACT FROM AUTHOR]

Transition paths characterize chemical reaction mechanisms. In this paper, we present a new method to find mean reaction paths based on the free energy. A nudged elastic band (NEB) is optimized using gradients and Hessians of the free energy, which are obtained from umbrella integration. The transition state can be refined by a Newton-Raphson search starting from the highest point of the NEB path. All optimizations are done using Cartesian coordinates. Independent molecular dynamics (MD) runs are performed at each image used to discretize the path. This makes the method intrinsically parallel. In contrast to other free energy methods, the algorithm does not become more expensive when including more degrees of freedom in the active space. The method is applied to the alanine-dipeptide as a test case and compared to pathways that have been derived from metadynamics and forward flux sampling. [ABSTRACT FROM AUTHOR]

The purpose of the article is to study the existence, regularity, stabilization and blow-up results of weak solution to the following parabolic (p , q) {(p,q)} -singular equation: ($\mathrm{P}_{t}$) { u t - Δ p ⁢ u - Δ q ⁢ u = ϑ ⁢ u - δ + f ⁢ (x , u) , u > 0 in ⁢ Ω × (0 , T) , u = 0 on ⁢ ∂ ⁡ Ω × (0 , T) , u ⁢ (x , 0) = u 0 ⁢ (x) in ⁢ Ω , \displaystyle{}\left\{\begin{aligned} \displaystyle{}u_{t}-\Delta_{p}u-\Delta_% {q}u&\displaystyle=\vartheta u^{-\delta}+f(x,u),\quad u>0&&\displaystyle% \phantom{}\text{in }\Omega\times(0,T),\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega\times(0,T),\\ \displaystyle u(x,0)&\displaystyle=u_{0}(x)&&\displaystyle\phantom{}\text{in }% \Omega,\end{aligned}\right. where Ω is a bounded domain in ℝ N {\mathbb{R}^{N}} with C 2 {C^{2}} boundary ∂ ⁡ Ω {\partial\Omega} , 1 < q < p < ∞ {1 0 {0<\delta,T>0} , N ≥ 2 {N\geq 2} and ϑ > 0 {\vartheta>0} is a parameter. Moreover, we assume that f : Ω × [ 0 , ∞) → ℝ {f:\Omega\times[0,\infty)\to\mathbb{R}} is a bounded below Carathéodory function, locally Lipschitz with respect to the second variable uniformly in x ∈ Ω {x\in\Omega} and u 0 ∈ L ∞ ⁢ (Ω) ∩ W 0 1 , p ⁢ (Ω) {u_{0}\in L^{\infty}(\Omega)\cap W^{1,p}_{0}(\Omega)}. We distinguish the cases as q-subhomogeneous and q-superhomogeneous depending on the growth of f (hereafter we will drop the term q). In the subhomogeneous case, we prove the existence and uniqueness of the weak solution to problem ( P t {\mathrm{P}_{t}}) for δ < 2 + 1 p - 1 {\delta<2+\frac{1}{p-1}}. For this, we first study the stationary problems corresponding to ( P t {\mathrm{P}_{t}}) by using the method of sub- and supersolutions and subsequently employing implicit Euler method, we obtain the existence of a solution to ( P t {\mathrm{P}_{t}}). Furthermore, in this case, we prove the stabilization result, that is, the solution u ⁢ (t) {u(t)} of ( P t {\mathrm{P}_{t}}) converges to u ∞ {u_{\infty}} , the unique solution to the stationary problem, in L ∞ ⁢ (Ω) {L^{\infty}(\Omega)} as t → ∞ {t\rightarrow\infty}. For the superhomogeneous case, we prove the local existence theorem by taking help of nonlinear semigroup theory. Subsequently, we prove finite time blow-up of solution to problem ( P t {\mathrm{P}_{t}}) for small parameter ϑ > 0 {\vartheta>0} in the case δ ≤ 1 {\delta\leq 1} and for all ϑ > 0 {\vartheta>0} in the case δ > 1 {\delta>1}. Moreover, we prove higher Sobolev integrability of the solution to purely singular problem corresponding to the steady state of ( P t {\mathrm{P}_{t}}), which is of independent interest. As a consequence of this, we improve the Sobolev regularity of solution to ( P t {\mathrm{P}_{t}}) for the case δ < 2 + 1 p - 1 {\delta<2+\frac{1}{p-1}}. [ABSTRACT FROM AUTHOR]

We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ (u) = 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ (Γ k) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}}. Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f. [ABSTRACT FROM AUTHOR]

Analytic energy gradients for the orbital-optimized third-order Mo\ller-Plesset perturbation theory (OMP3) [U. Bozkaya, J. Chem. Phys. 135, 224103 (2011)] are presented. The OMP3 method is applied to problematic chemical systems with challenging electronic structures. The performance of the OMP3 method is compared with those of canonical second-order Mo\ller-Plesset perturbation theory (MP2), third-order Mo\ller-Plesset perturbation theory (MP3), coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)] for investigating equilibrium geometries, vibrational frequencies, and open-shell reaction energies. For bond lengths, the performance of OMP3 is in between those of MP3 and CCSD. For harmonic vibrational frequencies, the OMP3 method significantly eliminates the singularities arising from the abnormal response contributions observed for MP3 in case of symmetry-breaking problems, and provides noticeably improved vibrational frequencies for open-shell molecules. For open-shell reaction energies, OMP3 exhibits a better performance than MP3 and CCSD as in case of barrier heights and radical stabilization energies. As discussed in previous studies, the OMP3 method is several times faster than CCSD in energy computations. Further, in analytic gradient computations for the CCSD method one needs to solve λ-amplitude equations, however for OMP3 one does not since λabij(1)=tijab(1) and λabij(2)=tijab(2). Additionally, one needs to solve orbital Z-vector equations for CCSD, but for OMP3 orbital response contributions are zero owing to the stationary property of OMP3. Overall, for analytic gradient computations the OMP3 method is several times less expensive than CCSD (roughly ∼4-6 times). Considering the balance of computational cost and accuracy we conclude that the OMP3 method emerges as a very useful tool for the study of electronically challenging chemical systems. [ABSTRACT FROM AUTHOR]

The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev- Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. Specifically, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the Ws;1-seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces. [ABSTRACT FROM AUTHOR]

This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

In this work, we proposed a new scheme based on Sobolev gradient approach for finding an approximate polynomial solution of certain first and second order ordinary differential equations. Continuous function instead of discretized differential operator is used to avoid numerical issues posed by the size of grid. For example, a simple first order equation is solved using different polynomial basis functions to illustrate the effectiveness of the algorithm. Then the theory of weighted Sobolev gradients is used for the singular Legendre's equation. Numerical experiments indicate that the new algorithm is more efficient than the previous algorithms discussed in the literature on the subject. [ABSTRACT FROM AUTHOR]

The paper concerns the optimal control problem with the full-range integral performance criterion for the nonlinear Schrödinger equation with the specific gradient summand and the complex potential when the performance criterion is the full-range integral. In this paper, the existence and uniqueness theorems regarding the solution of the optimal control problem under consideration are proven. [ABSTRACT FROM AUTHOR]

We study an inverse problem associated with an eddy current model. We first address the ill-posedness of the inverse problem by proving the compactness of the forward map with respect to the conductivity and the nonuniqueness of the recovery process. Then by virtue of nonradiating source conceptions, we establish a regularity result for the tangential trace of the true solution on the boundary, which is necessary to justify our subsequent mathematical formulation. After that, we formulate the inverse problem as a constrained optimization problem with an appropriate regularization and prove the existence and stability of the regularized minimizers. To facilitate the numerical solution of the nonlinear nonconvex constrained optimization, we introduce a feasible Lagrangian and its discrete variant. Then the gradient of the objective functional is derived using the adjoint technique. By means of the gradient, a nonlinear conjugate gradient method is formulated for solving the optimization system, and a Sobolev gradient is incorporated to accelerate the iterative process. Numerical examples are provided to demonstrate the feasibility of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We propose a new normalized Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in H¹ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem. [ABSTRACT FROM AUTHOR]

Accurate brain tumor segmentation plays a significant role in the area of radiotherapy diagnosis and in the proper treatment for brain tumor detection. Typically, the brain tumor has poor boundary and low contrast between normal and lesion soft tissues that makes segmentation of brain tumor in the CT images a challenging task. This paper presents a novel approach to brain image segmentation using pulse-coupled neural network (PCNN) and zero level set (ZL) with Sobolev gradient (SG) method. In this article, PCNN is designed to use as an edge mapper to provide a regional description for the ZL to segregate the CT images based on contour maps. The PCNN is used to estimate the exact threshold to obtain the prominent edges of the images. Resulting edges are utilized in the ZL to extract image contour from the source image. Due to the over-sensitivity of the ZL method on the initial contour, a level set with the SG has been equipped to overcome the limitation of the ZL method. The experimental results show satisfactory segmentation outcomes with excellent accuracy and acceleration in comparison with the state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Given any structure, we seek to find the solution of mechanical problem as precisely and efficiently as possible. Within this condition, the BEM is robust and promising development, standing out in the analysis of cases with occurrence of large stress gradients, as in problems of fracture mechanics. Moreover, in BEM the modeling of infinite means is performed naturally, without the use of approximations. For methods involving domain integration, such as FEM, this is not possible, as models with high number of elements are usually adopted and their ends are considered flexible supports. This paper deals with the development of numerical models based on the BEM for mechanical analysis of stiffened domains. In the modeling of hardeners, immersed in a medium defined by the BEM, we tried to use both the FEM, already present in the literature, and the BEM 1D, being a new formulation. The objective was to perform the coupling of BEM with FEM and BEM 1D for elements of any degree of approximation, evaluating both isotropic and anisotropic medium. In addition, a complementary objective was to verify the effects of the adoption of different discretization and approximation degrees on the stiffeners. However, the coupling with the BEM 1D leaded to more stable results and better approximations. It was observed that the FEM results were instable for many results, mainly in the quadratic approximations. When the approximation degree rises, the methods tend to converge to equivalent results, becoming very close in fourth degree approximation. Lastly, it was observed stress concentration in the stiffeners ends. In these regions, the discretization and the approximation degree have large influence to the numerical response. [ABSTRACT FROM AUTHOR]

We show the existence of an energetic solution to a model of shape-memory alloys in which the elastic energy is described by means of a gradient polyconvex functional. This allows us to show the existence of a solution based on weak continuity of nonlinear minors of deformation gradients in Sobolev spaces. Admissible deformations do not necessarily have integrable second derivatives. Under suitable assumptions, our model allows for solutions which are orientation preserving and globally injective everywhere in the domain representing the specimen. [ABSTRACT FROM AUTHOR]

The Sobolev gradient approach is an efficient way to construct preconditioned iterations for solving nonlinear problems. We extend this technique to be applicable for elliptic equations describing stationary states of reaction–diffusion problems if the nonlinearities have certain lack of differentiability. We derive convergence results of the Sobolev gradient method on an abstract level and then for our elliptic problem under different assumptions. Numerical tests show convergence as expected. [ABSTRACT FROM AUTHOR]

The predicted structures and electronic properties of CeO2 and Ce2O3 have been studied using conventional and hybrid density functional theory. The lattice constant and bulk modulus for CeO2 from local (LSDA) functionals are in good agreement with experiment, while the lattice parameter from a generalized gradient approximation (GGA) is too long. This situation is reversed for Ce2O3, where the LSDA lattice constant is much too short, while the GGA result is in reasonable agreement with experiment. Significantly, the screened hybrid HSE functional gives excellent agreement with experimental lattice constants for both CeO2 and Ce2O3. All methods give insulating ground states for CeO2 with gaps for the 4f band lying between 1.7 eV (LSDA) and 3.3 eV (HSE) and 6–8 eV for the conduction band. For Ce2O3 the local and GGA functionals predict a semimetallic ground state with small (0–0.3 eV) band gap but weak ferromagnetic coupling between the Ce+3 centers. By contrast, the HSE functional gives an insulating ground state with a band gap of 3.2 eV and antiferromagnetic coupling. Overall, the hybrid HSE functional gives a consistent picture of both the structural and electronic properties of CeO2 and Ce2O3 while treating the 4f band consistently in both oxides. [ABSTRACT FROM AUTHOR]

The analytic energy gradients of the optimized effective potential (OEP) method in density-functional theory are developed. Their implementation in the direct optimization approach of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)] and Wu and Yang [J. Theor. Comput. Chem. 2, 627 (2003)] are carried out and the validity is confirmed by comparison with corresponding gradients calculated via numerical finite difference. These gradients are then used to perform geometry optimizations on a test set of molecules. It is found that exchange-only OEP (EXX) molecular geometries are very close to the Hartree-Fock results and that the difference between the B3LYP and OEP-B3LYP results is negligible. When the energy is expressed in terms of a functional of Kohn-Sham orbitals, or in terms of a Kohn-Sham potential, the OEP becomes the only way to perform density-functional calculations and the present development in the OEP method should play an important role in the applications of orbital or potential functionals. [ABSTRACT FROM AUTHOR]

Carbon nanotubes are shown to be modulators of ionic gradients possibly impacting artificial nanophysiology. In contrast with wider nanotubes, the ion traffic inside narrow nanotubes (∼9.45 Å diameter) eventually reaches steady stoichiometries which may be dramatically different from bulk ratios. The narrow nanotube reduces the polarizability of confined water so that charge imbalances inside cannot be sustained. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We conduct finite element analysis to investigate the effect of sharp topography on surface ground deformation caused by pressure changes in a magma reservoir. Tilt data express the horizontal gradient of vertical displacement and therefore can emphasize small variations in deformation that go unnoticed using other methods. We find that the vertical displacement profile at a surface with a cliff can be thought of as the superposition of the deformation from shallow and deeper sources. This combination can create a small peak in vertical displacement that acts as a pseudo‐source, creating a reversal of the deformation gradient and therefore anomalous tilt magnitude and a rotation of up to 180°. We apply these models to Kīlauea Caldera and find that surface geometry creates a tilt rotation of ∼10°, partially explaining anomalous tilt that has been observed. Our analysis highlights the importance of considering topography when assessing tilt measurements at active volcanoes. Key Points: Finite element analysis is used to investigate the effect of sharp topography on deformationTilt magnitude and azimuth are drastically affected by the presence of a cliffAnomalous tilt at Kīlauea can be partially explained by the caldera topography [ABSTRACT FROM AUTHOR]

In this paper we present a complete iteration complexity analysis of inexact first-order Lagrangian and penalty methods for solving cone-constrained convex problems that have or may not have optimal Lagrange multipliers that close the duality gap. We first assume the existence of optimal Lagrange multipliers and study primal-dual first-order methods based on inexact information and augmented Lagrangian smoothing or Nesterov-type smoothing. For inexact (fast) gradient augmented Lagrangian methods, we derive an overall computational complexity of projections onto a simple primal set in order to attain an ε-optimal solution of the conic convex problem. For the inexact fast gradient method combined with Nesterov-type smoothing, we derive computational complexity projections onto the same set. Then, we assume that optimal Lagrange multipliers might not exist for the cone-constrained convex problem, and analyse the fast gradient method for solving penalty reformulations of the problem. For the fast gradient method combined with penalty framework, we also derive an overall computational complexity of projections onto a simple primal set to attain an ε-optimal solution for the original problem. [ABSTRACT FROM AUTHOR]

Abstract We consider the Griffith fracture model in two spatial dimensions, and prove existence of strong minimizers, with closed jump set and continuously differentiable deformation fields. One key ingredient, which is the object of the present paper, is a generalization to the vectorial situation of the decay estimate by De Giorgi, Carriero, and Leaci. This is based on replacing the coarea formula by a method to approximate S B D p functions with small jump set by Sobolev functions, and is restricted to two dimensions. The other two ingredients will appear in companion papers and consist respectively in regularity results for vectorial elliptic problems of the elasticity type and in a method to approximate in energy G S B D p functions by S B V p ones. [ABSTRACT FROM AUTHOR]

Abstract Given a connected Riemannian manifold N , an m -dimensional Riemannian manifold M which is either compact or the Euclidean space, p ∈ [ 1 , + ∞) and s ∈ (0 , 1 ] , we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space W s , p (M , N) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach–Steinhaus uniform boundedness principle in linear Banach spaces. [ABSTRACT FROM AUTHOR]