Sobolev Space Setting Research Articles

A mathematical framework for nonlinear wavefront reconstruction in adaptive optics systems with Fourier-type wavefront sensing

Advanced adaptive optics (AO) instruments have applications in ophthalmic imaging, free-space optical communications and the future generation of extremely large telescopes. These AO systems are designed to perform real-time corrections of dynamic wavefront aberrations. The corrections can be performed by converting wavefront measurements into deformable mirror (DM) actuator commands. The role of the DM is to mitigate aberrations by restoring a planar wavefront. Optimal DM actuator commands therefore require precise phase measurements across the entire wavefront. Reconstructing a wavefront from wavefront sensor (WFS) data is an inverse problem that depends on the type of WFS implemented. Nonlinear Fourier-type WFSs are included in the design of many current and upcoming AO systems. Conventionally, these sensors perform AO control based on simplifications and linearisations of the underlying models. However, in nonlinear regimes, approximation errors critically degrade image quality. This study looks at overcoming nonlinear wavefront sensing regimes by introducing a nonlinear, iterative algorithm for Fourier-type wavefront reconstruction. The algorithm used is well-known in the field of inverse problems. The underlying mathematical theory for modelling Fourier-type WFSs is provided, along with how these models can be used to perform nonlinear wavefront reconstruction. A significant advantage of the analysis presented is its generalised applicability to any Fourier-type sensor. The only input required is the mathematical expression for the optical element transfer function. The generalised and full mathematical model of Fourier-type WFSs is introduced in a Sobolev space setting. Necessary inputs are derived for the nonlinear iterative algorithms, such as Fréchet derivatives and adjoints. The generalised theory is then expanded to solve the inverse problem of wavefront reconstruction for all Fourier-type WFSs. Moreover, the study concentrates on the pyramid WFS (PWFS)—one of the most well-known Fourier-type WFSs—and shows a Hilbert transform representation of the amplitude of the incoming light on its detector. The developed theory is demonstrated using a simulated PWFS to measure an example wavefront.

Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces

This work is a contribution to the ongoing search for algebraic structures within a nonlinear setting. Here, we shall focus on the study of lineability of subsets of continuous functions on the one hand and within the setting of Sobolev spaces on the other (which represents a novelty in the area of research).

On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.

Short proofs for interpolation inequalities in Sobolev spaces with variable exponents

We present very short proofs for three versions of the Gagliardo–Nirenberg inequality in the setting of Sobolev spaces with variable exponents. These are formally expressed by $$ \|\nabla ^k f\|_{L^{r(\cdot )}(\mathbb {R}^d)} \le C(d,p,q,r,k,m) \, \|f\|_{

Anisotropic parabolic problem with variable exponent and regular data

In this paper, we study the existence of weak solutions for a class of nonlinear parabolic equations with regular data in the setting of variable exponent Sobolev spaces. We prove a "version" of a weak Lebesgue space estimate that goes back to "Lions J. L. Quelques méthodes de résolution des problèmes aux limites. Dunod, Paris (1969)" for parabolic equations with anisotropic constant exponents (pi(⋅)=pi).

Strain gradient elasto-plasticity model: 3D isogeometric implementation and applications to cellular structures

In the present work, we combine Mindlin’s strain gradient elasticity theory and Gudmundson–Gurtin–Anand strain gradient plasticity theory to form a unified framework. The gradient plasticity model is enriched by including the gradient of elastic strains into the expression of the internal virtual work and free energy. This augments the modelling capabilities by incorporating elasticity-related length scales along with plasticity-related energetic and dissipative ones. The strong form governing equations are derived via the principle of virtual work addressing a complete set of boundary conditions. The fourth-order boundary value problem of the gradient elasto-plasticity model is then formulated in a variational form within an H2 Sobolev space setting. Conforming Galerkin discretizations for numerical results are obtained utilizing an isogeometric approach with NURBS basis functions of degree p≥2 providing Cp−1-continuity. The implementation follows a viscoplastic constitutive framework and adopts the backward Euler time integration scheme. A set of benchmark examples is considered to illustrate convergence properties and to accomplish parameter studies. It is shown that the elastic length scale parameter controls the slope of the elastic part and causes an additional hardening in the plastic part of the material response curves. Finally, an illustrative example is considered in order to demonstrate the applicability of both the continuum model and the numerical method in capturing the size-dependent torsion response of cellular structures.

Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given.

On strongly quasilinear elliptic systems with weak monotonicity

Abstract In this paper, we prove existence results in the setting of Sobolev spaces for a strongly quasilinear elliptic system by means of Young measures and mild monotonicity assumptions.

Nonlinear elliptic equations with variable exponents involving singular nonlinearity

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and L1 datum in the setting of Sobolev spaces with variable exponents. We will prove that the lower order term has some regularizing effects on the solutions. This work generalizes some results given in [1–3].

The DB Boundary Condition in Electromagnetic Scattering Revisited

For the solution of the scattering problem for time-harmonic electromagnetic waves with boundary conditions for the normal components of both the electric and the magnetic field, an integral equation method proposed by Gülzow in 1988 is reconsidered. It is made more concise, made symmetric with respect to the electric and magnetic field, and also extended from the classical Hölder spaces to a contemporary Sobolev space setting. For this regime, new reciprocity principles for scattering of plane waves and dipole fields are established and a related far field operator is discussed. Finally, the corresponding inverse scattering problem to recover the shape of the scatterer from far field data is discussed with the main emphasis on uniqueness results.

Corrosion shape reconstruction of the mixed boundary in electrostatic imaging

We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green’s representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method.

Long time existence of solutions to an elastic flow of networks

The L2-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.

Local strong solutions to a quasilinear degenerate fourth-order thin-film equation

We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier–Stokes system, the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid’s shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents alpha ge 2 the corresponding initial boundary value problem fits into the abstract setting of Amann (Function Spaces, Differential Operators and Nonlinear Analysis, Vieweg Teubner Verlag, Stuttgart, 1993). Due to a lack of regularity this is not true for flow behaviour exponents alpha in (1,2). For this reason we prove an existence theorem for abstract quasilinear parabolic evolution problems with Hölder continuous dependence. This result provides existence of strong solutions to the non-Newtonian thin-film problem in the setting of fractional Sobolev spaces and (little) Hölder spaces. Uniqueness of strong solutions is derived by energy methods and by using the particular structure of the equation.

On some convergence results for fractional periodic Sobolev spaces

In this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series.

Decay estimates for a dissipative-dispersive linear semigroup and application to the viscous Boussinesq equation

At the core of this article is the new estimate for a class of dissipative-dispersive linear semigroup eΔt±ip(|∇|)t arising in the study of the viscous Boussinesq equation. We combine the decay estimate with introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our decay estimate, to show the global existence and asymptotic behavior of solutions to the viscous Boussinesq equation in Rn.

Filtered spectral differentiation method for numerical differentiation of periodic functions with application to heat flux estimation

Based on a truncated Fourier series technique to compute stable derivatives in a Sobolev space setting, we propose a method for numerical differentiation of periodic functions from a finite amount of noisy data. In our method, we construct stable approximations to high order derivatives by filtering high frequency components of spectral derivatives obtained through the Fourier differentiation matrix. We derive convergence rates for the approximate derivatives with essentially the same accuracy as those obtained by the truncated Fourier series technique. Our method is illustrated with numerical examples, focusing in particular, on the estimation of the heat flux distribution in coiled tubes from experimental temperature data.

Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovereda posterioriby a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateralL2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.

Effective energy integral functionalsfor thin films on curl-free vector fields in the Orlicz–Sobolev space setting

Effective energy integral functionalsfor thin films on curl-free vector fields in the Orlicz–Sobolev space setting

Global gradient estimates for asymptotically regular problems of p(x)-Laplacian type

We study an asymptotically regular problem of [Formula: see text]-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.

Kirchhoff–Love shells within strain gradient elasticity: Weak and strong formulations and an [formula omitted]-conforming isogeometric implementation

A strain gradient elasticity model for shells of arbitrary geometry is derived for the first time. The Kirchhoff–Love shell kinematics is employed in the context of a one-parameter modification of Mindlin’s strain gradient elasticity theory. The weak form of the static boundary value problem of the generalized shell model is formulated within an H3 Sobolev space setting incorporating first-, second- and third-order derivatives of the displacement variables. The strong form governing equations with a complete set of boundary conditions are derived via the principle of virtual work. A detailed description focusing on the non-standard features of the implementation of the corresponding Galerkin discretizations is provided. The numerical computations are accomplished with a conforming isogeometric method by adopting Cp−1-continuous NURBS basis functions of order p≥3. Convergence studies and comparisons to the corresponding three-dimensional solid element simulation verify the shell element implementation. Numerical results demonstrate the crucial capabilities of the non-standard shell model: capturing size effects and smoothening stress singularities.