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In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and interior W 2 , p -regularity of the solution, and the regularity of the free boundary ∂ ⁡ { u > ϕ } to the obstacle problem of the porous medium equation, u t = Δ ⁢ u m ( m > 1 ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ ⁡ { u > ϕ } and ∂ ⁡ { u > 0 } , we consider two cases on the initial data which make the free boundary ∂ ⁡ { u > ϕ } separate from the free boundary ∂ ⁡ { u > 0 } . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 -regularity of the free boundary ∂ ⁡ { u > ϕ } is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator. [ABSTRACT FROM AUTHOR]

In 2010, Qinian Jin considered a regularized Levenberg–Marquardt method in Hilbert spaces for getting stable approximate solution for nonlinear ill-posed operator equation F ⁢ (x) = y , where F : D ⁢ (F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y and obtained rate of convergence results under an appropriate source condition. In this paper, we propose a simplified Levenberg–Marquardt method in Hilbert spaces for solving nonlinear ill-posed equations in which sequence of iteration { x n δ } is defined as x n + 1 δ = x n δ - ( α n ⁢ I + F ′ ⁢ (x 0) * ⁢ F ′ ⁢ (x 0) ) - 1 ⁢ F ′ ⁢ (x 0) * ⁢ (F ⁢ (x n δ) - y δ) . Here { α n } is a decreasing sequence of nonnegative numbers which converges to zero, F ′ ⁢ (x 0) denotes the Fréchet derivative of F at an initial guess x 0 ∈ D ⁢ (F) for the exact solution x † and (F ′ ⁢ (x 0)) * denote the adjoint of F ′ ⁢ (x 0) . In our proposed method, we need to calculate Fréchet derivative of F only at an initial guess x 0 . Hence, it is more economic to use in numerical computations than the Levenberg–Marquardt method used in the literature. We have proved convergence of the method under Morozov-type stopping rule using a general tangential cone condition. In the last section of the paper, numerical examples are presented to demonstrate advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the iteratively regularized Gauss–Newton method with frozen derivative and formulate its convergence rates in the settings of Banach spaces. The convergence rates of iteratively regularized Gauss–Newton method with frozen derivative are well studied via generalized source conditions. We utilize the recently developed concept of conditional stability of the inverse mapping to derive the convergence rates. Also, in order to show the practicality of this paper, we show that our results are applicable on an ill-posed inverse problem. Finally, we compare the convergence rates derived in this paper with the existing convergence rates in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse coefficient problem of identifying both the mass density ρ ⁢ (x) > 0 and flexural rigidity r ⁢ (x) > 0 of a damped Euler–Bernoulli (cantilever) beam governed by the equation ρ ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T) , subject to boundary conditions u ⁢ (0 , t) = u x ⁢ (0 , t) = 0 , u x ⁢ x ⁢ (ℓ , t) = 0 , - (r ⁢ (x) ⁢ u x ⁢ x ⁢ (x , t)) x | x = ℓ = g ⁢ (t) , from the available measured boundary deflection ν ⁢ (t) := u ⁢ (ℓ , t) and rotation θ ⁢ (t) := u x ⁢ (ℓ , t) at the free end of the beam. The distinctive feature of the considered inverse coefficient problem is that not one, but two Neumann-to-Dirichlet operators have to be formally defined. The inverse problem is hence formulated as a system of nonlinear Neumann-to-Dirichlet operator equations with the right-hand sides consisting of the measured outputs. As a natural consequence of this approach, a vector-form Tikhonov functional is introduced whose components are squares of the L 2 -norm differences between predicted and measured outputs. We then prove existence of a quasi-solution of the inverse problem and derive explicit gradient formulae for the Fréchet derivatives of both components of the Tikhonov functional. These results are instrumental to any gradient based algorithms for reconstructing the two unknown coefficients of the considered damped Euler–Bernoulli beam. [ABSTRACT FROM AUTHOR]

In this paper, we studied the separability of the non-linear Schrodinger operator of the form S u x = − Δ u x + V x , u u x , where Δ u x = ∑ i = 1 n ∂ 2 u x ∂ x i 2 , with the non-linear matrix potential V x , u . We obtained the sufficient conditions for separability of this operator in the space L 2 R n ℓ and we established the suitable coercive inequalities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space (X , { | ⋅ | p } p ∈ Λ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 휏-Krein–Šmulian property, where 휏 is a weaker Hausdorff locally convex topology of 푋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 휏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ (R +) × C ⁢ (R +) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result. [ABSTRACT FROM AUTHOR]

We study minimisation problems in L ∞ for general quasiconvex first order functionals, where the class of admissible mappings is constrained by the sublevel sets of another supremal functional and by the zero set of a nonlinear operator. Examples of admissible operators include those expressing pointwise, unilateral, integral isoperimetric, elliptic quasilinear differential, Jacobian and null Lagrangian constraints. Via the method of L p approximations as p → ∞ , we illustrate the existence of a special L ∞ minimiser which solves a divergence PDE system involving certain auxiliary measures as coefficients. This system can be seen as a divergence form counterpart of the Aronsson PDE system which is associated with the constrained L ∞ variational problem. [ABSTRACT FROM AUTHOR]

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors' recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search. [ABSTRACT FROM AUTHOR]

We study the existence of solutions for the nonlinear elliptic problem - Δ ⁢ v - λ ⁢ g 1 ⁢ v + h 1 ⁢ (v) = f 1 in ⁢ V ∖ V 0 , v = 0 on ⁢ V 0 , where V is the Sierpiński gasket in ℝ N - 1 ( N ≥ 2 ), V 0 is its boundary (consisting of its N corners) and λ ∈ ℝ . Here f 1 , g 1 : V → ℝ , h 1 : ℝ → ℝ are the maps satisfying suitable hypotheses. [ABSTRACT FROM AUTHOR]

We propose a class of Newton-like methods with increasing convergence order for approximating the solutions of systems of nonlinear equations. Novelty of the methods is that in each step the order of convergence is increased by an amount of three at the cost of only one additional function evaluation. Another important feature is the single use of an inverse operator in each iteration, which makes the schemes attractive and computationally more efficient. Theoretical results regarding convergence and computational efficiency are verified through numerical examples including those arising from boundary value problems and integral equations. By way of comparison, it is shown that the present methods are more efficient than their existing counterparts, particularly when applied to solve the large systems of equations. [ABSTRACT FROM AUTHOR]

5 S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations 246 (2009), no. 7, 2958-2987. Keywords: Fully nonlinear operators; principal eigenvalue; Dirichlet problem; half-eigenvalues; uniqueness; 35J60; 35P30; 35B50 EN Fully nonlinear operators principal eigenvalue Dirichlet problem half-eigenvalues uniqueness 35J60 35P30 35B50 673 691 19 10/03/22 20221001 NES 221001 1 Introduction This article contributes to the study of eigenvalue problem of the form remark on the eigenvalue problem in a general smooth unbounded domain. Fully nonlinear operators, uniqueness, 35J60, 35J60, 35B50, principal eigenvalue, Dirichlet problem, half-eigenvalues, 35P30. [Extracted from the article]