Hilbert Scales Research Articles

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Solving Abel integral equations by regularisation in Hilbert scales

Integral operators of Abel type of order a>0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a family of appropriate Hilbert scales so that the operator is ill-posed of order a in the scale. We provide weak regularity assumptions on the kernel underlying the operator for the above to hold true. Our construction leads to a well-defined regularisation strategy by Tikhonov regularisation in Hilbert scales. We thereby generalise the results of Gorenflo and Yamamoto for a<1 to arbitrary a>0 and more general kernels. Thanks to tools from interpolation theory, we also show that the a priori information associated to the Hilbert scale formulates in terms of smoothness in usual Sobolev spaces up to boundary conditions, and that the regularisation term actually amounts to penalising derivatives. Finally, following the theoretical construction, we develop a comprehensive numerical approach, where the a priori smoothness is encoded in a single parameter rather than in a full operator. Several numerical examples are shown, both confirming the theoretical convergence rates and showing the general applicability of the method.

Final value problem governed by a class of time‐space fractional pseudo‐parabolic equations with weak nonlinearities

We study the final value problem involving a class of semilinear fractional pseudo‐parabolic equations, where the nonlinearity probably takes values in fractional Sobolev spaces. By establishing some estimates for resolvent operators and employing the embedding related to Hilbert scales and fractional Sobolev spaces, we are able to obtain the existence and uniqueness result to the mentioned problem. In addition, the behavior of solutions at initial time is analyzed with respect to the final data. It will be shown that various cases of the nonlinearity functions meet our setting, including functions with gradient term.

On regularity and asymptotic stability for semilinear nonlocal pseudo-parabolic equations

We deal with a class of nonlocal pseudo-parabolic equations involving strong nonlinearities. The questions on existence, regularity and stability of solutions are addressed by using local estimates, fixed point arguments, and the relation between the Hilbert scales and fractional Sobolev spaces.

Nonlinear Tikhonov regularization in Hilbert scales for inverse learning

In this paper, we study Tikhonov regularization scheme in Hilbert scales for a nonlinear statistical inverse problem with general noise. The regularizing norm in this scheme is stronger than the norm in the Hilbert space. We focus on developing a theoretical analysis for this scheme based on conditional stability estimates. We utilize the concept of the distance function to establish high probability estimates of the direct and reconstruction errors in the Reproducing Kernel Hilbert space setting. Furthermore, explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through an appropriate source condition. Our results improve upon and generalize previous results obtained in related settings.

On stability and regularity of solutions to generalized Rayleigh-Stokes equations involving delays in Hilbert scales

On stability and regularity of solutions to generalized Rayleigh-Stokes equations involving delays in Hilbert scales

Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

&lt;p&gt;In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.&lt;/p&gt;

The final value problem for anomalous diffusion equations involving weak-valued nonlinearities

We are concerned with the final value problem governed by a class of semilinear anomalous diffusion equations, where the nonlinearity may take values in Hilbert scales with negative orders. The solvability and Hölder regularity results are proved by establishing estimates in Hilbert scales for resolvent operators in connection with nonlinearity function. The obtained results are applicable for some concrete problems modeling subdiffusion phenomena.

On initial value problem for elliptic equation on the plane under Caputo derivative

Abstract In this article, we are interested to study the elliptic equation under the Caputo derivative. We obtain several regularity results for the mild solution based on various assumptions of the input data. In addition, we derive the lower bound of the mild solution in the appropriate space. The main tool of the analysis estimation for the mild solution is based on the bound of the Mittag-Leffler functions, combined with analysis in Hilbert scales space. Moreover, we provide a regularized solution for our problem using the Fourier truncation method. We also obtain the error estimate between the regularized solution and the mild solution. Our current article seems to be the first study to deal with elliptic equations with Caputo derivatives on the unbounded domain.

Analysis of the discrepancy principle for Tikhonov regularization with oversmoothing penalty term under low order source conditions

We study the application of Tikhonov regularization to ill-posed nonlinear operator equations. The objective of this work is to prove convergence rates for the discrepancy principle under low order source conditions of logarithmic type. We work within the framework of Hilbert scales and extend existing studies on this subject to the oversmoothing case. The latter means that the exact solution of the treated operator equation does not belong to the domain of definition of the penalty term. As a consequence, the Tikhonov functional fails to have a finite value.

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh, Appl. Math. Comput. 392, 125701, 2021] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

Small sample spaces for Gaussian processes

It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process X is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a “small” set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of X in the σ-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of X and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen–Loève theorem.

Interpolation inequalities for modified Sobolev spaces and their implications to inversion of Radon transform

The implications of interpolation inequality for Hilbert scales for obtaining estimates for the worst case error for ill-posed problems are well known. In this paper, we prove interpolation inequalities for the modified Sobolev scales, the general form of which was introduced by Sharafutdinov (Inverse Problems 33:025002, 2017), and make use of them along with the recently obtained modified Reshetnyak formulas for Radon transform to obtain error estimates for the worst case error associated with the Radon transform and appropriate source sets.

On Caputo fractional elliptic equation with nonlocal condition

This paper is first study for considering nonlocal elliptic equation with Caputo derivative. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We prove the non-correction of the problem in the sense of Hadamard. The main tool is the use of upper and lower bounds of the Mittag-Lefler function, combined with analysis in Hilbert scales space.

Final value problem for Rayleigh-Stokes type equations involving weak-valued nonlinearities

The final value problem governed by Rayleigh-Stokes type equations is investigated in the circumstance that the nonlinearity function may take values in Hilbert scales of negative order. We prove the existence and Hölder regularity results by analyzing the regularity of the resolvent operators and using the fixed point arguments.

On Global Solvability and Regularity for Generalized Rayleigh–Stokes Equations with History-Dependent Nonlinearities

We are concerned with the initial value problem governed by generalized Rayleigh–Stokes equations, where the nonlinearity depends on history states and takes values in Hilbert scales of negative order. The solvability and Hölder regularity of solutions are proved using fixed point arguments and embeddings of fractional Sobolev spaces. An application to a related inverse source problem is given.

Finite-time attractivity of strong solutions for generalized nonlinear abstract Rayleigh–Stokes equations

Abstract In the present paper, we address the global solvability and finite-time attractivity of strong solutions for an abstract Rayleigh–Stokes-type equation involving nonlinear perturbations in Hilbert spaces. Based on regularity estimates of resolvent operator, local estimates on Hilbert scales and fixed point arguments, we obtain some results on global existence and finite-time attractivity of strong solutions to our problem.

Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

&lt;p style='text-indent:20px;'&gt;In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.&lt;/p&gt;

Identifying a space-dependent source term in distributed order time-fractional diffusion equations

&lt;p style='text-indent:20px;'&gt;The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.&lt;/p&gt;

New results for variational regularization with oversmoothing penalty term in Banach spaces

In this article on variational regularization for ill-posed nonlinear problems, we are once again discussing the consequences of an oversmoothing penalty term. This means in our model that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales, but rarely in a Banach space setting. Our present results try to establish a theoretical justification of oversmoothing regularization in Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for Hölder-type smoothness and low order-type smoothness. An illustrative example is intended to indicate the specificity of occurring non-reflexive Banach spaces.