scholarly journals Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives

2021 ◽  
Vol 38 ◽  
pp. 36-53
Author(s):  
M. M. Turov ◽  
◽  
V. E. Fedorov ◽  
B. T. Kien ◽  
◽  
...  
Keyword(s):  
Inverse Problems ◽  
Fractional Derivative ◽  
Fractional Part ◽  
Well Posedness ◽  
Bounded Operators ◽  
Degenerate Problem ◽  
Linear Inverse Problems ◽  
Coefficient Problems ◽  
Degenerate Operator ◽  
Inverse Coefficient

The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

scholarly journals Inverse coefficient problems for nonlinear elliptic equations

2007 ◽  
Vol 49 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Runsheng Yang ◽  
Yunhua Ou
Keyword(s):  
Inverse Problems ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Unknown Coefficient ◽  
Nonlinear Elliptic ◽  
Inverse Coefficient Problems ◽  
Coefficient Problems ◽  
The Given ◽  
Inverse Coefficient

AbstractThis paper is devoted to a class of inverse coefficient problems for nonlinear elliptic equations. The unknown coefficient of the elliptic equations depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic equations are uniquely solvable for the given class of coefficients. Proof of the existence of a quasisolution of the inverse problems is obtained.

scholarly journals Regularization of linear inverse problems with total generalized variation

2014 ◽  
Vol 22 (6) ◽  
Author(s):  
Kristian Bredies ◽  
Martin Holler
Keyword(s):  
Inverse Problems ◽  
Minimization Problem ◽  
Regularization Parameter ◽  
Symmetric Tensor ◽  
Functions Of Bounded Variation ◽  
Well Posedness ◽  
Tensor Fields ◽  
Linear Inverse Problems ◽  
Total Generalized Variation ◽  
Generalized Variation

AbstractThe regularization properties of the total generalized variation (TGV) functional for the solution of linear inverse problems by means of Tikhonov regularization are studied. Considering the associated minimization problem for general symmetric tensor fields, the well-posedness is established in the space of symmetric tensor fields of bounded deformation, a generalization of the space of functions of bounded variation. Convergence for vanishing noise level is shown in a multiple regularization parameter framework in terms of the naturally arising notion of TGV-strict convergence. Finally, some basic properties, in particular non-equivalence for different parameters, are discussed for this notion.

Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative

2020 ◽  
Vol 28 (1) ◽  
pp. 53-61
Author(s):  
Vladimir Evgenyevich Fedorov ◽  
Natalia Dmitrievna Ivanova
Keyword(s):  
Inverse Problems ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Linear Inverse Problems ◽  
Linear Evolution ◽  
Solution Existence And Uniqueness ◽  
Sobolev System ◽  
Degenerate Operator ◽  
The Right

AbstractThe purpose of this work is to obtain sufficient conditions of a solution existence and uniqueness for a class of inverse problems for linear evolution equations with a degenerate operator at the derivative and with an unknown element in the right-hand side of the equation, which depends on the time variable. The overdetermination condition is given on the kernel of the operator at the derivative, the initial condition have the Cauchy form or the Showalter–Sidorov form. The obtained abstract results are applied to the investigation of linear inverse problems for the Sobolev system of equations and for the linearized Oskolkov system with overdetermination on the pressure gradient function.

Inverse problems for a class of degenerate evolution equations with Riemann – Liouville derivative

2019 ◽  
Vol 22 (2) ◽  
pp. 271-286 ◽  
Author(s):  
Vladimir E. Fedorov ◽  
Roman R. Nazhimov
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Cauchy Type ◽  
Liouville Derivative

Abstract Unique solvability and well-posedness issues are studied for linear inverse problems with a constant unknown parameter to fractional order differential equations with Riemann – Liouvlle derivative in Banach spaces. Firstly, well-posedness criteria for the inverse problem with the Cauchy type initial conditions to the differential equation in a Banach space that solved with respect to the fractional derivative is obtained. This result is applied to search of sufficient conditions for the unique solution existence of the inverse problem for equation with linear degenerate operator at the Riemann – Liouville fractional derivative. It is shown that the presence of the matching conditions for the data of the problem excludes the possibility of the well-posedness consideration for the degenerate inverse problem with the Cauchy type condition. But for the inverse problem with the Showalter – Sidorov type conditions it is found the criteria of the well-posedness. Abstract results are used to the search of conditions of the unique solvability for an inverse problem to a class of partial differential equations of time-fractional order with polynomials of elliptic differential operators with respect to the spatial variables.

scholarly journals AMP-Inspired Deep Networks for Sparse Linear Inverse Problems

2017 ◽  
Vol 65 (16) ◽  
pp. 4293-4308 ◽  
Author(s):  
Mark Borgerding ◽  
Philip Schniter ◽  
Sundeep Rangan
Keyword(s):  
Inverse Problems ◽  
Linear Inverse Problems ◽  
Deep Networks

scholarly journals The Regularized Weak Functional Matching Pursuit for linear inverse problems

2019 ◽  
Vol 27 (3) ◽  
pp. 317-340 ◽  
Author(s):  
Max Kontak ◽  
Volker Michel
Keyword(s):  
Inverse Problems ◽  
Matching Pursuit ◽  
A Priori ◽  
Computation Time ◽  
Point Of View ◽  
Computational Point ◽  
Linear Inverse Problems ◽  
Infinite Dimensional ◽  
Frame Condition ◽  
Ill Posed

Abstract In this work, we present the so-called Regularized Weak Functional Matching Pursuit (RWFMP) algorithm, which is a weak greedy algorithm for linear ill-posed inverse problems. In comparison to the Regularized Functional Matching Pursuit (RFMP), on which it is based, the RWFMP possesses an improved theoretical analysis including the guaranteed existence of the iterates, the convergence of the algorithm for inverse problems in infinite-dimensional Hilbert spaces, and a convergence rate, which is also valid for the particular case of the RFMP. Another improvement is the cancellation of the previously required and difficult to verify semi-frame condition. Furthermore, we provide an a-priori parameter choice rule for the RWFMP, which yields a convergent regularization. Finally, we will give a numerical example, which shows that the “weak” approach is also beneficial from the computational point of view. By applying an improved search strategy in the algorithm, which is motivated by the weak approach, we can save up to 90  of computation time in comparison to the RFMP, whereas the accuracy of the solution does not change as much.

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