Regularization of linear operator equations

‎A matrix LSQR algorithm for solving constrained linear operator equations

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

Applied Harmonic Analysis meets Sparse Regularization of Operator Equations

Sparse regularization of operator equations has already shown its effectiveness both theoretically and practically. The area of applied harmonic analysis offers a variety of systems such as wavelet systems which provide sparse approximations within certain model situations which then allows to apply this general approach provided that the solution belongs to this model class. However, many impo...

Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian ...

Solvability of Systems of Linear Operator Equations

Let G be a semigroup of commuting linear operators on a linear space S with the group operation of composition. The solvability of the system of equations /,/ = ,■, i = I, ... , r, where /, e G and j e S , was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions Ijfr = Irfj , i ^ j , are necessar...

Hypergeometric series solutions of linear operator equations