# Compact linear operators and Krylov subspace methods

#### by Jan Marthedal Rasmussen - jmr@imm.dtu.dk

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# [ Summary: ]

This thesis deals with linear ill-posed problems related to compact operators, and iterative Krylov subspace methods for solving discretized versions of these.

Linear compact operators in infinite dimensional Hilbert spaces will be investigated and several results on the singular values and eigenvalues for such will be presented. A large subset of linear compact operators consists of integral operators and many results will be based on the kernel of such operators.

Finite dimensional approximations to these operators will be considered by using Galerkin discretization. Several results will be shown stating how singular values and eigenvalues (and corresponding vectors) of infinite dimensional operators and their finite dimensional approximations are related.

Krylov subspace methods, with focus on GMRES, will be investigated in relation to discrete ill-posed problems, that is, linear finite dimensional systems of equations that originate from ill-posed problems. By using the spectral decomposition of the coefficient matrix, results on the convergence of GMRES are derived.

Keywords:

Compact linear operator, Krylov subspace method, eigenvalues, singular values.

# [ Summary in Danish: ]

Denne afhandling omhandler lineære `ill-posed' problemer relateret til kompakte operatorer, og iterative Krylov underrums metoder til at løse diskretiserede udgaver af disse.

Lineære kompakte operatorer i Hilbert rum af uendelig dimension vil blive undersøgt og flere resultater vedrørende singulære værdier og egenværdier af sådanne vil blive præsenteret. En stor delmængde af lineære kompakte operatorer udgøres af integraloperatorer og mange resultater vil blive baseret på kernen af sådanne operatorer.

Endelig-dimensionale tilnærmelser til disse operatorer vil blive betragtet ved brug af Galerkin diskretisering. Adskellige resultater vil blive vist omhandlende hvorledes singulære værdier og egenværdier (og tilhørende vektorer) af uendelig-dimensionale operatorer og deres endelig-dimensionale tilnærmelser er relaterede.

Krylov underrums metoder, med fokus på GMRES, vil blive undersøgt i relation til diskrete ill-posed problemer, dvs. linære endeligt-dimensionale ligningssystemer der kommer fra ill-posed problemer. Ved at benytte diagonalisering af koefficientmatricen vil resultater omkring GMRES' konvergens blive udledt.

Keywords:

Nøgleord: Kompakte lineære operatorer, Krylov underrums metoder, egenværdier, singulære værdier.