arXiv:1710.09244v3 [math.NA] 3 Jul 2018 [email protected] ecnie ier l-oe nes rbesi h omo operator of form the in problems inverse ill-posed linear, consider We Introduction 1 ∗ ozsr 61,303Gotne,Gray b.sprung@mat Germany, G¨ottingen, 37083 16-18, Lotzestr. ihrodrcnegnertsfrBregman for rates convergence order Higher trtdvrainlrglrzto finverse of regularization variational iterated Keywords: theoret Our sp demonstrated. experiments. Besov numerical be of in can terms confirmed rates are in the interpreted of ps are optimality elliptic VSCs the with the regularization where res entropy operators These for VSC. order discussed third p ther under and fidelity rates convergence data derive general we with v spaces iterated lead Banach Bregman which in For order ularization spaces. high Hilbert arbitrarily in cons rates of be convergence VSCs to introduce have we regularization paper variational converge s bee of order the versions has higher to even ated VSCs up For order rates regularization. convergence higher Tikhonov obtained of low- towards who only step (2013) yielding first Grasmair by of A disadvantage su rates. the of have vergence formulations spaces other S Banach as well inequalities. in con as variational (VSCs) of source conditions form such source the terms in determin penalty formulated is or often convergence fidelity source this called data of solution rate non-quadratic the on The conditions noi 0. smoothness the stract to as tends spaces side Banach in hand equations operator ill-posed ear nrp euaiain aitoa oreconditions MSC: source variational regularization, entropy esuytecnegneo aitoal euaie solu regularized variationally of convergence the study We 52,65J22 65J20, ejmnSprung Benjamin nes rbes aitoa euaiain convergen regularization, variational problems, inverse etme 7 2018 27, September problems f T Abstract ∗ = 1 g obs hrtnHohage Thorsten eudodifferential odtos For conditions. c variational uch drd nthis In idered. c ae,iter- rates, nce rainlreg- ariational ei h right the in se hconditions ch nlyterms, enalty h.uni-goettingen.de, lsaefur- are ults in olin- to tions iin are ditions clresults ical ooptimal to re con- order ∗ db ab- by ed aturation csand aces taken n erates, ce equations (1) with a bounded linear operator T : between Banach spaces and . We will assume that T is injective, but thatX →T Y−1 : T ( ) is not continuous.X Y The exact solution will be denoted by f †, and the noisyX → observed X data by gobs , assuming the standard deterministic noise model ∈ Y
gobs Tf † δ (2) k − k≤ with noise level δ > 0. To obtain a stable estimator of f † from such data we will consider generalized Tikhonov regularization of the form
1 fˆ arg min (Tf gobs)+ (f) (P ) α ∈ αS − R 1 f∈X with a convex, lower semi-continuous penalty functional : R , R , a regularization parameter α> 0 and a data fidelityR termX → ∪ {∞} 6≡ ∞ (g) := 1 g q (3) S q k kY for some q> 1. The aim of regularization theory is to bound the reconstruction error fˆα f † in terms of the noise level δ. Classically, in a Hilbert space setting, conditionsk − implyingk such bounds have been formulated in terms of the spectral calculus of the operator T ∗T ,
f † (T ∗T )ν/2( ) (4) ∈ X for some ν > 0 and some initial guess f0. In fact, using spectral theory it is easy to show that (4) yields
† ν f fˆ = δ ν+1 (5) − α O