Weighting operators for sparsity regularization

Standard regularization methods typically favor solutions which are in, or close to, the orthogonal complement of the null space of the forward operator/matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖠</m:mi> </m:math> {{A}} . This particular biasedness might not be desirable in applications and can lead to severe challenges when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖠</m:mi> </m:math> {{A}} is non-injective. We have therefore, in a series of papers, investigated how to “remedy” this fact, relative to a chosen basis and in a certain mathematical sense: Based on a weighting procedure, it turns out that it is possible to modify both Tikhonov and sparsity regularization such that each member of the chosen basis can be almost perfectly recovered from their image under <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖠</m:mi> </m:math> {{A}} . In particular, we have studied this problem for the task of using boundary data to identify the source term in an elliptic PDE. However, this weighting procedure involves <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>𝖠</m:mi> <m:mo>†</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mi>𝖠</m:mi> </m:mrow> </m:math> {{A}^{}{A}} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝖠</m:mi> <m:mo>†</m:mo> </m:msup> </m:math> {{A}^{}} denotes the pseudo-inverse of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖠</m:mi> </m:math> {{A}} , and can thus be CPU-demanding and lead to undesirable error amplification. We therefore, in this paper, study alternative weighting approaches and prove that some of the recovery results established for the methodology involving <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝖠</m:mi> <m:mo>†</m:mo> </m:msup> </m:math> {{A}^{}} hold for a broader class of weighting schemes. In fact, it turns out that “any” linear operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖡</m:mi> </m:math> {{B}} has an associated proper weighting defined in terms of images under <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖡𝖠</m:mi> </m:math> {{B}{A}} . We also present a series of numerical experiments, employing different choices of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝖡</m:mi> </m:math> {{B}} .