Filter based methods for statistical linear inverse problems

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory

Alignment of U1-viperitoxin-Dr1b amino acid sequence with U1-viperitoxin-Dr1a.

<p>Sequences were obtained from UniProt database and are presented with unique identification numbers and entry names. In residues marked as ‘*’ are single fully conserved residues. At the positions highlighted in yellow, ‘:’ and ‘.’ denote positions with conservation between groups of strongly similar properties (scoring > 0.5 in the Gonnet PAM 250 matrix) and, conservation between groups of weakly similar properties (scoring = < 0.5 in the Gonnet PAM 250 matrix) respectively.</p

Clinical and epidemiological data of the 245 patients recruited for the present study.

<p>Clinical and epidemiological data of the 245 patients recruited for the present study.</p

Intact protein analysis chromatogram of MALDI-TOF: the intact mass of U1-viperitoxin-Dr1b is 13.564 kDa.

<p>Intact protein analysis chromatogram of MALDI-TOF: the intact mass of U1-viperitoxin-Dr1b is 13.564 kDa.</p

Regeneration of the entire human epidermis using transgenic stem cells

Junctional epidermolysis bullosa (JEB) is a severe and often lethal genetic disease caused by mutations in genes encoding the basement membrane component laminin-332. Surviving patients with JEB develop chronic wounds to the skin and mucosa, which impair their quality of life and lead to skin cancer. Here we show that autologous transgenic keratinocyte cultures regenerated an entire, fully functional epidermis on a seven-year-old child suffering from a devastating, lifethreatening form of JEB. The proviral integration pattern was maintained in vivo and epidermal renewal did not cause any clonal selection. Clonal tracing showed that the human epidermis is sustained not by equipotent progenitors, but by a limited number of long-lived stem cells, detected as holoclones, that can extensively self-renew in vitro and in vivo and produce progenitors that replenish terminally differentiated keratinocytes. This study provides a blueprint that can be applied to other stem cell-mediated combined ex vivo cell and gene therapies