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This paper addresses the inverse scattering problem in a domain ε. The input data, measured out- side ε, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside ε. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial di!erential equations. Subsequently, we develop the Carleman contraction mapping method to e!ectively tackle this system. It is noteworthy to highlight this method’s ro- bustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive. Some numerical examples are presented. 

Enterotoxigenic Escherichia coli (ETEC) is the primary etiologic agent of traveler’s diarrhea and a major cause of diarrheal disease and death worldwide, especially in infants and young children. Despite significant efforts over the past several decades, an affordable vaccine that appreciably decreases mortality and morbidity associated with ETEC infection among children under the age of 5 years remains an unmet aspirational goal. Here, we describe robust, cost-effective biosynthetic routes that leverage glycoengineered strains of non-pathogenic E. coli or their cell-free extracts for producing conjugate vaccine candidates against two of the most prevalent O serogroups of ETEC, O148 and O78. Specifically, we demonstrate site-specific installation of O-antigen polysaccharides (O-PS) corresponding to these serogroups onto licensed carrier proteins using the oligosaccharyltransferase PglB from Campylobacter jejuni. The resulting conjugates stimulate strong O-PS-specific humoral responses in mice and elicit IgG antibodies that possess bactericidal activity against the cognate pathogens. We also show that one of the prototype conjugates decorated with serogroup O148 O-PS reduces ETEC colonization in mice, providing evidence of vaccine-induced mucosal protection. We anticipate that our bacterial cell-based and cell-free platforms will enable creation of multivalent formulations with the potential for broad ETEC serogroup protection and increased access through low-cost biomanufacturing.