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The Banff International Research Station will host the "Reconstruction Methods for Inverse Problems" workshop in Banff from June 23, 2019 to June 28, 2019. Inverse problems require to determine the cause from a set of indirect observations. Such problems appear in medical imaging, non destructive testing of materials, computerized tomography, source reconstructions in acoustics, computer vision and geophysics, to mention but a few. The 21st century is the golden age of computer imaging: Measurement devices have become enormously powerful and huge amounts of data are recorded at every eye glimpse. Moreover, computer technology has developed to such a high degree of efficiency that the evaluation of such an enormous amount of data has become possible *if* adequate mathematical and computational tools are used. Recently, the community has been exposed to fundamentally new mathematical models (such as learning), which stimulated exciting theoretical developments and new computational algorithms for solving complicated large scale inverse problems. This workshop will survey modern and identify new mathematical and computational developments for tackling such problems. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada-s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta-s Advanced Education and Technology, and Mexico-s Consejo Nacional de Ciencia y Tecnología (CONACYT).