{"604337":{"#nid":"604337","#data":{"type":"event","title":"DCL Seminar Series: Fred Daum","body":[{"value":"\u003Cp\u003EWe derive a new exact stochastic particle flow filter, using a theorem of Gromov. Our\u003Cbr \/\u003E\r\nfilter is many orders of magnitude faster than standard particle filters for high dimensional\u003Cbr \/\u003E\r\nproblems, and our filter beats the extended Kalman filter accuracy by orders of magnitude\u003Cbr \/\u003E\r\nfor difficult nonlinear problems. Our theory uses particle flow to compute Bayes\u0026rsquo; rule,\u003Cbr \/\u003E\r\nrather than a pointwise multiply. We do not use resampling of particles or proposal\u003Cbr \/\u003E\r\ndensities or any MCMC method. But rather, we design the particle flow with the solution\u003Cbr \/\u003E\r\nof a linear first order highly underdetermined PDE. We solve this PDE as an exact formula\u003Cbr \/\u003E\r\nwhich is valid for arbitrary smooth nowhere vanishing densities. Gromov proves that there\u003Cbr \/\u003E\r\nexists a \u0026ldquo;nice\u0026rdquo; solution to a linear constant coefficient PDE for smooth functions if and\u003Cbr \/\u003E\r\nonly if the number of unknowns is sufficiently large (at least the number of linearly\u003Cbr \/\u003E\r\nindependent equations plus the dimension of the state vector). A \u0026ldquo;nice\u0026rdquo; solution of the\u003Cbr \/\u003E\r\nPDE means that we do not need any integration, and hence it is very fast. To dispel the\u003Cbr \/\u003E\r\nmystery of Gromov\u0026rsquo;s theorem we show the simplest non-trivial example. We also show\u003Cbr \/\u003E\r\nseveral generalizations of Gromov\u0026rsquo;s theorem. Particle flow is similar to optimal transport,\u003Cbr \/\u003E\r\nbut it is much simpler and faster because we avoid solving a variational problem. Optimal\u003Cbr \/\u003E\r\ntransport is (almost always) deterministic whereas our particle flow is stochastic, like all\u003Cbr \/\u003E\r\nsuch algorithms that actually work robustly for difficult high dimensional problems\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"Gromov\u2019s method for particle flow filters"}],"uid":"34578","created_gmt":"2018-03-27 19:40:31","changed_gmt":"2018-03-27 19:40:31","author":"awatson46","boilerplate_text":"","field_publication":"","field_article_url":"","field_event_time":{"event_time_start":"2018-03-29T15:00:00-04:00","event_time_end":"2018-03-29T16:00:00-04:00","event_time_end_last":"2018-03-29T16:00:00-04:00","gmt_time_start":"2018-03-29 19:00:00","gmt_time_end":"2018-03-29 20:00:00","gmt_time_end_last":"2018-03-29 20:00:00","rrule":null,"timezone":"America\/New_York"},"extras":[],"groups":[{"id":"66191","name":"Decision and Control Lab (DCL)"}],"categories":[],"keywords":[{"id":"1808","name":"graduate students"}],"core_research_areas":[],"news_room_topics":[],"event_categories":[],"invited_audience":[{"id":"78761","name":"Faculty\/Staff"},{"id":"78771","name":"Public"},{"id":"78751","name":"Undergraduate students"}],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[],"email":[],"slides":[],"orientation":[],"userdata":""}}}