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Texas A&M University
Mathematics

Numerical Analysis Seminar

Date: February 19, 2020

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Maciej Paszynski, AGH University of Science and Technology, Krakow, Poland

  

Title: Supermodeling of a tumor dynamics employing isogeometric analysis solvers with piece-wise constant test functions

Abstract: In this talk, we show that it is possible to obtain reliable numerical prognoses about cancer dynamics by creating the supermodel of cancer, which consists of several coupled instances (the sub-models) of a generic cancer model, developed with isogeometric analysis (IGA). Its integration with real data can be achieved by employing a prediction/correction learning scheme focused on fitting several values of coupling coefficients between submodels, instead of matching scores (even hundreds) of tumor model parameters as it is in the classical data adaptation techniques. We also show how to speed up the tumor simulations by employing the piece-wise constant test functions in IGA framework. Namely, we show that the rows of the system of linear equations can be combined, and the test functions can be sum up to 1 using the partition of unity property at the quadrature points. Thus, the test functions in higher continuity IGA can be set to piece-wise constants. This formulation is equivalent to testing with piece-wise constant basis functions, with supports span over some parts of the domain. The resulting method is Petrov-Galerkin's kind. This observation has the following consequences. The numerical integration cost can be reduced because we do not need to evaluate the test functions since they are equal to one. The resulting method is equivalent to a linear combination of the collocations at points and with weights resulting from applied quadrature over the spans defined by supports of the piece-wise constant test functions.