Advective partial differential equations can be used to describe many scientific processes. Two significant sources of error that can cause difficulties in inferring parameters from experimental data on these processes include (i) noise from the measurement and collection of experimental data and (ii) numerical error in approximating the forward solution to the advection equation. How this second source of error alters parameter estimation and uncertainty quantification during an inverse problem methodology is not well understood. As a step towards a better understanding of this problem, we present both analytical and computational results concerning how a least squares cost function and parameter estimator behave in the presence of numerical error in approximating solutions to the underlying advection equation. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on our results and their general nature, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.

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Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

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Chris Smith, a first-year student from the BioFrontiers Institute鈥檚 Interdisciplinary Quantitative Biology PhD Program, received a 2015 fellowship. Second-year IQ Biology student, John Nardini, was given an honorable mention.

Smith is rostered in the Department of Ecology and Evolutionary Biology. His interests are focused on evolutionary genomics and how they are affected by local adaptations and speciation. He plans to use his fellowship funding to support him in the development of computational methods for biologists to use when researching genomics.

The NSF fellowships provide three years of financial support over a five-year period. This is split into a $34,000 annual stipend and a $12,000 cost-of-education allowance to the graduate institution. The fellowships also provide opportunities to participate in international research collaborations and access to NSF-supported research infrastructure. From over 16,500 applicants, a total of 2,000 Graduate Fellows were awarded in 2015, 23 of which went to CU-麻豆影院 students. Over the past five years, 30 percent of students in the IQ Biology program at BioFrontiers have either received a GRFP award or been given an honorable mention.

The University of Colorado, 麻豆影院 students won 30 NSF fellowships, and CU-麻豆影院 was among the top 20 universities with NSF fellows last year. BioFrontiers and the IQ Biology program are honored to have students involved in the NSF Graduate Research Fellowship Program, which is one of the most prestigious awards available for student researchers. 

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