Microsoft Word - SustainabilityReport_BCC.doc

healthy/unhealthy plants are challenging, particularly using messy real-world

data. Thus, understanding how the health of indicator species can give early

warning of problems in an ecosystem will require new mathematical and

statistical tools.

Finally, we need better methods to figure out just how good our

mathematical models are. For example, we have models that describe how

substances move between the atmosphere, land and oceans over time, using a

series of non-linear ordinary differential equations. This cycle is critical for marine

life, which relies on the balance between nutrients, phytoplankton, and

zooplankton, which are in turn influenced by temperature, light, and resource

availability. The equations of the model are, of course, approximations, but

standard analyses commonly ignore this, and the result is that the model doesn’t

properly predict the size distribution of phytoplankton and zooplankton. Similar

difficulties arise in the highly complex problem of modeling air quality. If air

quality is substantially worse than our large reaction-diffusion models predict, we

need to know whether that is because the underlying dynamics are different from

what we thought, because of inherent randomness in the system, or because our

models captured those dynamics badly. Models have become vastly more

complex over time, making existing methods of evaluation inadequate.

Mathematical sciences challenges in the area of measuring and

monitoring progress toward sustainability include development of the

following:

• Tools for uncertainty quantification via probabilistic modeling
  approaches, including tools to deal with the following challenges in this
  area alone:
  9 Characterizing the bias or discrepancy between models and reality
  (data);
  9 Recognizing that cost constraints often mean that models can only
  be run for certain combinations of input parameters, requiring
  extrapolation of model output to other input parameters;
  9 Accounting for uncertainties in the initial conditions;
  9 Estimating unknown parameters in the process models;
  9 Accommodating stochastic features of the process models;
  9 Producing predictions that arise by combining models and
  observational data, as might occur via data assimilation methods.