The main goals of the first three years have been achieved, i.e., the development of particle-based and continuum-based algorithms for cross-scaleup-scale analysis of complex fluid flows. The U. Minnesota team has focused on particle-based methods, wavelets (Rustad et al., 2001) and visualization and has had great success with the dissipative and fluid particle dynamics algorithms, as applied to colloidal, polymeric and biological systems, wavelet filtering and visualization endeavors. We have organized two sessions in nonlinear geophysics at the A.G.U. Fall Meeting (2000,2002), which have indeed synergetically stimulated the community and promoted cross-disciplinary efforts in the geosciences. The LANL team has succeeded with continuum-based algorithms, in particular, fractal interpolating functions (fif). These have been applied to 1-D flow and transport equations (Travis, 2000; 2002) as a proof of principle, providing solutions that capture dynamics at all scales. In addition, the fif representations can be integrated to provide sub-grid-scale homogenization, which can be used in more traditional finite difference or finite element solutions of porous flow and transport. Another useful tool for fluid flow problems is the ability to solve inverse problems, that is, given present-time observations of a fluid flow, what was the initial state of that fluid system? We havemore » demonstrated this capability for a large-scale problem of 3-D flow in the Earth's crust (Bunge, Hagelberg & Travis, 2002). Use of the adjoint method for sensitivity analysis (Marchuk, 1995) to compute derivatives of models makes the large-scale inversion feasible in 4-D, , space and time. Further, a framework for simulating complex fluid flow in the Earth's crust has been implemented (Dutrow et al, 2001). The remaining task of the first three-year campaign is to extend the implementation of the fif formalism to our 2-D and 3-D computer codes, which is straightforward, but involved.« less