The past and future trends for raw computing power and for software productivity are examined. The impact of these trends will be to greatly enlarge the scope of scientific applications for which PSEs are feasible. The scientific challenges to create for problem solving environments (PSEs) these applications are described and some potential approaches presented. Available computing power (determined by hardware and algorithms) directly determines the feasibility of these approaches. In recent decades the hardware power has grown at an astounding rate. There has been equally astounding advances in algorithm power; the combined effect of these advances is to increase computational power by 12 to 20 orders of magnitude for many real applications. Programming is the third key component in computational applications and increases in the productivity of writing computer codes in Fortran, C, Java, etc., are very low. The result is that programming is now the principal cost in computation and the challenge for PSEs is to deliver this power in an efficient, readily used way.

Obvious challenges arise in extending the dimensionality, refining the scales, using better models, parallel computing, and in creating better algorithms. This talk will focus on the vaguer challenges in PSEs to address multi-physics and multi-scale phenomena, to improve software productivity, to validate results, to incorporate computational intelligence, and, finally, to provide a language for computational sciences.