The history of stellar seismology suggests that observation and theory often take turns advancing our understanding. The recent tripling of the sample of pulsating white dwarfs generated by the Sloan Digital Sky Survey represents a giant leap on the observational side. The time is ripe for a comparable advance on the theoretical side. There are basically two ways we can improve our theoretical understanding of pulsating stars: we can improve the fundamental ingredients of the models, or we can explore the existing models in greater computational detail. For pulsating white dwarfs, much progress has recently been made on both fronts: models now exist that connect the interior structure to its complete evolutionary history, while a method of using parallel computers for global exploration of relatively simple models has also been developed. Future advances in theoretical white dwarf asteroseismology will emerge by combining these two approaches, yielding unprecedented insight into the physics of diffusion, nuclear burning, and mixing.

8 pages, 1 fig, Communications in Asteroseismology (invited review)