Why do you think your paper is highly cited?

Many physical phenomena are modeled by (systems of) partial differential equations.  Closed form solutions of these equations are generally not possible and therefore they must be solved by numerical methods.  There are two approaches to the numerical solution of partial differential equations.  The one (e.g. standard finite element methods) sets the computational scheme in advance.  The other (adaptive methods) is an iterative scheme that tries to take advantage of computations already done to determine what the next iteration should be.  Heuristically, adaptive methods should be able to adapt to the singularities in the unknown solution and thereby obtain better resolution, which is essential for large problems. Adaptive methods are often used in practice with excellent numerical results.  While experience gives strong evidence for their ability of reducing the computational complexity in a significant way, even to the extent that demanding problems become tractable, very little has been known about a rigorous foundation in terms of convergence and complexity estimates that allow one to judge the performance of such methods and their potential. This paper gives the first rigorous convergence analysis for elliptic problems for a new adaptive scheme that turns out to be optimal, in the sense that it achieves a desired target accuracy at a computational amount of work that stays proportional to the smallest number of degrees of freedom, needed to recover the unknown solution within target accuracy.

  Does it describe a new discovery or a new methodology that's useful to others?

The paper presents several new discoveries.  Of these, the paradigm for how to construct a successful adaptive algorithm may be the most important. The specific algorithms constructed in the paper may prove to be numerically useful. Some of the techniques utilized in the adaptive algorithm (such as fast matrix vector multiplication, coarsening, and thresholding) should be useful in other contexts.  In addition, the paper puts forward a mathematical theory based on concepts of approximation that will be useful in judging the performance of any adaptive algorithm.

  What were some of the circumstances that led you to do this research?

We were aware that no previous work had proven the advantages of adaptive methods for partial differential equations.  We found the lack of such estimates very unsatisfactory since adaptive concepts are so important.  We were not content with numerical evidence but felt a rigorous foundation for adaptive methods was needed.  So we found it an exciting mathematical challenge to try to prove such estimates and settle once and for all the question of whether adaptive methods are really more effective than traditional methods.  We had some insight into the problem from our previous work in approximation theory, computational harmonic analysis, signal and image processing, and numerical simulation.

  Could you summarize the significance of your paper in layman's terms?

Physical phenomena are most often modeled as (systems of) partial differential equations.  These equations typically cannot be solved in exact form and therefore must be solved numerically on a computer.  One might think that we can achieve the accuracy we want by simple increases in computational power (larger computers).  But this is far from being the case and in fact the largest advances (and the only real hope in many problems) is to devise clever numerical schemes.  One therefore seeks numerical methods which obtain a given target accuracy with the fewest number of computations.  There have been two general approaches.  One sets the numerical scheme in advance and does not alter it during the computation.  The other (adaptive methods) tries to adapt the scheme during the computational process.  The question then arises whether such adaptive schemes offer any real advantage over non-adaptive schemes and, secondly, whether one can construct optimal adaptive schemes which can be proven to be most efficient.  This paper shows that this is indeed the case for a certain class of differential equations (elliptic equations) and thereby shows once and for all that adaptive schemes are indeed beneficial.