INFORMATION-BASED COMPLEXITY - RECENT RESULTS AND OPEN PROBLEMS

Information-based complexity (IBC) studies the computational complexity of infinite dimensional problems. These are problems where either the input or output are elements of infinite dimensional spaces. Such problems commonly arise in the continuous mathematical models used in science and engineering. Examples include optimization, integration, approximation, ordinary and partial differential equations, and integral equations. See the research monograph [1] for a comprehensive treatment of IBC and an extensive bibliography. A recent expository account may be found in [2]. Since digital computers can handle only finite sets of numbers, infinite dimensional inputs such as multivariate functions on the reals must be replaced by finite sets of numbers. Thus the computer input has only partial information about the mathematical input. Furthermore, the computer input is contaminated with errors such as round-off error and measurement error. IBC studies problems where the computer input is partial and/or contaminated. Although the focus of IBC is on infinite dimensional problems, it has been applied to finite dimensional problems where the computer input is partial and/or contaminated. Example include synchronizing clocks in a distributed system, large linear systems, and large eigenvalue problems. Since only a partial and/or contaminated input is available, the original mathematical problem can be only approximately solved. The goal of IBC is to compute the approximation at minimal cost. It has been proven that the complexity of many continuous problems grows exponentially with dimension in the worst case deterministic setting. Thus these problems are intractable. This may be contrasted with combinatorial complexity where exponential growth is only conjectured for many problems. Note that for continuous problems, dimension plays the role that number of objects plays in discete problems. Very high dimensional problems ocur in numerous applications in disciplines such as physics, chemistry, economics, and statistics.