An Evolution Model for Integration Problems

In this paper a general evolution model for integration problems is introduced and applied to the solution of the Rendering Equation which is a multidimensional integral equation modelling radiant light transfer. Often it is not possible to solve integral equations in a closed analytical form. Then an estimate for the integral has to be determined by numerical or stochastic approximation. Unfortunately stochastic integration techniques are extremely slow to converge. Numerical techniques are usually faster in convergence but their success strongly depends on function requirements. The evolution model presented here overcomes the disadvantages of these methods, i.e. loss of direction and missing innovation. It is far more flexible and produces reliable estimates even in the presence of multimodal integrands. Furthermore it procides a general framework for the description of a wide variety of adaptive sampling techniques.