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We review recent advances in fast algorithms for fast integral equation solvers that are useful for IC applications. We review fast solvers for Laplace's equation, which is about 10 times faster than the conventional fast multipole method. Then we review the physics of low-frequency electromagnetics, and the relevant low-frequency method of moments. We describe a fast solver that allows us to solve over one million unknowns on a workstation recently. In addition, we demonstrate the applications of these fast integral equation solvers to the lithography problem. In addition, we propose a scheme whereby we first characterize blocks of linear circuits with network S, Y, or Z parameters. Then a fast real-time convolution scheme is used to calculate the interaction of a linear circuit with nonlinear terminations such as transistors and diodes. Such a scheme requires no model-order reduction of the circuits.