Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.
[1] Anderes E., Coram M.A., Two-dimensional density estimation using smooth invertible transformation, J. Statist. Plann. Inference, 2011, 141(3), 1183–1193 http://dx.doi.org/10.1016/j.jspi.2010.09.019
[2] Badea L., Daripa P., On a boundary control approach to domain embedding method, SIAM J. Control Optim., 2001, 40(2), 421–449 http://dx.doi.org/10.1137/S0363012999357380
[3] Badea L., Daripa P., A fast algorithm for two-dimensional elliptic problems, Numer. Algorithms, 2002, 30(3–4), 199–239 http://dx.doi.org/10.1023/A:1020176803736
[4] Badea L., Daripa P., On a Fourier method of embedding domains using an optimal distributed control, Numer. Algorithms, 2003, 32(2–4), 261–273 http://dx.doi.org/10.1023/A:1024002802603
[5] Badea L., Daripa P., A domain embedding method using the optimal distributed control and a fast algorithm, Numer. Algorithms, 2004, 36(2), 95–112 http://dx.doi.org/10.1023/B:NUMA.0000033094.75324.48
[6] Borges L., Daripa P., A parallel version of a fast algorithm for singular integral transforms, Numer. Algorithms, 2000, 23(1), 71–96 http://dx.doi.org/10.1023/A:1019143832124
[7] Borges L., Daripa P., A fast parallel algorithm for the Poisson equation on a disk, J. Comput. Phys., 2001, 169(1), 151–192 http://dx.doi.org/10.1006/jcph.2001.6720
[8] Daripa P., On applications of a complex variable method in compressible flows, J. Comput. Phys., 1990, 88(2), 337–361 http://dx.doi.org/10.1016/0021-9991(90)90183-2
[9] Daripa P., A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane, SIAM J. Sci. Statist. Comput., 1992, 13(6), 1418–1432 http://dx.doi.org/10.1137/0913080
[10] Daripa P., A fast algorithm to solve the Beltrami equation with applications to quasiconformal mappings, J. Comput. Phys., 1993, 106(2), 355–365
[11] Daripa P., Dash R.K., A numerical study of pulsatile blood flow in an eccentric catheterized artery using a fast algorithm, J. Engrg. Math., 2002, 42(1), 1–22 http://dx.doi.org/10.1023/A:1014332225766
[12] Daripa P., Mashat D., Singular integral transforms and fast numerical algorithms, Numer. Algorithms, 1998, 18(2), 133–157 http://dx.doi.org/10.1023/A:1019117414918
[13] Daripa P., Mashat D., An efficient and novel numerical method for quasiconformal mappings of doubly connected domains, Numer. Algorithms, 1998, 18(2), 159–178 http://dx.doi.org/10.1023/A:1019169431757
[14] Du K., A simple numerical method for complex geometrical optics solutions to the conductivity equation, SIAM J. Sci. Comput., 2011, 33(1), 328–341 http://dx.doi.org/10.1137/100802256
[15] Golberg M.A. (Ed.), Solution Methods for Integral Equations, Math. Concepts Methods Sci. Engrg., 18, Plenum Press, New York, 1978
[16] Golberg M.A. (Ed.), Numerical Solution of Integral Equations, Math. Concepts Methods Sci. Engrg., 42, Plenum Press, New York, 1990
[17] Greengard L., The Rapid Evaluation of Potential Fields in Particle Systems, ACM Disting. Diss., MIT Press, Cambridge, 1988
[18] Greengard L., Kropinski M.C., Mayo A., Integral equation methods for Stokes flow and isotropic elasticity in the plane, J. Comput. Phys., 1996, 125(2), 403–414 http://dx.doi.org/10.1006/jcph.1996.0102
[19] Greengard L., Rokhlin V., A fast algorithm for particle simulations, J. Comput. Phys., 1987, 73(2), 325–348 http://dx.doi.org/10.1016/0021-9991(87)90140-9
[20] Greengard L., Rokhlin V., A new version of the fast multipole method for the Laplace equation in three dimensions, In: Acta Numer., 6, Cambridge University Press, Cambridge, 1997, 229–269
[21] Hackbusch W., Integral Equations, Internat. Ser. Numer. Math., 120, Birkhäuser, Basel, 1995 http://dx.doi.org/10.1007/978-3-0348-9215-5
[22] Hwang K., Advanced Computer Architecture: Parallelism, Scalability, Programmability, McGraw-Hill, New York, 1993