[1] R.A., Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2] L.V., Ahlfors, Complex Analysis, 3rd edn, McGraw-Hill, New York, 1985.
[3] M., Ahues, A., Largillier and B.V., Limaye, Spectral Computations for Bounded Operators, Chapman and Hall/CRC, London, 2001.
[4] B.K., Alpert, A class of bases in L2 for the sparse representation of integral operators, SIAM Journal on Mathematical Analysis, 24 (1993), 246.
[5] B., Alpert, G., Beylkin, R., Coifman and V., Rokhlin, Wavelet-like bases for the fast solution of second-kind integral equations, SIAM Journal on Scientific Computing, 14 (1993), 159.
[6] P.M., Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, NJ, 1971.
[7] K. E., Atkinson, Numerical solution of Fredholm integral equation of the second kind, SIAM Journal on Numerical Analysis, 4 (1967), 337.
[8] K.E., Atkinson, The numerical solution of the eigenvalue problem for compact integral operators, Transactions of the American Mathematical Society, 129 (1967), 458.
[9] K.E., Atkinson, Iterative variants of the Nyström method for the numerical solution of integral equations, Numerische Mathematik, 22 (1973), 17.
[10] K.E., Atkinson, The numerical evaluation of fixed points for completely continuous operators, SIAM Journal on Numerical Analysis, 10 (1973), 799.
[11] K.E., Atkinson, Convergence rates for approximate eigenvalues of compact integral operators, SIAM Journal on Numerical Analysis, 12 (1975), 213.
[12] K.E., Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimension, in M., Golberg (ed.), Numerical Solution of Integral Equations, Plenum Press, New York, 1990.
[13] K.E., Atkinson, A survey of numerical methods for solving nonlinear integral equations, Journal of Integral Equations and Applications, 4 (1992), 15.
[14] K.E., Atkinson, The numerical solution of a nonlinear boundary integral equation on smooth surfaces, IMA Journal of Numerical Analysis, 14 (1994), 461.
[15] K.E., Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.
[16] K.E., Atkinson and A., Bogomolny, The discrete Galerkin method for integral equations, Mathematics of Computation, 48 (1987), 595.
[17] K.E., Atkinson and G., Chandler, Boundary integral equation methods for solving Laplace's equation with nonlinear boundary conditions: The smooth boundary case, Mathematics of Computation, 55 (1990), 451.
[18] K.E., Atkinson and G., Chandler, The collocation method for solving the radiosity equation for unocluded surfaces, Journal of Integral Equations and Applications, 10 (1998), 253.
[19] K.E., Atkinson and D., Chien, Piecewise polynomial collocation for boundary integral equations, SIAM Journal on Scientific Computing, 16 (1995), 651.
[20] K.E., Atkinson and J., Flores, The discrete collocation method for nonlinear integral equations, IMA Journal of Numerical Analysis, 13 (1993), 195.
[21] K.E., Atkinson, I., Graham and I., Sloan, Piecewise continuous collocation for integral equations, SIAM Journal on Numerical Analysis, 20 (1987), 172.
[22] K.E., Atkinson and W., Han, Theoretical Numerical Analysis, Springer-Verlag, New York, 2001.
[23] K.E., Atkinson and F.A., Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM Journal on Numerical Analysis, 24 (1987), 1352–1373.
[24] K.E., Atkinson and F.A., Potra, On the discrete Galerkin method for Fredholm integral equations of the second kind, IMA Journal of Numerical Analysis, 9 (1989), 385.
[25] K.E., Atkinson and I.H., Sloan, The numerical solution of the first kind logarithmic kernel integral equations on smooth open curves, Mathematics of Computation, 56 (1991), 119.
[26] I., Babuška and J.E., Osborn, Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues, SIAM Journal on Numerical Analysis, 24 (1987), 1249–1276.
[27] I., Babuška and J.E., Osborn, Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Mathematics of Computation, 52 (1989), 275.
[28] G., Beylkin, R., Coifman and V., Rokhlin, Fast wavelet transforms and numerical algorithms I, Communications on Pure and Applied Mathematics, 44 (1991), 141–183.
[29] R., Bialecki and A., Nowak, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions, Applied Mathematical Modelling, 5 (1981), 417.
[30] S., Börm, L., Grasedyck and W., Hackbusch, Introduction to hierarchical matrices with applications, Engineering Analysis with Boundary Elements, 27 (2003), 405–422.
[31] J.H., Bramble and J.E., Osborn, Rate of convergence estimates for nonselfadjoint eigenvalue approximations, Mathematics of Computation, 27 (1973), 525.
[32] C., Brebbia, J., Telles and L., Wrobel, Boundary Element Technique: Theory and Applications in Engineering, Springer-Verlag, Berlin, 1984.
[33] M., Brenner, Y., Jiang and Y., Xu, Multiparameter regularization for Volterra kernel identification via multiscale collocation methods, Advances in Computational Mathematics, 31 (2009), 421.
[34] H., Brunner, On the numerical solution of nonlinear Volterra–Fredholm integral equations by collocation methods, SIAM Journal on Numerical Analysis, 27 (1990), 987.
[35] H., Brunner, On implicitly linear and iterated collocation methods for Hammerstein integral equations, Journal of Integral Equations and Applications, 3 (1991), 475.
[36] H.J., Bungartz and M., Griebel, Sparse, grids, Acta Numerica, 13 (2004), 147–269.
[37] H., Cai and Y., Xu, A fast Fourier–Galerkin method for solving singular boundary integral equations, SIAM Journal on Numerical Analysis, 46 (2008), 1965–1984.
[38] Y., Cao, T., Herdman and Y., Xu, A hybrid collocation method for Volterra integral equations with weakly singular kernels, SIAM Journal on Numerical Analysis, 41 (2003), 264.
[39] Y., Cao, M., Huang, L., Liu and Y., Xu, Hybrid collocation methods for Fredholm integral equations with weakly singular kernels, Applied Numerical Mathematics, 57 (2007), 549.
[40] Y., Cao, B., Wu and Y., Xu, A fast collocation method for solving stochastic integral equations, SIAM Journal on Numerical Analysis, 47 (2009), 3744–3767.
[41] Y., Cao and Y., Xu, Singularity preserving Galerkin methods for weakly singular Fredholm integral equations, Journal of Integral Equations and Applications, 6 (1994), 303.
[42] T., Carleman, Über eine nichtlineare Randwertaufgabe bei der Gleichung Δu=0, Mathematische Zeitschrift, 9 (1921), 35.
[43] A., Cavaretta, W., Dahmen and C.A., Micchelli, Stationary subdivision, Memoirs of the American Mathematical Society, No. 453, 1991.
[44] F., Chatelin, Spectral Approximation of Linear Operators, Academic Press, New York, 1983.
[45] J., Chen, Z., Chen and S., Cheng, Multilevel augmentation methods for solving the sine-Gordon equation, Journal of Mathematical Analysis and Applications, 375 (2011), 706.
[46] J., Chen, Z., Chen and Y., Zhang, Fast singularity preserving methods for integral equations with non-smooth solutions, Journal of Integral Equations and Applications, 24 (2012), 213.
[47] M., Chen, Z., Chen and G., Chen, Approximate Solutions of Operator Equations, World Scientific, Singapore, 1997.
[48] Q., Chen, C.A., Micchelli and Y., Xu, On the matrix completion problem for multivariate filter bank construction, Advances in Computational Mathematics, 26 (2007), 173.
[49] Q., Chen, T., Tang and Z., Teng, A fast numerical method for integral equations of the first kind with logarithmic kernel using mesh grading, Journal of Computational Mathematics, 22 (2004), 287.
[50] X., Chen, Z., Chen and B., Wu, Multilevel augmentation methods with matrix compression for solving reformulated Hammerstein equations, Journal of Integral Equations and Applications, 24 (2012), 513.
[51] X., Chen, Z., Chen, B., Wu and Y., Xu, Fast multilevel augmentation methods for nonlinear boundary integral equations, SIAM Journal on Numerical Analysis, 49 (2011), 2231–2255.
[52] X., Chen, Z., Chen, B., Wu and Y., Xu, Fast multilevel augmentation methods for nonlinear boundary integral equations II: efficient implementation, Journal of Integral Equations and Applications, 24 (2012), 545.
[53] X., Chen, R., Wang and Y., Xu, Fast Fourier–Galerkin methods for nonlinear boundary integral equations, Journal of Scientific Computing, 56 (2013), 494–514.
[54] Z., Chen, S., Cheng, G., Nelakanti and H., Yang, A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization. International Journal of Computer Mathematics, 87 (2010), 565.
[55] Z., Chen, S., Cheng and H., Yang, Fast multilevel augmentation methods with compression technique for solving ill-posed integral equations, Journal of Integral Equations and Applications, 23 (2011), 39.
[56] Z., Chen, S., Ding, Y., Xu and H., Yang, Multiscale collocation methods for illposed integral equations via a coupled system, Inverse Problems, 28 (2012), 025006.
[57] Z., Chen, S., Ding and H., Yang, Multilevel augmentation algorithms based on fast collocation methods for solving ill-posed integral equations, Computers & Mathematics with Applications, 62 (2011), 2071–2082.
[58] Z., Chen, Y., Jiang, L., Song and H., Yang, A parameter choice strategy for a multi-level augmentation method solving ill-posed operator equations, Journal of Integral Equations and Applications, 20 (2008), 569.
[59] Z., Chen, J., Li and Y., Zhang, A fast multiscale solver for modified Hammerstein equations, Applied Mathematics and Computation, 218 (2011), 3057–3067.
[60] Z., Chen, G., Long and G., Nelakanti, The discrete multi-projection method for Fredholm integral equations of the second kind, Journal of Integral Equations and Applications, 19 (2007), 143.
[61] Z., Chen, G., Long and G., Nelakanti, Richardson extrapolation of iterated discrete projection methods for eigenvalue approximation, Journal of Computational and Applied Mathematics, 223 (2009), 48.
[62] Z., Chen, G., Long, G., Nelakanti and Y., Zhang, Iterated fast collocation methods for integral equations of the second kind, Journal of Scientific Computing, 57 (2013), 502.
[63] Z., Chen, Y., Lu, Y., Xu and H., Yang, Multi-parameter Tikhonov regularization for linear ill-posed operator equations, Journal of Computational Mathematics, 26 (2008), 37.
[64] Z., Chen, C.A., Micchelli and Y., Xu, The Petrov–Galerkin methods for second kind integral equations II: Multiwavelet scheme, Advances in Computational Mathematics, 7 (1997), 199.
[65] Z., Chen, C.A., Micchelli and Y., Xu, A construction of interpolating wavelets on invariant sets, Mathematics of Computation, 68 (1999), 1569–1587.
[66] Z., Chen, C.A., Micchelli and Y., Xu, Hermite interpolating wavelets, in L., Li, Z., Chen and Y., Zhang (eds), Lecture Notes in Scientific Computation, International Culture Publishing, Beiging, 2000, pp. 31–39.
[67] Z., Chen, C.A., Micchelli and Y., Xu, A multilevel method for solving operator equations, Journal of Mathematical Analysis and Applications, 262 (2001), 688–699.
[68] Z., Chen, C.A., Micchelli and Y., Xu, Discrete wavelet Petrov–Galerkin methods, Advances in Computational Mathematics, 16 (2002), 1.
[69] Z., Chen, C.A., Micchelli and Y., Xu, Fast collocation method for second kind integral equations, SIAM Journal on Numerical Analysis, 40 (2002), 344.
[70] Z., Chen, G., Nelakanti, Y., Xu and Y., Zhang, A fast collocation method for eigenproblems of weakly singular integral operators, Journal of Scientific Computing, 41 (2009), 256.
[71] Z., Chen, B., Wu and Y., Xu, Multilevel augmentation methods for solving operator equations, Numerical Mathematics.A Journal of Chinese Universities, 14 (2005), 31.
[72] Z., Chen, B., Wu and Y., Xu, Error control strategies for numerical integrations in fast collocation methods, Northeastern Mathematical Journal, 21(2) (2005), 233–252.
[73] Z., Chen, B., Wu and Y., Xu, Multilevel augmentation methods for differential equations, Advances in Computational Mathematics, 24 (2006), 213.
[74] Z., Chen, B., Wu and Y., Xu, Fast numerical collocation solutions of integral equations, Communications on Pure and Applied Mathematics, 6 (2007), 649–666.
[75] Z., Chen, B., Wu and Y., Xu, Fast collocation methods for high-dimensional weakly singular integral equations, Journal of Integral Equations and Applications, 20 (2008), 49.
[76] Z., Chen, B., Wu and Y., Xu, Fast multilevel augmentation methods for solving Hammerstein equations, SIAM Journal on Numerical Analysis, 47 (2009), 2321–2346.
[77] Z., Chen and Y., Xu, The Petrov–Galerkin and integrated Petrov–Galerkin methods for second kind integral equations, SIAM Journal on Numerical Analysis, 35 (1998), 406.
[78] Z., Chen, Y., Xu and H., Yang, A multilevel augmentation method for solving ill-posed operator equations, Inverse Problems, 22 (2006), 155.
[79] Z., Chen, Y., Xu and H., Yang, Fast collocation methods for solving ill-posed integral equations of the first kind, Inverse Problems, 24 (2008), 065007.
[80] Z., Chen, Y., Xu and J., Zhao, The discrete Petrov–Galerkin method for weakly singular integral equations, Journal of Integral Equations and Applications, 11 (1999), 1.
[81] D., Chien and K.E., Atkinson, A discrete Galerkin method for hypersingular boundary integral equations, IMA Journal of Numerical Analysis, 17 (1997), 463–478.
[82] C.K., Chui and J.Z., Wang, A cardinal spline approach to wavelets, Proceedings of the American Mathematical Society, 113 (1991), 785.
[83] K.C., Chung and T.H., Yao, On lattices admitting unique Lagrange interpolation, SIAM Journal on Numerical Analysis, 14 (1977), 735.
[84] A., Cohen, W., Dahmen and R., DeVore, Multiscale decomposition on bounded domains, Transactions of the American Mathematical Society, 352 (2000), 3651–3685.
[85] M., Cohen and J., Wallace, Radiosity and Realistic Image Synthesis, Academic Press, New York, 1993.
[86] J.B., Convey, A Course in Functional Analysis, Springer-Verlag, New York, 1990.
[87] F., Cucker and S., Smale, On the mathematical foundation of learning, Bulletin of the American Mathematical Society, 39 (2002), 1.
[88] W., Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica, 6 (1997), 55.
[89] W., Dahmen, H., Harbrecht and R., Schneider, Compression techniques for boundary integral equations – asymptotically optimal complexity estimates, SIAM Journal on Numerical Analysis, 43 (2006), 2251–2271.
[90] W., Dahmen, H., Harbrecht and R., Schneider, Adaptive methods for boundary integral equations: Complexity and convergence estimates, Mathematics of Computation, 76 (2007), 1243–1274.
[91] W., Dahmen, A., Kunoth and R., Schneider, Operator equations, multiscale concepts and complexity, in The Mathematics of Numerical Analysis (Park City, UT, 1995), pp. 225–261 [Lecture in Applied Mathematics, No. 32, American Mathematical Society, Providence, RI, 1996].
[92] W., Dahmen and C.A., Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM Journal on Numerical Analysis, 30 (1993), 507.
[93] W., Dahmen and C.A., Micchelli, Biorthogonal wavelet expansions, Constructive Approximation, 13 (1997), 293.
[94] W., Dahmen, S., Prössdorf and R., Schneider, Wavelet approximation methods for pseudodifferential equations I: Stability and convergence, Mathematische Zeitschrift, 215 (1994), 583.
[95] W., Dahmen, S., Prössdorf and R., Schneider, Wavelet approximation methods for pseudodifferential equations II: Matrix compression and fast solutions, Advances in Computational Mathematics, 1 (1993), 259.
[96] W., Dahmen, R., Schneider and Y., Xu, Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation, Numerische Mathematik, 86 (2000), 49.
[97] I., Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909.
[98] I., Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics No. 61, SIAM, Philadelphia, PA, 1992.
[99] K., Deimling and D., Klaus, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
[100] R., DeVore, B., Jawerth and V., Popov, Compression of wavelet decompositions, American Journal of Mathematics, 114 (1992), 737.
[101] R., DeVore and B., Lucier, Wavelets, Acta Numerica, 1 (1991), 1.
[102] J., Dicka, P., Kritzerb, F.Y., Kuoc and I.H., Sloan, Lattice-Nyström method for Fredholm integral equations of the second kind with convolution type kernels, Journal of Complexity, 23 (2007), 752.
[103] V., Dicken and P., Maass, Wavelet-Galerkin methods for ill-posed problems. Journal of Inverse and Ill-Posed Problems, 4 (1996), 203.
[104] S., Ding and H., Yang, Multilevel augmentation methods for nonlinear ill-posed problems, International Journal of Computer Mathematics, 88 (2011), 3685–3701.
[105] S., Ehrich and A., Rathsfeld, Piecewise linear wavelet collocation, approximation of the boundary manifold, and quadrature, Electronic Transactions on Numerical Analysis, 12 (2001), 149.
[106] H.W., Engl, M., Hanke and A., Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
[107] W., Fang and M., Lu, A fast collocation method for an inverse boundary value problem, International Journal for Numerical Methods in Engineering, 59 (2004), 1563–1585.
[108] W., Fang, F., Ma and Y., Xu, Multilevel iteration methods for solving integral equations of the second kind, Journal of Integral Equations and Applications, 14 (2002), 355.
[109] W., Fang, Y., Wang and Y., Xu, An implementation of fast wavelet Galerkin methods for integral equations of the second kind, Journal of Scientific Computing, 20, 277.
[110] I., Fenyö and H., Stolle, Theorie und Praxis der Linearen Integralgleichungen, Vols 1–4, Birkháuser-Verlag, Berlin, 1981–84.
[111] N.J., Ford, M.L., Morgado and M., Rebelo, Nonpolynomial collocation approximation of solutions to fractional differential equations, Fractional Calculus and Applied Analysis, 16 (2013), 874.
[112] N.J., Ford, M.L., Morgado and M., Rebelo, High order numerical methods for fractional terminal value problems, Computational Methods in Applied Mathematics, 14 (2014), 55.
[113] W.F., Ford, Y., Xu and Y., Zhao, Derivative correction for quadrature formulas, Advances in Computational Mathematics, 6 (1996), 139.
[114] L., Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, MA, 1988.
[115] L., Greengard and V., Rokhlin, A fast algorithm for particle simulation, Journal of Computational Physics, 73 (1987), 325.
[116] C.W., Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Research Notes in Mathematics No. 105, Pitman, Boston, MA, 1984.
[117] C.W., Groetsch, Uniform convergence of regularization methods for Fredholm equations of the first kind, Journal of Australian Mathematical Society, Series A, 39 (1985), 282.
[118] C.W., Groetsch, Convergence analysis of a regularized degenerate kernel method for Fredholm equations of the first kind, Integral Equations and Operator Theory, 13 (1990), 67.
[119] C.W., Groetsch, Linear inverse problems, in O., Scherze (ed.), Handbook of Mathematical Methods in Imaging, pp 3–41, Springer-Verlag, New York, 2011.
[120] W., Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, Berlin, 1985.
[121] W., Hackbusch, Integral Equations: Theory and Numerical Treatment [translated and revised by the author from the 1989 German original], International Series of Numerical Mathematics No. 120, Birháuser-Verlag, Basel, 1995.
[122] W., Hackbusch, A sparse matrix arithmetic based on H-matrices, I. Introduction to H-matrices, Computing, 62 (1999), 89.
[123] W., Hackbusch and B., Khoromskij, A sparse H-matrix arithmetic: General complexity estimates, Numerical Analysis 2000, Vol. VI, Ordinary differential equations and integral equations, Journal of Computational and Applied Mathematics, 125 (2000), 479.
[124] W., Hackbusch and Z., Nowak, A multilevel discretization and solution method for potential flow problems in three dimensions, in E.H., Hirschel (ed.), Finite Approximations in Fluid Mechanics, Notes on Numerical Fluid Mechanics No. 14, Vieweg, Braunschweig, 1986.
[125] W., Hackbusch and Z., Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numerische Mathematik, 54 (1989), 463–491.
[126] T., Haduong, La methode de Schenck pour la resolution numerique du probleme de radiation acoustique, Bull. Dir. Etudes Recherces, Ser.C, Math.Inf. Service Inf.Math. Appl., 2 (1979), 15.
[127] U., Hámarik, On the discretization error in regularized projection methods with parameter choice by discrepancy principle, in A.N., Tikhonov (ed.), Ill-posed problems in Natural Sciences, VSP, Utrecht/TVP, Moscow, 1992, pp. 24–29.
[128] U., Hámarik, Quasioptimal error estimate for the regularized Ritz–Galerkin method with the a-posteriori choice of the parameter, Acta et Commentationes Universitatis Tartuensis, 937 (1992), 63.
[129] U., Hámarik, On the parameter choice in the regularized Ritz–Galerkin method, Eesti Teaduste Akadeemia Toimetise d. Fsika. Matemaatika, 42 (1993), 133.
[130] A., Hammerstein, Nichtlineare integralgleichungen nebst anwendungen, Acta Mathematica, 54 (1930), 117.
[131] G., Han, Extrapolation of a discrete collocation-type method of Hammerstein equations, Journal of Computational and Applied Mathematics, 61 (1995), 73–86.
[132] G., Han and J., Wang, Extrapolation of Nystrom solution for two-dimensional nonlinear Fredholm integral equations, Journal of Scientific Computing, 14 (1999), 197.
[133] M., Hanke and C.R., Vogel, Two-level preconditioners for regularized inverse problems I: Theory, Numerische Mathematik, 83 (1999), 385.
[134] M., Hanke and C.R., Vogel, Two-level preconditioners for regularized inverse problems II: Implementation and numerical results, preprint.
[135] H., Harbrecht, U., Káhler and R., Schneider, Wavelet matrix compression for boundary integral equations, in Parallel Algorithms and Cluster Computing, pp. 129–149, Lecture Notes in Computational Science and Engineering No. 52, Springer-Verlag, Berlin, 2006.
[136] H., Harbrecht, M., Konik and R., Schneider, Fully discrete wavelet Galerkin schemes, Engineering Analysis with Boundary Elements, 27 (2003), 423.
[137] H., Harbrecht, S., Pereverzev and R., Schneider, Self-regularization by projection for noisy pseudodifferential equations of negative order, Numerische Mathematik, 95 (2003), 123.
[138] H., Harbrecht and R., Schneider, Wavelet Galerkin schemes for 2D-BEM, in Operator Theory: Advances and Applications, Vol. 121, Birkháuser-Verlag, Berlin, 2001.
[139] H., Harbrecht and R., Schneider, Wavelet Galerkin schemes for boundary integral equations – implementation and quadrature, SIAM Journal on Scientific Computing, 27 (2006), 1347–1370.
[140] H., Harbrecht and R., Schneider, Rapid solution of boundary integral equations by wavelet Galerkin schemes, in Multiscale, Nonlinear and Adaptive Approximation, pp. 249–294, Springer-Verlag, Berlin, 2009.
[141] E., Hille and J., Tamarkin, On the characteristic values of linear integral equations, Acta Mathematica 57 (1931), 1.
[142] R.A., Horn and C.R., Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[143] G.C., Hsiao and A., Rathsfeld, Wavelet collocation methods for a first kind boundary integral equation in acoustic scattering, Advances in Computational Mathematics, 17 (2002), 281.
[144] G.C., Hsiao andW. L., Wendland, Boundary Integral Equations, Springer-Varlag, Berlin, 2008.
[145] C., Huang, H., Guo and Z., Zhang, A spectral collocation method for eigenvalue problems of compact integral operators, Journal of Integral Equations and Applications, 25 (2013), 79.
[146] M., Huang, Wavelet Petrov–Galerkin algorthims for Fredholm integral equations of the second kind, PhD thesis, Academia Sinica (in Chinese), 2003.
[147] M., Huang, A construction of multiscale bases for Petrov–Galerkin methods for integral equations, Advances in Computational Mathematics, 25 (2006), 7.
[148] J.E., Hutchinson, Fractals and self similarity, Indiana University Mathematics Journal, 30 (1981), 713.
[149] M., Jaswon and G., Symm, Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London, 1977.
[150] Y., Jeon, An indirect boundary integral equation method for the biharmonic equation, SIAM Journal on Numerical Analysis, 31 (1994), 461.
[151] Y., Jeon, New boundary element formulas for the biharmonic equation, Advances in Computational Mathematics, 9 (1998), 97.
[152] Y., Jeon, New indirect scalar boundary integral equation formulas for the biharmonic equation, Journal of Computational and Applied Mathematics, 135 (2001), 313.
[153] Y., Jeon and W., McLean, A new boundary element method for the biharmonic equation with Dirichlet boundary conditions, Advances in Computational Mathematics, 19 (2003), 339.
[154] Y., Jiang, B., Wang and Y., Xu, A fast Fourier–Galerkin method solving a boundary integral equation for the biharmonic equation, SIAM Journal on Numerical Analysis, 52 (2014), 2530–2554.
[155] Y., Jiang and Y., Xu, Fast Fourier Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition, Journal of Computational and Applied Mathematics, 234 (2010), 2792–2807.
[156] Q., Jin and Z., Hou, On an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems, Numerische Mathematik, 83 (1999), 139.
[157] B., Kaltenbacher, On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems, 17 (2001), 767.
[158] H., Kaneko, K., Neamprem and B., Novaprateep, Wavelet collocation method and multilevel augmentation method for Hammerstein equations, SIAM Journal on Scientific Computing, 34 (2012), A309–A338.
[159] H., Kaneko, R., Noren and B., Novaprateep, Wavelet applications to the Petrov–Galerkin method for Hammerstein equations, Applied Numerical Mathematics, 45 (2003), 255.
[160] H., Kaneko, R.D., Noren and P.A., Padilla, Superconvergence of the iterated collocation methods for Hammerstein equations, Journal of Computational and Applied Mathematics, 80 (1997), 335.
[161] H., Kaneko, R., Noren and Y., Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, Journal of Integral Equations and Applications, 4 (1992), 391.
[162] H., Kaneko, R., Noren and Y., Xu, Regularity of the solution of Hammerstein equations with weakly singular kernel, Integral Equation Operator Theory, 13 (1990), 660.
[163] H., Kaneko and Y., Xu, Degenerate kernel method for Hammerstein equations, Mathematics of Computation, 56 (1991), 141.
[164] H., Kaneko and Y., Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Mathematics of Computation, 62 (1994), 739.
[165] H., Kaneko and Y., Xu, Superconvergence of the iterated Galerkin methods for Hammerstein equations, SIAM Journal on Numerical Analysis, 33 (1996), 1048–1064.
[166] T., Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, Journal d'Analyse Mathématique, 6 (1958), 261.
[167] T., Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1976.
[168] C.T., Kelley, A fast two-grid method for matrix H-equations, Transport Theory and Statistical Physics, 18 (1989), 185.
[169] C.T., Kelley, A fast multilevel algorithm for integral equations, SIAM Journal on Numerical Analysis, 32 (1995), 501.
[170] C.T., Kelley and E.W., Sachs, Fast algorithms for compact fixed point problems with inexact function evaluations, SIAM Journal on Scientific and Statistical Computing, 12 (1991), 725.
[171] M., Kelmanson, Solution of nonlinear elliptic equations with boundary singularities by an integral equation method, Journal of Computational Physics, 56 (1984), 244.
[172] D., Kincaid and W., Cheney, Numerical Analysis: Mathematics of Scientific Computing, 3rd edn, American Mathematical Society, Providence, RI, 2002.
[173] A., Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn, Applied Mathematical Sciences No. 120, Springer-Verlag, New York, 2011.
[174] E., Klann, R., Ramlau and L., Reichel, Wavelet-based multilevel methods for linear ill-posed problems, BIT Numerical Mathematics, 51 (2011), 669.
[175] M.A., Kransnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York, 1964.
[176] M.A., Kransnosel'skii, G.M., Vainikko, P.P., Zabreiko, Ya.B., Rutiskii and V.Ya., Stetsenko, Approximate Solution of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972.
[177] R., Kress, Linear Integral Equations, Springer-Verlag, Berlin, 1989.
[178] R., Kress, Numerical Analysis, Graduate Texts in Mathematics No. 181, Springer-Verlag, New York, 1998.
[179] S., Kumar, Superconvergence of a collocation-type method for Hammerstein equations, IMA Journal of Numerical Analysis, 7 (1987), 313.
[180] S., Kumar, A discrete collocation-type method for Hammerstein equation, SIAM Journal on Numerical Analysis, 25 (1988), 328.
[181] S., Kumar and I.H., Sloan, A new collocation-type method for Hammerstein integral equations, Mathematics of Computation, 48 (1987), 585.
[182] L.J., Lardy, A variation of Nystrom's method for Hammerstein equations, Journal of Integral Equations, 3 (1981), 43.
[183] P.D., Lax, Functional Analysis, Wiley-Interscience, New York, 2002.
[184] F., Li, Y., Li and Z., Li, Existence of solutions to nonlinear Hammerstein integral equations and applications, Journal of Mathematical Analysis and Applications, 323 (2006), 209.
[185] E., Lin, Multiscale approximation for eigenvalue problems of Fredholm integral equations, Journal of Applied Functional Analysis, 2 (2007), 461.
[186] G.G., Lorentz, Approximation Theory and Functional Analysis, Academic Press, Boston, MA, 1991.
[187] Y., Lu, L., Shen and Y., Xu, Shadow block iteration for solving linear systems obtained from wavelet transforms, Applied and Computational Harmonic Analysis, 19 (2005), 359.
[188] Y., Lu, L., Shen and Y., Xu, Multi-parameter regularization methods for highresolution image reconstruction with displacement errors, IEEE Transactions on Circuits and Systems I, 54 (2007), 1788–1799.
[189] Y., Lu, L., Shen and Y., Xu, Integral equation models for image restoration: High accuracy methods and fast algorithms, Inverse Problems, 26 (2010) 045006.
[190] M.A., Lukas, Comparisons of parameter choice methods for regularization with discrete noisy data, Inverse Problems, 14 (1998), 161.
[191] X., Luo, L., Fan, Y., Wu and F., Li, Fast multilevel iteration methods with compression technique for solving ill-posed integral equations, Journal of Computational and Applied Mathematics, 256 (2014), 131.
[192] X., Luo, F., Li and S., Yang, A posteriori parameter choice strategy for fast multiscale methods solving ill-posed integral equations, Advances in Computational Mathematics, 36 (2012), 299.
[193] P., Maass, S.V., Pereverzev, R., Ramlau and S.G., Solodky, An adaptive discretization for Tikhonov–Phillips regularization with a posteriori parameter selection, Numerische Mathematik, 87 (2001), 485.
[194] P., Mathé, Saturation of regularization methods for linear ill-posed problems in Hilbert spaces, SIAM Journal on Numerical Analysis, 42 (2004), 968.
[195] P., Mathé and S.V., Pereverzev, Discretization strategy for linear ill-posed problems in variable Hilbert scales, Inverse Problems, 19 (2003), 1263–1277.
[196] C.A., Micchelli, Using the refinable equation for the construction of pre-wavelets, Numerical Algorithm, 1 (1991), 75.
[197] C.A., Micchelli and M., Pontil, Learning the kernel function via regularization, Journal of Machine Learning Research, 6 (2005), 1099–1125.
[198] C.A., Micchelli, T., Sauer and Y., Xu, A construction of refinable sets for interpolating wavelets, Results in Mathematics, 34 (1998), 359.
[199] C.A., Micchelli, T., Sauer and Y., Xu, Subdivision schemes for iterated function systems, Proceedings of the American Mathematical Society, 129 (2001), 1861–1872.
[200] C.A., Micchelli and Y., Xu, Using the matrix refinement equation for the construction of wavelets on invariant sets, Applied and Computational Harmonic Analysis, 1 (1994), 391.
[201] C.A., Micchelli and Y., Xu, Reconstruction and decomposition algorithms for biorthogonal multiwavelets, Multidimensional Systems and Signal Processing, 8 (1997), 31.
[202] C.A., Micchelli, Y., Xu and Y., Zhao, Wavelet Galerkin methods for secondkind integral equations, Journal of Computational and Applied Mathematics, 86 (1997), 251.
[203] S.G., Mikhlin, Mathematical Physics, an Advanced Course, North-Holland, Amsterdam, 1970.
[204] G., Monegato and I.H., Sloan, Numerical solution of the generalized airfoil equation for an airfoil with a flap, SIAM Journal on Numerical Analysis, 34 (1997), 2288–2305.
[205] M.T., Nair, On strongly stable approximations, Journal of Australian Mathematical Society, Series A, 52 (1992), 251.
[206] M.T., Nair, A unified approach for regularized approximation methods for Fredholm integral equations of the first kind, Numerical Functional Analysis and Optimization, 15 (1994), 381.
[207] M.T., Nair and S.V., Pereverzev, Regularized collocation method for Fredholm integral equations of the first kind, Journal of Complexity, 23 (2007), 454.
[208] J., Nedelec, Approximation des Equations Integrales en Mecanique et en Physique, Lecture Notes, Centre Math.Appl., Ecole Polytechnique, Palaiseau, France, 1977.
[209] G., Nelakanti, A degenerate kernel method for eigenvalue problems of compact integral operators, Advances in Computational Mathematics, 27 (2007), 339–354.
[210] G., Nelakanti, Spectral Approximation for Integral Operators, PhD thesis, Indian Insitute of Technology, Bombay, 2003.
[211] D.W., Nychka and D.D., Cox, Convergence rates for regularized solutions of integral equations from discrete noisy data, Annals of Statistics, 17 (1989), 556–572.
[212] J.E., Osborn, Spectral approximation for compact operators, Mathematics of Computation, 29 (1975), 712.
[213] R., Pallav and A., Pedas, Quadratic spline collocation method for weakly singular integral equations and corresponding eigenvalue problem, Mathematical Modelling and Analysis, 7 (2002), 285.
[214] B.L., Panigrahi and G., Nelakanti, Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem, Journal of Applied Mathematics and Computing, 43 (2013), 175.
[215] B.L., Panigrahi and G., Nelakanti, Richardson extrapolation of iterated discrete Galerkin method for eigenvalue problem of a two dimensional compact integral operator, Journal of Scientific Computing, 51 (2012), 421.
[216] S.V., Pereverzev and E., Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM Journal on Numerical Analysis, 43 (2005), 2060–2076.
[217] D.L., Phillips, A technique for the numerical solution of certain integral equations of the first kind, Journal of the Association for Computing Machinery, 9 (1962), 84.
[218] M., Pincus, Gaussian processes and Hammerstein integral equations, Transactions of the American Mathematical Society, 134 (1968), 193.
[219] R., Plato, On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations, Numerische Mathematik, 75 (1996), 99.
[220] R., Plato, The Galerkin scheme for Lavrentiev's m-times iterated method to solve linear accretive Volterra integral equations of the first kind, BIT. Numerical Mathematics, 37 (1997), 404.
[221] R., Plato and U., Hámarik, On pseudo-optimal parameter choices and stopping rules for regularization methods in Banach spaces, Numerical Functional Analysis and Optimization, 17 (1996), 181.
[222] R., Plato and G., Vainikko, On the regularization of projection methods for solving ill-posed problems, Numerische Mathematik, 57 (1990), 63.
[223] R., Prazenica, R., Lind and A., Kurdila, Uncertainty estimation from Volterra kernels for robust flutter analysis, Journal of Guidance, Control, and Dynamics, 26 (2003), 331.
[224] M.P., Rajan, Convergence analysis of a regularized approximation for solving Fredholm integral equations of the first kind, Journal of Mathematical Analysis and Applications, 279 (2003), 522.
[225] A., Rathsfeld, A wavelet algorithm for the solution of the double layer potential equation over polygonal boundaries, Journal of Integral Equations and Applications, 7 (1995), 47.
[226] A., Rathsfeld, A wavelet algorithm for the solution of a singular integral equation over a smooth two-dimensional manifold, Journal of Integral Equations and Applications, 10 (1998), 445.
[227] A., Rathsfeld and R., Schneider, On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations, Mathematical Methods in the Applied Sciences, 26 (2003), 937.
[228] T., Raus, About regularization parameter choice in case of approximately given errors of data, Acta et Commentationes Universitatis Tartuensis, 937 (1992), 77–89.
[229] M., Rebeloa and T., Diogob, A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel, Journal of Computational and Applied Mathematics, 234 (2010), 2859–2869.
[230] L., Reichel and A., Shyshkov, Cascadic multilevel methods for ill-posed problems, Journal of Computational and Applied Mathematics, 233 (2010), 1314–1325.
[231] A., Rieder, A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization, Numerische Mathematik, 75 (1997), 501.
[232] S.D., Riemenschneider and Z., Shen, Wavelets and pre-wavelets in low dimensions, Journal of Approximation Theory, 71 (1992), 18.
[233] K., Riley, Two-level preconditioners for regularized ill-posed problems, PhD thesis, Montana State University, 1999.
[234] F., Rizzo, An integral equation approach to boundary value problems of classical elastostatics, Quarterly Journal of Applied Mathematics, 25 (1967), 83.
[235] V., Rokhlin, Rapid solution of integral equations of classical potential theory, Journal of Computational Physics, 60 (1983), 187.
[236] H.L., Royden, Real Analysis, Macmillan, New York, 1963.
[237] L., Rudin, S., Osher and E., Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259.
[238] T., Runst and W., Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, de Gryuter, Berlin, 1996.
[239] K., Ruotsalainen and W., Wendland, On the boundary element method for some nonlinear boundary value problems, Numerische Mathematik, 53 (1988), 299–314.
[240] J., Saranen, Projection methods for a class of Hammerstein equations, SIAM Journal on Numerical Analysis, 27 (1990), 1445–1449.
[241] R., Schneider, Multiskalen- und wavelet-Matrixkompression: Analyiss-sasierte Methoden zur effizienten Lösung groβer vollbesetzter Gleichungs-systeme, Habilitationsschrift, Technische Hochschule Darmstadt, 1995.
[242] C., Schwab, Variable order composite quadrature of singular and nearly singular integrals, Computing, 53 (1994), 173.
[243] Y., Shen and W., Lin, Collocation method for the natural boundary integral equation, Applied Mathematics Letters, 19 (2006), 1278–1285.
[244] F., Sillion and C., Puech, Radiosity and Global Illumination, Morgan Kaufmann, San Francisco, CA, 1994.
[245] I.H., Sloan, Iterated Galerkin method for eigenvalue problem, SIAM Journal on Numerical Analysis, 13 (1976), 753.
[246] I.H., Sloan, Superconvergence, in M., Golberg (ed.), Numerical Solution of Integral Equations, Plenum, New York, 1990, pp. 35–70.
[247] I.H., Sloan and V., Thomee, Superconvergence of the Galerkin iterates for integral equations of the second kind, Journal of Integral Equations, 9 (1985), 1.
[248] S.G., Solodky, On a quasi-optimal regularized projection method for solving operator equations of the first kind, Inverse Problems, 21 (2005), 1473–1485.
[249] G.W., Stewart, Fredholm, Hilbert, Schmidt: Three Fundamental Papers on Integral Equations, translated with commentary by G.W., Stewat, 2011. Available at www.cs.und.edu/stewart/FHS.pdf.
[250] J., Tausch, The variable order fast multipole method for boundary integral equations of the second kind, Computing, 72 (2004), 267.
[251] J., Tausch and J., White, Multiscale bases for the sparse representation of boundary integral operators on complex geometry, SIAM Journal on Scientific Computation, 24 (2003), 1610–1629.
[252] A.N., Tikhonov, Solution of incorrectly formulated problems and the regularization method, Doklady Akademii Nauk SSSR, 151 (1963), 501–504 [translated in Soviet Mathematics 4: 1035–1038].
[253] F.G., Tricomi, Integral Equations, Dover Publications, New York, 1985.
[254] A.E., Taylor and D.C., Lay, Introduction to Functional Analysis, 2nd edn, John Wiley & Sons, New York, 1980.
[255] G., Vainikko, A perturbed Galerkin method and the general theory of approximate methods for nonlinear equations, ž.Vyčisl.Mat. i Mat.Fiz., 7 (1967), 723–751 [English translation, U.S.S.R. Computational Mathematics and Mathematical Physics, 7 (1967), 723–751].
[256] G., Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin, 1993.
[257] G., Vainikko, A., Pedes and P., Uba, Methods of Solving Weakly Singular Integral Equations (in Russian), Tartu University, 1984.
[258] G., Vainikko and P., Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, Journal of Australian Mathematical Society, Series B, 22 (1981), 431.
[259] C.R., Vogel and M.E., Oman, Fast, robust total variation-based reconstruction of noisy, blurred images, IEEE Transactions on Image Processing, 7 (1998), 813–824.
[260] T., von Petersdorff and C., Schwab, Wavelet approximation of first kind integral equations in a polygon, Numerische Mathematik, 74 (1996), 479.
[261] T., von Petersdordd, R., Schneider and C., Schwab, Multiwavelets for second kind integral equations, SIAM Journal on Numerical Analysis, 34 (1997), 2212–2227.
[262] G., Wahba, Spline Models for Observational Data, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990.
[263] B., Wang, R., Wang and Y., Xu, Fast Fourier–Galerkin methods for first-kind logarithmic-kernel integral equations on open arcs, Science China Mathematics, 53 (2010), 1.
[264] Y., Wang and Y., Xu, A fast wavelet collocation method for integral equations on polygons, Journal of Integral Equations and Applications, 17 (2005), 277.
[265] W., Wendland, On some mathematical aspects of boundary element methods for elliptic problems, in J., Whiteman (ed.), The Mathematics of Finite Elements and Applications, Academic Press, London, 1985, pp. 230–257.
[266] W.-J., Xie and F.-R., Lin, A fast numerical solution method for two dimensional Fredholm integral equations of the second kind, Applied Numerical Mathematics, 59 (2009), 1709–1719.
[267] Y., Xu, H.L., Chen and Q., Zou, Limit values of derivatives of the Cauchy integrals and computation of the logarithmic potentials, Computing, 73 (2004), 295.
[268] Y., Xu and H., Zhang, Refinable kernels, Journal of Machine Learning Research, 8 (2007), 2083–2120.
[269] Y., Xu and Y., Zhao, Quadratures for improper integrals and their applications in integral equations, Proceedings of Symposia in Applied Mathematics, 48 (1994), 409–413.
[270] Y., Xu and Y., Zhao, Quadratures for boundary integral equations of the first kind with logarithmic kernels, Journal of Integral Equations and Applications, 8 (1996), 239.
[271] Y., Xu and Y., Zhao, An extrapolation method for a class of boundary integral equations, Mathematics of Computation, 65 (1996), 587.
[272] Y., Xu and A., Zhou, Fast Boolean approximation methods for solving integral equations in high dimensions, Journal of Integral Equations and Applications, 16 (2004), 83.
[273] Y., Xu and Q., Zou, Adaptive wavelet methods for elliptic operator equations with nonlinear terms, Advances in Computational Mathematics, 19 (2003), 99.
[274] H., Yang and Z., Hou, Convergence rates of regularized solutions and parameter choice strategy for positive semi-definite operator equation, Numerical References Mathematics. A Journal of Chinese Universities (Chinese Edition), 20 (1998), 245–251.
[275] H., Yang and Z., Hou, A posteriori parameter choice strategy for nonlinear monotone operator equations, Acta Mathematicae Applicatae Sinica, 18 (2002), 289–294.
[276] K., Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965.
[277] E., Zeidler, Nonlinear Functional Analysis and its Applications I, II/B, Springer-Verlag, New York, 1990.
[278] M., Zhong, S., Lu and J., Cheng, Multiscale analysis for ill-posed problems with semi-discrete Tikhonov regularization, Inverse Problems, 28 (2012) 065019.
[279] A., Zygmund, Trigonometric Series, Cambridge University Press, New York, 1959.