In the paper "Fast simulation of Gaussian random fields”, a typo occurred. Instead of it should read in Algorithm3.1, Remarkd). For convenience of the reader we reproduce below the complete corrected algorithm

Lang, Annika

Potthoff, Jürgen

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Monte Carlo Methods Appl. 19 (2013), 73–75DOI 10.1515/mcma-2013-0003 © de Gruyter 2013ErratumFast simulation of Gaussian random fields[Monte Carlo Methods Appl. 17 (2011), 195–214]Annika Lang and Jürgen PotthoffAbstract. In the paper “Fast simulation of Gaussian random fields”, a typo occurred. In-stead of .pk1kd /i D ki=li it should read .pk1kd /i D .ki  Ni=2/=li in Algorithm 3.1,Remark d). For convenience of the reader we reproduce below the complete correctedalgorithms.Keywords. Gaussian random fields, fast Fourier transform, simulation.2010 Mathematics Subject Classification. 65C05, 65C50, 60G60, 60H40.Algorithm 3.1.Remarksa) The functions FFT and FFT 1 include all necessary rescaling depending onthe used FFT algorithm and the integers Ni .b) A is a d -dimensional complex-valued array, B is real-valued.c) xk1kd denotes the grid point corresponding to the integers .k1; : : : ; kd /.The grid points are distributed equidistantly in each direction, i.e., the dis-tance of two arbitrary neighbor grid points in direction ei is given by a con-stant xi .d) The points pk1kd in the Fourier domain are given by.pk1kd /i D .ki  Ni=2/=li for i D 1; : : : ; d .Inputa) D a d -dimensional rectangular region with l1; : : : ; ld lengths of the edges,b) N1; : : : ; Nd the numbers of discretization points in each direction, all even,c) 1=2 a symmetric, positive function on Rd ,d) R. / a function that generates independentN .0; jN j 1/-distributed randomnumbers.74 A. Lang and J. PotthoffOutputGRF B on D, where the covariance is given by the Fourier transform of  .for ki D 0; : : : ; Ni   1; i D 1; : : : ; d doB.k1; : : : ; kd / R. /Iend forA FFTBIfor ki D 0; : : : ; Ni   1; i D 1; : : : ; d doA.k1; : : : ; kd / A.k1; : : : ; kd /  .pk1kd /1=2  jDj 1Iend forB  FFT 1AIAlgorithm 3.2.Remarks Same as in Algorithm 3.1.All calculations have to be done modulo Ni in the i -th direction.Input Same as in Algorithm 3.1.Output Same as in Algorithm 3.1.for ki D 0; : : : ; Ni   1, i D 1; : : : ; d   1, kd D 0; : : : ; Nd=2 doif ki 2 ¹0;Ni=2º, for all i D 1; : : : ; d thenReA.k1; : : : ; kd / R. /  .pk1kd /1=2  jDj 1IImA.k1; : : : ; kd / 0IelseReA.k1; : : : ; kd / 2 1=2R. /  .pk1kd /1=2  jDj 1IImA.k1; : : : ; kd / 2 1=2R. /  .pk1kd /1=2  jDj 1IReA.N1   k1; : : : ; Nd   kd / ReA.k1; : : : ; kd /IImA.N1   k1; : : : ; Nd   kd /  ImA.k1; : : : ; kd /Iend ifend forB  FFT 1AIAlgorithm 3.3.Remarks Same as in Algorithm 3.2.Input Same as in Algorithm 3.2.Output Same as in Algorithm 3.2.function create_random_field(D;N1; : : : ; Nd )A complex_array.N1; : : : ; Nd /;Erratum: Fast simulation of Gaussian random fields 75set_array(¹N1; : : : ; Nd º, ¹ º, A);B  FFT 1AIend functionfunction set_array(¹N1; : : : ; Nj º, ¹kjC1; : : : ; kd º, A)for ki D 0; : : : ; Ni ; i D 1; : : : ; j   1, kj D 1; : : : ; Nj =2   1 doReA.k1; : : : ; kd / 2 1=2R. /  .pk1kd /1=2  jDj 1IImA.k1; : : : ; kd / 2 1=2R. /  .pk1kd /1=2  jDj 1IReA.N1   k1; : : : ; Nd   kd / ReA.k1; : : : ; kd /IImA.N1   k1; : : : ; Nd   kd /  ImA.k1; : : : ; kd /Iif (j¹N1; : : : ; Nj ºj D 1) thenReA.0; : : : ; kd / R. /  .p0k2kd /1=2  jDj 1IImA.0; : : : ; kd / 0IReA.N1=2; : : : ; kd / R. /  .pN1=2k2kd /1=2  jDj 1IImA.N1=2; : : : ; kd / 0Ielseset_array(¹N1; : : : ; Nj 1º, ¹0; kjC1; : : : ; kd º, A);set_array(¹N1; : : : ; Nj 1º, ¹Nj =2; kjC1; : : : ; kd º, A);end ifend forend functionReceived February 2, 2013; accepted February 18, 2013.Author informationAnnika Lang, Seminar für Angewandte Mathematik, ETH Zürich,Rämistrasse 101, 8092 Zürich, Switzerland.E-mail: annika.lang@sam.math.ethz.chJürgen Potthoff, Institut für Mathematik, Universität Mannheim,68131 Mannheim, Germany.E-mail: potthoff@math.uni-mannheim.de

Erratum: Fast simulation of Gaussian random fields[Monte Carlo Methods Appl. 17 (2011), 195-214]

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