LEADER 03389nam 22004215a 450 001 9910151929403321 005 20100703234500.0 010 $a3-03719-584-3 024 70$a10.4171/084 035 $a(CKB)3710000000953864 035 $a(CH-001817-3)118-100703 035 $a(PPN)178155799 035 $a(EXLCZ)993710000000953864 100 $a20100703j20100703 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aTractability of Multivariate Problems$b[electronic resource] $eVolume II: Standard Information for Functionals /$fErich Novak, Henryk Woz?niakowski 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2010 215 $a1 online resource (675 pages) 225 0 $aEMS Tracts in Mathematics (ETM)$v12 330 $aThis three-volume set is a comprehensive study of the tractability of multivariate problems. The present second volume deals with algorithms using standard information consisting of function values for the approximation of linear and selected nonlinear functionals. An important example is numerical multivariate integration. The proof techniques used in volumes I and II are quite different. It is especially hard to establish meaningful lower error bounds for the approximation of functionals by using finitely many function values. Here, the concept of decomposable reproducing kernels is helpful, allowing it to find matching lower and upper error bounds for some linear functionals. It is then possible to conclude tractability results from such error bounds. Tractability results even for linear functionals are very rich in variety. There are infinite-dimensional Hilbert spaces for which the approximation with an arbitrarily small error of all linear functionals requires only one function value. There are Hilbert spaces for which all nontrivial linear functionals suffer from the curse of dimensionality. This holds for unweighted spaces, where the role of all variables and groups of variables is the same. For weighted spaces one can monitor the role of all variables and groups of variables. Necessary and sufficient conditions on the decay of the weights are given to obtain various notions of tractability. The text contains extensive chapters on discrepancy and integration, decomposable kernels and lower bounds, the Smolyak/sparse grid algorithms, lattice rules and the CBC (component-by-component) algorithms. This is done in various settings. Path integration and quantum computation are also discussed. The book is of interest for researchers working in computational mathematics, especially in approximation of high-dimensional problems. It is also well suited for graduate courses and seminars. 61 open problems... 606 $aMathematical theory of computation$2bicssc 606 $aNumerical analysis$2msc 606 $aComputer science$2msc 615 07$aMathematical theory of computation 615 07$aNumerical analysis 615 07$aComputer science 686 $a65-xx$a68-xx$2msc 700 $aNovak$b Erich$056070 702 $aWoz?niakowski$b Henryk 801 0$bch0018173 906 $aBOOK 912 $a9910151929403321 996 $aTractability of Multivariate Problems$92565018 997 $aUNINA LEADER 01055nam a2200265 i 4500 001 991000182769707536 005 20020509163223.0 008 000220s1975 fr ||| | fre 035 $ab11322421-39ule_inst 035 $aPARLA203555$9ExL 040 $aDip.to Filosofia$bita 082 0 $a501 100 1 $aCantin, André$0385038 245 14$aLes sciences séculières et la foi :$bles deux voies de la science au jugement de s. Pierre Damien (1007-1072) 260 $aSpoleto :$bCentro italiano di studi sull'alto Medioevo,$c1975 300 $a633 p. ;$c22 cm 490 0 $aCentro italiano di studi sull'alto Medioevo ;$v5 650 4$aPier Damiani (santo)$xConcezione delle scienze 650 4$aScienze$xTeorie$yMedioevo 907 $a.b11322421$b01-03-17$c01-07-02 912 $a991000182769707536 945 $aLE005IF XXII A 30$g1$i2005000049989$lle005$o-$pE0.00$q-$rl$s- $t0$u0$v0$w0$x0$y.i11493495$z01-07-02 996 $aSciences séculières et la foi$9824191 997 $aUNISALENTO 998 $ale005$b01-01-00$cm$da $e-$ffre$gfr $h4$i1