{"@context":"https:\/\/schema.org\/","@type":"PublicationVolume","@id":"#volume546","volumeNumber":4401,"name":"Dagstuhl Seminar Proceedings, Volume 4401","dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"Periodical","@id":"#series119","name":"Dagstuhl Seminar Proceedings","issn":"1862-4405","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume546"},"hasPart":[{"@type":"ScholarlyArticle","@id":"#article961","name":"04401 Abstracts Collection \u2013 Algorithms and Complexity for Continuous","abstract":"From 26.09.04 to 01.10.04, the Dagstuhl Seminar ``Algorithms and Complexity for Continuous Problems'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.\r\nDuring the seminar, several participants presented their current\r\nresearch, and ongoing work and open problems were discussed. Abstracts of\r\nthe presentations given during the seminar as well as abstracts of\r\nseminar results and ideas are put together in this paper. The first section\r\ndescribes the seminar topics and goals in general.\r\nLinks to extended abstracts or full papers are provided, if available.","keywords":["Complexity and regularization of ill-posed problems","nonlinear approximation","tractability of high-dimensional numerical problems quantum computing","stochastic computation and quantization","global optimization","differential and integral equations"],"author":[{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Traub, Joseph F.","givenName":"Joseph F.","familyName":"Traub"}],"position":1,"pageStart":1,"pageEnd":21,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Traub, Joseph F.","givenName":"Joseph F.","familyName":"Traub"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.1","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article962","name":"04401 Summary \u2013 Algorithms and Complexity for Continuous Problems","abstract":"The goal of this workshop was to bring together\r\nresearchers from different communities working on\r\ncomputational aspects of continuous problems.\r\nContinuous computational problems arise in many\r\nareas of science and engineering. Examples include\r\npath and multivariate integration, function\r\napproximation, optimization, as well as\r\ndifferential, integral, and operator equations.\r\nUnderstanding the complexity of such problems and\r\nconstructing efficient algorithms is both\r\nimportant and challenging.\r\nThe workshop was of a very interdisciplinary\r\nnature with invitees from, e.g., computer science,\r\nnumerical analysis, discrete, applied, and pure\r\nmathematics, physics, statistics, and scientific\r\ncomputation. Many of the lectures were presented\r\nby Ph.D. students.\r\nCompared to earlier meetings, several very active\r\nresearch areas received more emphasis. These\r\ninclude Quantum Computing, Complexity and\r\nTractability of high-dimensional problems,\r\nStochastic Computation, and Quantization, which\r\nwas an entirely new field for this workshop.\r\nDue to strong connections between the topics\r\ntreated at this workshop many of the participants\r\ninitiated new cooperations and research projects.\r\nFor more details, see the pdf-file with the same\r\ntitle.","keywords":["Complexity and Regularization of Ill-Posed Problems","Non-Linear Approximation","Tractability of High-Dimensional Numerical Problems Quasi-Monte Carlo Methods","Quantum Computing","Stochastic Computation and Quantization","Global Optimization","Differential and Integral Equation"],"author":[{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Traub, Joseph F.","givenName":"Joseph F.","familyName":"Traub"}],"position":2,"pageStart":0,"pageEnd":0,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Traub, Joseph F.","givenName":"Joseph F.","familyName":"Traub"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.2","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article963","name":"Fast Component-By-Component Construction of Rank-1 Lattice Rules for (Non-)Primes (Part II)","abstract":"Part I: (this part of the talk by Ronald Cools)\r\nWe restate our previous result which showed that it is possible to\r\nconstruct the generating vector of a rank-1 lattice rule in a fast way,\r\ni.e. O(s n log(n)), with s the number of dimensions and n the number of\r\npoints assumed to be prime. Here we explicitly use basic facts from\r\nalgebra to exploit the structure of a matrix \u2013 which introduces the\r\ncrucial cost in the construction \u2013 to get a matrix-vector\r\nmultiplication in time O(n log(n)) instead of O(n^2). We again stress\r\nthe fact that the algorithm works for any tensor product reproducing\r\nkernel Hilbert space.\r\n\r\nPart II: (this part of the talk by Dirk Nuyens)\r\nIn the second part we generalize the tricks used for primes to\r\nnon-primes, by basically falling back to algebraic group theory. In\r\nthis way it can be shown that also for a non-prime number of points,\r\nthis crucial matrix-vector multiplication can be done in time O(n\r\nlog(n)). We conclude that the construction of rank-1 lattice rules in\r\nan arbitrary r.k.h.s. for an arbitrary amount of points can be done in a\r\nfast way of O(s n log(n)).","keywords":["numerical integration","cubature\/quadrature","rank-1 lattice","component-by-component construction","fast algorithm"],"author":[{"@type":"Person","name":"Nuyens, Dirk","givenName":"Dirk","familyName":"Nuyens"},{"@type":"Person","name":"Cools, Ronald","givenName":"Ronald","familyName":"Cools"}],"position":3,"pageStart":1,"pageEnd":26,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Nuyens, Dirk","givenName":"Dirk","familyName":"Nuyens"},{"@type":"Person","name":"Cools, Ronald","givenName":"Ronald","familyName":"Cools"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.3","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article964","name":"Functional Quantization and Entropy for Stochastic Processes","abstract":"Let X be a Gaussian process and let U denote the \r\nStrassen ball of X. A precise link between the \r\nL^2-quantization error of X and the Kolmogorov \r\n(metric) entropy of U in a Hilbert space setting\r\nis established. In particular, the sharp \r\nasymptotics of the Kolmogorov entropy problem is \r\nderived. The condition imposed is regular \r\nvariation of the eigenvalues of the covariance \r\noperator. Good computable quantizers for Gaussian \r\nand diffusion processes and their numerical \r\nefficieny are discussed. \r\nThis is joint work with G. Pag\u00c3\u0192\u00c2\u00a8s, Universit\u00c3\u0192\u00c2\u00a9 de Paris 6.","keywords":["Functional quantization","entropy","product quantizers"],"author":[{"@type":"Person","name":"Luschgy, Harald","givenName":"Harald","familyName":"Luschgy"},{"@type":"Person","name":"Pag\u00e8s, Gilles","givenName":"Gilles","familyName":"Pag\u00e8s"}],"position":4,"pageStart":1,"pageEnd":15,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Luschgy, Harald","givenName":"Harald","familyName":"Luschgy"},{"@type":"Person","name":"Pag\u00e8s, Gilles","givenName":"Gilles","familyName":"Pag\u00e8s"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.4","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article965","name":"Information-Based Nonlinear Approximation: An Average Case Setting","abstract":"Nonlinear approximation has usually been studied\r\nunder deterministic assumption and complete\r\ninformation about the underlying functions. \r\nWe assume only partial information and we are \r\ninterested in the average case error and \r\ncomplexity of approximation. It turns out that \r\nthe problem can be essentially split into two \r\nindependent problems related to average case \r\nnonlinear (restricted) approximation from \r\ncomplete information, and average case \r\nunrestricted approximation from partial \r\ninformation. The results are then applied to \r\naverage case piecewise polynomial approximation, \r\nand to average case approximation of real \r\nsequences.","keywords":["average case setting","nonlinear approximation","information-based comlexity"],"author":{"@type":"Person","name":"Plaskota, Leszek","givenName":"Leszek","familyName":"Plaskota"},"position":5,"pageStart":1,"pageEnd":1,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Plaskota, Leszek","givenName":"Leszek","familyName":"Plaskota"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.5","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article966","name":"Lower Bounds and Non-Uniform Time Discretization for Approximation of Stochastic Heat Equations","abstract":"We study algorithms for approximation of the mild solution \r\nof stochastic heat equations on the spatial domain ]0,1[^d. \r\nThe error of an algorithm is defined in L_2-sense.\r\nWe derive lower bounds for the error of every algorithm\r\nthat uses a total of N evaluations of one-dimensional components \r\nof the driving Wiener process W. For equations with additive \r\nnoise we derive matching upper bounds and we construct \r\nasymptotically optimal algorithms. The error bounds depend on \r\nN and d, and on the decay of eigenvalues of the covariance of W\r\nin the case of nuclear noise. In the latter case the use of \r\nnon-uniform time discretizations is crucial.","keywords":["Stochastic heat equation","Non-uniform time discretization","minimal errors","upper and lower bounds"],"author":[{"@type":"Person","name":"Ritter, Klaus","givenName":"Klaus","familyName":"Ritter"},{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"}],"position":6,"pageStart":1,"pageEnd":37,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Ritter, Klaus","givenName":"Klaus","familyName":"Ritter"},{"@type":"Person","name":"M\u00fcller-Gronbach, Thomas","givenName":"Thomas","familyName":"M\u00fcller-Gronbach"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.6","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article967","name":"Monte Carlo solution for the Poisson equation on the base of spherical processes with shifted centres","abstract":"We consider a class of spherical processes rapidly \r\nconverging to the boundary (so called Decentred \r\nRandom Walks on Spheres or spherical processes \r\nwith shifted centres) in comparison with the \r\nstandard walk on spheres. The aim is to compare \r\ncosts of the corresponding Monte Carlo estimates \r\nfor the Poisson equation. Generally, these costs \r\ndepend on the cost of simulation of one trajectory \r\nand on the variance of the estimate.\r\nIt can be proved that for the Laplace equation the \r\nlimit variance of the estimation doesn't depend on \r\nthe kind of spherical processes. Thus we have very \r\neffective estimator based on the decentred random \r\nwalk on spheres. As for the Poisson equation, it \r\ncan be shown that the variance is bounded by a \r\nconstant independent of the kind of spherical \r\nprocesses (in standard form or with shifted \r\ncentres). We use simulation for a simple model \r\nexample to investigate variance behavior in more \r\ndetails.","keywords":["Poisson equation","Laplace operator","Monte Carlo solution","spherical process","random walk on spheres","rate of convergence"],"author":{"@type":"Person","name":"Golyandina, Nina","givenName":"Nina","familyName":"Golyandina"},"position":7,"pageStart":1,"pageEnd":8,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Golyandina, Nina","givenName":"Nina","familyName":"Golyandina"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.7","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article968","name":"Numerical Approximation of Parabolic Stochastic Partial Differential Equations","abstract":"The topic of the talk were the time approximation\r\nof quasi linear stochastic partial differential\r\nequations of parabolic type. The framework were\r\nin the setting of stochastic evolution equations.\r\nAn error bounds for the implicit Euler scheme was\r\ngiven and the stability of the scheme were considered.","keywords":["Stochastic Partial Differential Equations","Stochastic evolution Equations","Numerical Approximation","implicit Euler scheme"],"author":{"@type":"Person","name":"Hausenblas, Erika","givenName":"Erika","familyName":"Hausenblas"},"position":8,"pageStart":1,"pageEnd":2,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Hausenblas, Erika","givenName":"Erika","familyName":"Hausenblas"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.8","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article969","name":"On the Complexity of Computing Multi-Homogeneous B\u00c3\u0192\u00c2\u00a9zout Numbers","abstract":"We study the question how difficult it is to compute the multi-homogeneous\r\nB\\'ezout number for a polynomial system of given number $n$ of variables\r\nand given support $A$ of monomials with non-zero coefficients.\r\nWe show that this number is NP-hard to compute. It cannot even be efficiently \r\napproximated within an arbitrary, fixed factor unless P = NP. \r\n\r\nThis is joint work with Gregorio Malajovich.","keywords":["multi-homogeneous B\u00c3\u0192\u00c2\u00a9zout numbers","number of roots of polynomials","approximation algorithms"],"author":[{"@type":"Person","name":"Meer, Klaus","givenName":"Klaus","familyName":"Meer"},{"@type":"Person","name":"Malajovich, Gregorio","givenName":"Gregorio","familyName":"Malajovich"}],"position":9,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Meer, Klaus","givenName":"Klaus","familyName":"Meer"},{"@type":"Person","name":"Malajovich, Gregorio","givenName":"Gregorio","familyName":"Malajovich"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.9","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article970","name":"On the Complexity of Parabolic Initial Value Problems with Variable Drift","abstract":"We consider linear parabolic initial value \r\nproblems of second order in several dimensions. \r\nThe initial condition is supposed to be fixed \r\nand we investigate the comutational complexity if \r\nthe coefficients of the parabolic equations\r\nmay vary in certain function spaces. Using the \r\nparametrix method (or Neumann series), we prove \r\nthat lower bounds for the error of numerical \r\nmethods are related to lower bounds for \r\nintegration problems. On the other hand, \r\napproximating the Neumann series with Smolyak's \r\nmethod, we show that the problem is not much \r\nharder than a certain approximation problem. For \r\nH\u00c3\u0192\u00c2\u00b6lder classes on compact sets, e.g., lower and \r\nupper bounds are close together, such that we have \r\nan almost optimal method.","keywords":["Partial differential equations","parabolic problems","Smolyak method","optimal methods"],"author":[{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Ritter, Klaus","givenName":"Klaus","familyName":"Ritter"}],"position":10,"pageStart":1,"pageEnd":24,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Petras, Knut","givenName":"Knut","familyName":"Petras"},{"@type":"Person","name":"Ritter, Klaus","givenName":"Klaus","familyName":"Ritter"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.10","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article971","name":"Optimal algorithms for global optimization in case of unknown Lipschitz constant","abstract":"We consider a family of function classes which \r\nallow functions with several minima and which \r\ndemand only Lipschitz continuity for smoothness.\r\n\r\nWe present an algorithm almost optimal for each of \r\nthese classes.","keywords":["Global optimization","Lipschitz functions","optimal rate of convergence","complexity"],"author":{"@type":"Person","name":"Horn, Matthias U.","givenName":"Matthias U.","familyName":"Horn"},"position":11,"pageStart":1,"pageEnd":22,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Horn, Matthias U.","givenName":"Matthias U.","familyName":"Horn"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.11","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article972","name":"Optimal Approximation of Elliptic Problems by Linear and Nonlinear Mappings","abstract":"We study the optimal approximation of the solution \r\nof an operator equation Au=f by linear mappings of\r\nrank n and compare this with the best n-term \r\napproximation with respect to an optimal Riesz \r\nbasis. We consider worst case errors, where f \r\nis an element of the unit ball of a Hilbert space. \r\nWe apply our results to boundary value problems \r\nfor elliptic PDEs on an arbitrary bounded \r\nLipschitz domain. Here we prove that approximation \r\nby linear mappings is as good as the best n-term \r\napproximation with respect to an optimal Riesz \r\nbasis. Our results are concerned with\r\napproximation, not with computation. \r\nOur goal is to understand better the possibilities \r\nof nonlinear approximation.","keywords":["elliptic operator equation","worst case error","linear approximation method","nonlinear approximation method","best n-term approximation Bernstein widths","manifold widths"],"author":[{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Sickel, Winfried","givenName":"Winfried","familyName":"Sickel"}],"position":12,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Sickel, Winfried","givenName":"Winfried","familyName":"Sickel"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.12","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article973","name":"Optimal Approximation of Elliptic Problems II: Wavelet Methods","abstract":"This talk is concerned with optimal approximations\r\nof the solutions of elliptic boundary value\r\nproblems. After briefly recalling the fundamental \r\nconcepts of optimality, we shall especially\r\ndiscuss best n-term approximation schemes based\r\non wavelets. We shall mainly be concerned with\r\nthe Poisson equation in Lipschitz domains. It\r\nturns out that wavelet schemes are suboptimal\r\nin general, but nevertheless they are superior to\r\nthe usual uniform approximation methods. \r\nMoreover, for specific domains, i.e., for\r\npolygonal domains, wavelet methods are\r\nin fact optimal. These results are based on\r\nregularity estimates of the exact solution\r\nin a specific scale of Besov spaces.","keywords":["Elliptic operator equations","worst case error","best n-term approximation","wavelets","Besov regularity"],"author":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Sickel, Winfried","givenName":"Winfried","familyName":"Sickel"}],"position":13,"pageStart":1,"pageEnd":4,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Dahlke, Stephan","givenName":"Stephan","familyName":"Dahlke"},{"@type":"Person","name":"Novak, Erich","givenName":"Erich","familyName":"Novak"},{"@type":"Person","name":"Sickel, Winfried","givenName":"Winfried","familyName":"Sickel"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.13","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article974","name":"Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights; Worst Case Setting","abstract":"We present polynomial-time algorithms for multivariate problems defined over tensor product\r\nspaces of finite order. That is, algorithms that solve the problem with error not exceeding \\epsilon\r\nat cost bounded by C d^q \\epsilon^{-1}, where \r\nd denotes the number of variables.","keywords":["Multivariate integration","multivariate approximation","complexity","polynomial-time algorithms","finite-order weights"],"author":[{"@type":"Person","name":"Wasilkowski, Gregorz W.","givenName":"Gregorz W.","familyName":"Wasilkowski"},{"@type":"Person","name":"Wozniakowski, Henryk","givenName":"Henryk","familyName":"Wozniakowski"}],"position":14,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Wasilkowski, Gregorz W.","givenName":"Gregorz W.","familyName":"Wasilkowski"},{"@type":"Person","name":"Wozniakowski, Henryk","givenName":"Henryk","familyName":"Wozniakowski"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.14","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article975","name":"Quantization of self-similar Probabilities","abstract":"The asymptotic behaviour of the quantization errors for self-similar probabilities is determined.","keywords":["quantization","self-similar probabilities"],"author":[{"@type":"Person","name":"Graf, Siegfried","givenName":"Siegfried","familyName":"Graf"},{"@type":"Person","name":"Luschgy, Harald","givenName":"Harald","familyName":"Luschgy"}],"position":15,"pageStart":1,"pageEnd":0,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Graf, Siegfried","givenName":"Siegfried","familyName":"Graf"},{"@type":"Person","name":"Luschgy, Harald","givenName":"Harald","familyName":"Luschgy"}],"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.15","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"},{"@type":"ScholarlyArticle","@id":"#article976","name":"Upper Error Bounds for Approximations of Stochastic Differential Equations with Markovian Switching","abstract":"We consider stochastic differential equations with\r\nMarkovian switching (SDEwMS). An SDEwMS is a\r\nstochastic differential equation with drift and\r\ndiffusion coefficients depending not only on the\r\ncurrent state of the solution but also on the \r\ncurrent state of a right-continuous Markov chain \r\ntaking values in a finite state space. \r\nConsequently, an SDEwMS can be viewed as the \r\nresult of a finite number of different scenarios \r\nswitching from one to the other according to the \r\nmovement of the Markov chain. The generator of the \r\nMarkov chain is given by transition probabilities \r\ninvolving a parameter which controls the intensity \r\nof switching from one state to another. We \r\nconstruct numerical schemes for the approximation \r\nof SDE'swMS and present upper error bounds for \r\nthese schemes. Our numerical schemes are based on \r\na time discretization with constant step-size and \r\non the values of a discrete Markov chain at the \r\ndiscretization points. It turns out that for the \r\nEuler scheme a similar upper bound as in the case \r\nof stochastic ordinary differential equations can \r\nbe obtained, while for the Milstein scheme there \r\nis a strong connection between the power of the \r\nstep-size appearing in the upper bound and the \r\nintensity of the switching.","keywords":["stochastic differential equations with Markovian switching","Markov chains","numerical methods","Euler scheme","Milstein scheme"],"author":{"@type":"Person","name":"Hofmann, Norbert","givenName":"Norbert","familyName":"Hofmann"},"position":16,"pageStart":1,"pageEnd":2,"dateCreated":"2005-04-19","datePublished":"2005-04-19","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Hofmann, Norbert","givenName":"Norbert","familyName":"Hofmann"},"copyrightYear":"2005","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/DagSemProc.04401.16","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":"#volume546"}]}