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Q3: Algorithms for high dimensional systems

Principal Investigators: Novak, Wipf

Functional integrals in lattice gauge theories are high dimensional (∼ 106) integrals which have to be approximated as accurately as possible. The choice of methods and their quality depend strongly on the integrand. For quantum field theories with fermions which lead to nonlocal and strongly varying integrands, effective algorithms shall be developed and applied as well as effective inverters for occuring matrices. The solvability of high dimensional problems in quantum theory with defensible numerical complexity shall be investigated.

further information (german)

Examples of Ph.D. topics

  1. Welche geometrischen Integrationsprobleme sind tractable für randomisierte Algorithmen?
  2. Funktionenklassen, die durch Feldtheorien auf dem Gitter motiviert sind: optimale Berechnung der Integrale in Abhängigkeit der Klassenparameter
  3. Schnell mischende Markovketten für Spin-Modelle, Hard-Core-Modelle, Gitterfeldtheorien
  4. Optimale Dichten für Importance Sampling
  5. Mischungseigenschaften des Slice Samplers


Josef Dick, Daniel Rudolf, Houying Zhu
Discrepancy bounds for uniformly ergodic Markov chain quasi-Monte Carlo
arXiv:1303.2423 [stat.CO], Mar 2013

Daniel Rudolf, Mario Ullrich
Positivity of hit-and-run and related algorithms
arXiv:1212.4512 [math.PR], Dec 2012

Daniel Rudolf
Hit-and-run for numerical integration
arXiv:1212.4486 [math.PR], Dec 2012

M. Ullrich
Rapid mixing of Swendsen-Wang and single-bond dynamics in two dimensions
arXiv:1202.6321 [math.PR], Mar 2012

M. Ullrich
Swendsen-Wang is faster than single-bond dynamics
arXiv:1201.5793 [math.PR], Jan 2012

D. Rudolf
Explicit error bounds for Markov chain Monte Carlo
arXiv:1108.3201 [math.PR], Aug 2011

M. Ullrich
Comparison of Swendsen-Wang and Heat-Bath Dynamics
arXiv:1105.3665v1 [math.PR], May 2011

M. Ullrich
Exact Sampling for the Ising Model at all Temperatures
arXiv:1012.3944v2 [math.PR], Dec 2010

D. Rudolf
Error bounds for computing the expectation by Markov chain Monte Carlo
Monte Carlo Methods Appl. 16 (2010), no. 3-4, 323-342, arXiv:0906.2359 [math.NA], Jun 2009

D. Rudolf
Explicit error bounds for lazy reversible Markov Chain Monte Carlo
J. Complexity 25 (2009), no. 1, 11-24, Dissertationes Math. *485* (2012), 93 pp, arXiv:0805.3587v2 [math.NA], May 2008