Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen OPUS4-5183 Masterarbeit / Diplomarbeit Lie, Han Cheng A Meshless Method for Conformational Analysis 2012 Numerical Mathematics OPUS4-6014 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J. Cameron--Martin theorems for sequences of Cauchy-distributed random variables Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. 2016 arXiv 1608.03784 Numerical Mathematics OPUS4-6023 misc Lie, Han Cheng; Sullivan, T. J. Cameron--Martin theorems for sequences of Cauchy-distributed random variables Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. 2016 urn:nbn:de:0297-zib-60230 Numerical Mathematics OPUS4-6428 misc Quer, Jannes; Lie, Han Cheng Some connections between importance sampling and enhanced sampling methods in molecular dynamics Enhanced sampling methods play an important role in molecular dynamics, because they enable the collection of better statistics of rare events that are important in many physical phenomena. We show that many enhanced sampling methods can be viewed as methods for performing importance sampling, by identifying important correspondences between the language of molecular dynamics and the language of probability theory. We illustrate these connections by highlighting the similarities between the rare event simulation method of Hartmann and Schütte (J. Stat. Mech. Theor. Exp., 2012), and the enhanced sampling method of Valsson and Parrinello (Phys. Rev. Lett. 113, 090601). We show that the idea of changing a probability measure is fundamental to both enhanced sampling and importance sampling. 2017 urn:nbn:de:0297-zib-64289 Numerical Mathematics OPUS4-6377 Dissertation Lie, Han Cheng On a strongly convex approximation of a stochastic optimal control problem for importance sampling of metastable diffusions 2016 urn:nbn:de:kobv:188-fudissthesis000000101680-4 Numerical Mathematics OPUS4-6632 misc Lie, Han Cheng; Sullivan, T. J.; Teckentrup, Aretha Random forward models and log-likelihoods in Bayesian inverse problems We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations. 2018 29 SIAM/ASA Journal on Uncertainty Quantification 6 4 1600 1629 urn:nbn:de:0297-zib-66324 10.1137/18M1166523 Numerical Mathematics OPUS4-6657 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J.; Stuart, Andrew Strong convergence rates of probabilistic integrators for ordinary differential equations Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. 2019 18 Statistics and Computing 29 6 1265 1283 10.1007/s11222-019-09898-6 Numerical Mathematics OPUS4-6660 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J. Equivalence of weak and strong modes of measures on topological vector spaces Modes of a probability measure on an infinite-dimensional Banach space X are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of X, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces. 2018 Inverse Problems 34 11 115013 10.1088/1361-6420/aadef2 Numerical Mathematics OPUS4-6241 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J. Quasi-invariance of countable products of Cauchy measures under non-unitary dilations 2018 6 Electronic Communications in Probability 23 8 1 6 10.1214/18-ECP113 Numerical Mathematics OPUS4-5879 misc Bojarovski, Stefan; Hege, Hans-Christian; Lie, Han Cheng; Weber, Marcus Topological analysis and visualization of scalar functions characterizing conformational transitions of molecules on multiple time-scales Molecular processes such as protein folding or ligand-receptor-binding can be understood by analyzing the free energy landscape. Those processes are often metastable, i.e. the molecular systems remain in basins around local minima of the free energy landscape, and in rare cases undergo gauche transitions between metastable states by passing saddle-points of this landscape. By discretizing the configuration space, this can be modeled as a discrete Markov process. One way to compute the transition rates between conformations of a molecular system is by utilizing Transition Path Theory and the concept of committor functions. A fundamental problem from the computational point of view is that many time-scales are involved, ranging from 10^(-14) sec for the fastest motion to 10^(-6) sec or more for conformation changes that cause biological effects. The goal of our work is to provide a better understanding of such transitions in configuration space on various time-scales by analyzing characteristic scalar functions topologically and geometrically. We are developing suitable visualization and interaction techniques to support our analysis. For example, we are analyzing a transition rate indicator function by computing and visualizing its Reeb graph together with the sets of molecular states corresponding to maxima of the transition rate indicator function. A particular challenge is the high dimensionality of the domain which does not allow for a straightforward visualization of the function. The computational topology approach to the analysis of the transition rate indicator functions for a molecular system allows to explore different time scales of the system by utilizing coarser or finer topological partitioning of the function. A specific goal is the development of tools for analyzing the hierarchy of these partitionings. This approach tackles the analysis of a complex and sparse dataset from a different angle than the well-known spectral analysis of Markov State Models. 2015 Shape Up 2015 - Exercises in Materials Geometry and Topology, 14-18 Sept. 2015, Berlin, Germany Numerical Mathematics