On different notions of timescales in molecular dynamics
|
|
- Doreen Chapman
- 5 years ago
- Views:
Transcription
1 On different notions of timescales in molecular dynamics Origin of Scaling Cascades in Protein Dynamics June 8, 217 IHP Trimester Stochastic Dynamics out of Equilibrium
2 Overview 1. Motivation 2. Definition of the timescales of interest: convergence rates and mean first exit times 3. Review of results for overdamped Langevin equation 4. Approach for linear but irreversible systems: entropy production
3 Origin of the scaling cascades in protein dynamics Proteinstructure (Source: MaxPlanckForschung 4/23) Observation: single point mutations large non-local effects Trying to understand the observations: study parameter sensitivies of timescales study relations between timescales
4 A model to simulate molecular dynamics Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t where X R 3N : positions of the atoms V : R 3N R is the interaction potential B t is standard N dimensional Brownian motion β 1 R + is the temperature.
5 Illustration of the model Overdamped Langevin equation dx t = V (X t )dt + 2β 1 db t Associated probability density ρ t and equilibrium distribution ρ e V β
6 Quantities of interest Convergence to equilibrium: ρ t ρ c γ(t) Mean first exit times (MFET): E(τ x ( D)) = E(inf {t > : X t / D})
7 Decay towards equilibrium in L 2 µ 1 Generator: L = β 1 2 V dx t = V (X t )dt + 2β 1 db t describes evolution of expectation values and probability densities is self-adjoint wrt dµ = 1 Z e βv (x) dx, i.e. Lf, g µ = f, Lg µ spec( L) {} [λ, ) this implies convergence towards equilibrium in L 2, i.e. µ 1 ρ t ρ 2 L 2 µ 1 Problems: = λ is not known ρ L 2 µ 1 ρ t (x) ρ (x) 2 ρ 1 (x)dx e 2λt ρ ρ 2 L 2 µ 1 ρ (x) 2 e βv (x) dx <
8 Mean first exit times and eigenvalues of L Theorem [Bovier et al. 24]: Assumptions V has n minima x 1,..., x n ordering of minima according to energy barrier height possible Define τ xk (S k 1 ) = inf {t : X t S k 1, X = x k }, S k 1 = k 1 B 1 (x j). β Then L has n eigenvalues = Λ 1 > Λ 2 >... > Λ n and δ > such that 1 ( Λ k = 1 + O(1 + e βδ ) ) E(τ xk (S k 1 )) = c exp ( β (x k, {x 1,..., x k 1 })) (1 + O(1 + β 1 ) log β 1 ), c >. j=1
9 What we would like to do Convergence in terms of relative entropy instead of L 2 : µ 1 Relative entropy H(ρ t ρ ) = ( ρt (x) ρ t (x) log ρ (x) ) dx Relation to L 1 norm via Csizsàr-Kullback/Pinsker inequality: ρ t ρ L 1 2H(ρ t ρ ) L 1 is the natural norm for probability densities relation to measurable quantities applicable for any process that admits a pdf, not only reversible processes H is computable from simulation data
10 Linear but possibly irreversible processes Conditions dx t = AX t dt + σ 2β 1 db t, X R n, A R n n, (i) spec(a) C = {λ C : R(λ) < } (ii) A and σ fulfill the Kalman rank condition rank( [ σ, Aσ,..., A n 1 σ ] ) = n σ R n m, B t R m, m n. = existence of unique positive invariant measure ρ = N (, Σ). Theorem[Arnold, Erb 214] H(ρ t ρ ) c H(ρ ρ )e 2λ A t, λ A = min { R(λ) : λ spec(a)}, c 1.
11 Mean first exit times and eigenvalues of the covariance dx t = AX t dt + σ 2β 1 db t Interested in: τ x ( D) = inf {t > : X t / D}, D = {x : x < 1}. Theorem [Zabczyk 85] Assume that conditions (i) and (ii) are fulfilled s. th. ρ = N (, Σ) exists. Let λ Σ = max {λ : λ spec(σ)} >, E = {v : Σv = λ Σ v}. Then for large β, i.e. small temperature lim β β 1 log E(τ x ( D)) = 1 2λ exit time Σ and for any η > lim P(dist(X τ x ( D), E) η) = 1 exit path β We can show that λ A (2λ Σ) 1 λ + σ, λ + σ = min{λ > : λ spec(σσ T )}.
12 Analysis of relaxation behaviour Splitting up: H(ρ t ρ ) = ( ρt (x) log ρ (x) =a(t) ) ρ t (x) dx = 1 [ Tr(Σt Σ 1 ) n Tr(log(Σ t Σ 1 )) + µ T t Σ 1 ] µ t. 2 }{{}}{{} =b(t) } {{ } Covariance Same structure in all terms: z T e AT t Σ 1 e At z. For a(t) and b(t) : z = (Σ Σ ) 1 2, for c(t) : z = x. } {{ } } =c(t) {{ } Mean
13 Some examples for different relaxation behaviour high temperature low temperature, x = EVec(A) H/H relative entropy H()*e -2 t a(t) b(t) c(t) H/H relative entropy H()*e -2 t a(t) b(t) c(t) time time low temperature, x EVec(A) low temperature, A = A T H/H relative entropy H()*e -2 t c*h()*e -2 t a(t) b(t) c(t) H/H relative entropy H()*e -2 t a(t) b(t) c(t) time time
14 Understanding plateaus in the entropy decay Necessary and sufficient condition for the existence of a plateau: degeneracy of the noise, i.e. det σσ T =. For c(t) this translates to: ċ(t) = ż p = z moves along contour lines of the potential p Here z i (t) = e λ i t (Sx ) i, p(z) = z T S T Σ 1 S 1 z with S such that SAS 1 = diag(λ 1,..., λ n ). det(σσ T ) = H/H a b c c at discrete times S time z at discrete times S 1 15
15 From degenerate to isotropic noise c(t) = z T S T Σ 1 S 1 z, z i = e λ i t z i (), λ 1 = 1, λ 2 = 1. det(σσ T ) = H/H a b c c at discrete times S time z at discrete times S 1 15 det(σσ T ) = 1 H/H a b c c at discrete times time S z at discrete times S
16 Identification of slow and fast? Can we identify slow and fast dof? Can we get estimates on the marginals? Can we get hierarchichal order of timescales?
2012 NCTS Workshop on Dynamical Systems
Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz 2012 NCTS Workshop on Dynamical Systems National Center for Theoretical Sciences, National Tsing-Hua University Hsinchu,
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationResidence-time distributions as a measure for stochastic resonance
W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly sis u n d S to ch a stik Period of Concentration: Stochastic Climate Models MPI Mathematics in the Sciences, Leipzig, 23 May 1 June 2005 Barbara
More informationEffective dynamics for the (overdamped) Langevin equation
Effective dynamics for the (overdamped) Langevin equation Frédéric Legoll ENPC and INRIA joint work with T. Lelièvre (ENPC and INRIA) Enumath conference, MS Numerical methods for molecular dynamics EnuMath
More informationIntertwinings for Markov processes
Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013
More informationNon equilibrium thermodynamic transformations. Giovanni Jona-Lasinio
Non equilibrium thermodynamic transformations Giovanni Jona-Lasinio Kyoto, July 29, 2013 1. PRELIMINARIES 2. RARE FLUCTUATIONS 3. THERMODYNAMIC TRANSFORMATIONS 1. PRELIMINARIES Over the last ten years,
More informationNumerical methods in molecular dynamics and multiscale problems
Numerical methods in molecular dynamics and multiscale problems Two examples T. Lelièvre CERMICS - Ecole des Ponts ParisTech & MicMac project-team - INRIA Horizon Maths December 2012 Introduction The aim
More informationmacroscopic view (phenomenological) microscopic view (atomistic) computing reaction rate rate of reactions experiments thermodynamics
Rate Theory (overview) macroscopic view (phenomenological) rate of reactions experiments thermodynamics Van t Hoff & Arrhenius equation microscopic view (atomistic) statistical mechanics transition state
More informationLorenz like flows. Maria José Pacifico. IM-UFRJ Rio de Janeiro - Brasil. Lorenz like flows p. 1
Lorenz like flows Maria José Pacifico pacifico@im.ufrj.br IM-UFRJ Rio de Janeiro - Brasil Lorenz like flows p. 1 Main goals The main goal is to explain the results (Galatolo-P) Theorem A. (decay of correlation
More informationSpeeding up Convergence to Equilibrium for Diffusion Processes
Speeding up Convergence to Equilibrium for Diffusion Processes G.A. Pavliotis Department of Mathematics Imperial College London Joint Work with T. Lelievre(CERMICS), F. Nier (CERMICS), M. Ottobre (Warwick),
More informationThe Kramers problem and first passage times.
Chapter 8 The Kramers problem and first passage times. The Kramers problem is to find the rate at which a Brownian particle escapes from a potential well over a potential barrier. One method of attack
More informationNEW FUNCTIONAL INEQUALITIES
1 / 29 NEW FUNCTIONAL INEQUALITIES VIA STEIN S METHOD Giovanni Peccati (Luxembourg University) IMA, Minneapolis: April 28, 2015 2 / 29 INTRODUCTION Based on two joint works: (1) Nourdin, Peccati and Swan
More informationStatistical Mechanics of Active Matter
Statistical Mechanics of Active Matter Umberto Marini Bettolo Marconi University of Camerino, Italy Naples, 24 May,2017 Umberto Marini Bettolo Marconi (2017) Statistical Mechanics of Active Matter 2017
More informationOn thermodynamics of stationary states of diffusive systems. Giovanni Jona-Lasinio Coauthors: L. Bertini, A. De Sole, D. Gabrielli, C.
On thermodynamics of stationary states of diffusive systems Giovanni Jona-Lasinio Coauthors: L. Bertini, A. De Sole, D. Gabrielli, C. Landim Newton Institute, October 29, 2013 0. PRELIMINARIES 1. RARE
More informationOnsager theory: overview
Onsager theory: overview Pearu Peterson December 18, 2006 1 Introduction Our aim is to study matter that consists of large number of molecules. A complete mechanical description of such a system is practically
More informationTheoretical Tutorial Session 2
1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationHypoelliptic multiscale Langevin diffusions and Slow fast stochastic reaction diffusion equations.
Hypoelliptic multiscale Langevin diffusions and Slow fast stochastic reaction diffusion equations. Wenqing Hu. 1 (Joint works with Michael Salins 2 and Konstantinos Spiliopoulos 3.) 1. Department of Mathematics
More information16. Working with the Langevin and Fokker-Planck equations
16. Working with the Langevin and Fokker-Planck equations In the preceding Lecture, we have shown that given a Langevin equation (LE), it is possible to write down an equivalent Fokker-Planck equation
More informationRelative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague joint work
More informationImportance splitting for rare event simulation
Importance splitting for rare event simulation F. Cerou Frederic.Cerou@inria.fr Inria Rennes Bretagne Atlantique Simulation of hybrid dynamical systems and applications to molecular dynamics September
More informationDynamical systems with Gaussian and Levy noise: analytical and stochastic approaches
Dynamical systems with Gaussian and Levy noise: analytical and stochastic approaches Noise is often considered as some disturbing component of the system. In particular physical situations, noise becomes
More informationDerivation of the GENERIC form of nonequilibrium thermodynamics from a statistical optimization principle
Derivation of the GENERIC form of nonequilibrium thermodynamics from a statistical optimization principle Bruce Turkington Univ. of Massachusetts Amherst An optimization principle for deriving nonequilibrium
More informationOPTIMAL CONTROL SYSTEMS
SYSTEMS MIN-MAX Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University OUTLINE MIN-MAX CONTROL Problem Definition HJB Equation Example GAME THEORY Differential Games Isaacs
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationFree Entropy for Free Gibbs Laws Given by Convex Potentials
Free Entropy for Free Gibbs Laws Given by Convex Potentials David A. Jekel University of California, Los Angeles Young Mathematicians in C -algebras, August 2018 David A. Jekel (UCLA) Free Entropy YMC
More informationKolmogorov Equations and Markov Processes
Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define
More informationBridging the Gap between Center and Tail for Multiscale Processes
Bridging the Gap between Center and Tail for Multiscale Processes Matthew R. Morse Department of Mathematics and Statistics Boston University BU-Keio 2016, August 16 Matthew R. Morse (BU) Moderate Deviations
More informationIntroduction to Random Diffusions
Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationStochastic contraction BACS Workshop Chamonix, January 14, 2008
Stochastic contraction BACS Workshop Chamonix, January 14, 2008 Q.-C. Pham N. Tabareau J.-J. Slotine Q.-C. Pham, N. Tabareau, J.-J. Slotine () Stochastic contraction 1 / 19 Why stochastic contraction?
More informationON FRACTAL DIMENSION OF INVARIANT SETS
ON FRACTAL DIMENSION OF INVARIANT SETS R. MIRZAIE We give an upper bound for the box dimension of an invariant set of a differentiable function f : U M. Here U is an open subset of a Riemannian manifold
More informationLecture 12: Detailed balance and Eigenfunction methods
Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility), 4.2-4.4 (explicit examples of eigenfunction methods) Gardiner
More informationx. Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ 2 ).
.8.6 µ =, σ = 1 µ = 1, σ = 1 / µ =, σ =.. 3 1 1 3 x Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ ). The Gaussian distribution Probably the most-important distribution in all of statistics
More informationDecay rates for partially dissipative hyperbolic systems
Outline Decay rates for partially dissipative hyperbolic systems Basque Center for Applied Mathematics Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ Numerical Methods
More informationStrong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term
1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes
More informationSome appications of free stochastic calculus to II 1 factors.
Some appications of free stochastic calculus to II 1 factors. Dima Shlyakhtenko UCLA Free Entropy Dimension (Voiculescu) G = S nitely generated discrete group, τ G : CF S C. ω Mn n = M 1 k=1 k k/{(x k
More informationOn a class of stochastic differential equations in a financial network model
1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationEmergence of collective dynamics from a purely stochastic orogin
Emergence of collective dynamics from a purely stochastic orogin Yann Brenier CNRS-Centre de Mathématiques Laurent SCHWARTZ Ecole Polytechnique FR 91128 Palaiseau. Transport phenomena in collective dynamics,
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 1, 216 Statistical properties of dynamical systems, ESI Vienna. David
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationEnergy Barriers and Rates - Transition State Theory for Physicists
Energy Barriers and Rates - Transition State Theory for Physicists Daniel C. Elton October 12, 2013 Useful relations 1 cal = 4.184 J 1 kcal mole 1 = 0.0434 ev per particle 1 kj mole 1 = 0.0104 ev per particle
More informationEQUITY MARKET STABILITY
EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference,
More informationA Lévy-Fokker-Planck equation: entropies and convergence to equilibrium
1/ 22 A Lévy-Fokker-Planck equation: entropies and convergence to equilibrium I. Gentil CEREMADE, Université Paris-Dauphine International Conference on stochastic Analysis and Applications Hammamet, Tunisia,
More informationlim n C1/n n := ρ. [f(y) f(x)], y x =1 [f(x) f(y)] [g(x) g(y)]. (x,y) E A E(f, f),
1 Part I Exercise 1.1. Let C n denote the number of self-avoiding random walks starting at the origin in Z of length n. 1. Show that (Hint: Use C n+m C n C m.) lim n C1/n n = inf n C1/n n := ρ.. Show that
More informationNumerical methods for conservation laws with a stochastically driven flux
Numerical methods for conservation laws with a stochastically driven flux Håkon Hoel, Kenneth Karlsen, Nils Henrik Risebro, Erlend Briseid Storrøsten Department of Mathematics, University of Oslo, Norway
More informationPostulates of quantum mechanics
Postulates of quantum mechanics Armin Scrinzi November 22, 2012 1 Postulates of QM... and of classical mechanics 1.1 An analogy Quantum Classical State Vector Ψ from H Prob. distr. ρ(x, p) on phase space
More informationStochastic differential equations in neuroscience
Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans
More informationString method for the Cahn-Hilliard dynamics
String method for the Cahn-Hilliard dynamics Tiejun Li School of Mathematical Sciences Peking University tieli@pku.edu.cn Joint work with Wei Zhang and Pingwen Zhang Outline Background Problem Set-up Algorithms
More informationStatistical Mechanics and Thermodynamics of Small Systems
Statistical Mechanics and Thermodynamics of Small Systems Luca Cerino Advisors: A. Puglisi and A. Vulpiani Final Seminar of PhD course in Physics Cycle XXIX Rome, October, 26 2016 Outline of the talk 1.
More informationAsymptotic Analysis 88 (2014) DOI /ASY IOS Press
Asymptotic Analysis 88 4) 5 DOI.333/ASY-4 IOS Press Smoluchowski Kramers approximation and large deviations for infinite dimensional gradient systems Sandra Cerrai and Michael Salins Department of Mathematics,
More informationarxiv: v2 [math.pr] 9 Mar 2018
arxiv:1701.00985v [math.pr] 9 Mar 018 DIRICHLET S AND THOMSON S PRINCIPLES FOR NON-SELFADJOINT ELLIPTIC OPERATORS WITH APPLICATION TO NON-REVERSIBLE METASTABLE DIFFUSION PROCESSES. C. LANDIM, M. MARIANI,
More informationThe glass transition as a spin glass problem
The glass transition as a spin glass problem Mike Moore School of Physics and Astronomy, University of Manchester UBC 2007 Co-Authors: Joonhyun Yeo, Konkuk University Marco Tarzia, Saclay Mike Moore (Manchester)
More informationL 1 stability of conservation laws for a traffic flow model
Electronic Journal of Differential Equations, Vol. 2001(2001), No. 14, pp. 1 18. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationLinear Response and Onsager Reciprocal Relations
Linear Response and Onsager Reciprocal Relations Amir Bar January 1, 013 Based on Kittel, Elementary statistical physics, chapters 33-34; Kubo,Toda and Hashitsume, Statistical Physics II, chapter 1; and
More informationOPTIMAL CONTROL AND ESTIMATION
OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION
More informationA DIFFUSIVE MODEL FOR MACROSCOPIC CROWD MOTION WITH DENSITY CONSTRAINTS
A DIFFUSIVE MODEL FOR MACROSCOPIC CROWD MOTION WITH DENSITY CONSTRAINTS Abstract. In the spirit of the macroscopic crowd motion models with hard congestion (i.e. a strong density constraint ρ ) introduced
More information5 Applying the Fokker-Planck equation
5 Applying the Fokker-Planck equation We begin with one-dimensional examples, keeping g = constant. Recall: the FPE for the Langevin equation with η(t 1 )η(t ) = κδ(t 1 t ) is = f(x) + g(x)η(t) t = x [f(x)p
More informationLecture 12: Detailed balance and Eigenfunction methods
Miranda Holmes-Cerfon Applied Stochastic Analysis, Spring 2015 Lecture 12: Detailed balance and Eigenfunction methods Readings Recommended: Pavliotis [2014] 4.5-4.7 (eigenfunction methods and reversibility),
More informationfor all subintervals I J. If the same is true for the dyadic subintervals I D J only, we will write ϕ BMO d (J). In fact, the following is true
3 ohn Nirenberg inequality, Part I A function ϕ L () belongs to the space BMO() if sup ϕ(s) ϕ I I I < for all subintervals I If the same is true for the dyadic subintervals I D only, we will write ϕ BMO
More informationIf we want to analyze experimental or simulated data we might encounter the following tasks:
Chapter 1 Introduction If we want to analyze experimental or simulated data we might encounter the following tasks: Characterization of the source of the signal and diagnosis Studying dependencies Prediction
More informationDependence. MFM Practitioner Module: Risk & Asset Allocation. John Dodson. September 11, Dependence. John Dodson. Outline.
MFM Practitioner Module: Risk & Asset Allocation September 11, 2013 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y
More informationEEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as
L30-1 EEL 5544 Noise in Linear Systems Lecture 30 OTHER TRANSFORMS For a continuous, nonnegative RV X, the Laplace transform of X is X (s) = E [ e sx] = 0 f X (x)e sx dx. For a nonnegative RV, the Laplace
More informationGeorgia Institute of Technology Nonlinear Controls Theory Primer ME 6402
Georgia Institute of Technology Nonlinear Controls Theory Primer ME 640 Ajeya Karajgikar April 6, 011 Definition Stability (Lyapunov): The equilibrium state x = 0 is said to be stable if, for any R > 0,
More informationNonlinear Observers. Jaime A. Moreno. Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México
Nonlinear Observers Jaime A. Moreno JMorenoP@ii.unam.mx Eléctrica y Computación Instituto de Ingeniería Universidad Nacional Autónoma de México XVI Congreso Latinoamericano de Control Automático October
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationDensity of States for Random Band Matrices in d = 2
Density of States for Random Band Matrices in d = 2 via the supersymmetric approach Mareike Lager Institute for applied mathematics University of Bonn Joint work with Margherita Disertori ZiF Summer School
More informationFokker-Planck Equation on Graph with Finite Vertices
Fokker-Planck Equation on Graph with Finite Vertices January 13, 2011 Jointly with S-N Chow (Georgia Tech) Wen Huang (USTC) Hao-min Zhou(Georgia Tech) Functional Inequalities and Discrete Spaces Outline
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky
ELEC 3035, Lecture 3: Autonomous systems Ivan Markovsky Equilibrium points and linearization Eigenvalue decomposition and modal form State transition matrix and matrix exponential Stability ELEC 3035 (Part
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationFormulas for probability theory and linear models SF2941
Formulas for probability theory and linear models SF2941 These pages + Appendix 2 of Gut) are permitted as assistance at the exam. 11 maj 2008 Selected formulae of probability Bivariate probability Transforms
More informationLangevin Methods. Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 10 D Mainz Germany
Langevin Methods Burkhard Dünweg Max Planck Institute for Polymer Research Ackermannweg 1 D 55128 Mainz Germany Motivation Original idea: Fast and slow degrees of freedom Example: Brownian motion Replace
More informationPartial Differential Equations, Winter 2015
Partial Differential Equations, Winter 215 Homework #2 Due: Thursday, February 12th, 215 1. (Chapter 2.1) Solve u xx + u xt 2u tt =, u(x, ) = φ(x), u t (x, ) = ψ(x). Hint: Factor the operator as we did
More informationCheng Soon Ong & Christian Walder. Canberra February June 2018
Cheng Soon Ong & Christian Walder Research Group and College of Engineering and Computer Science Canberra February June 2018 Outlines Overview Introduction Linear Algebra Probability Linear Regression
More informationInput-to-state stability and interconnected Systems
10th Elgersburg School Day 1 Input-to-state stability and interconnected Systems Sergey Dashkovskiy Universität Würzburg Elgersburg, March 5, 2018 1/20 Introduction Consider Solution: ẋ := dx dt = ax,
More information5 Handling Constraints
5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest
More informationBayesian inverse problems with Laplacian noise
Bayesian inverse problems with Laplacian noise Remo Kretschmann Faculty of Mathematics, University of Duisburg-Essen Applied Inverse Problems 2017, M27 Hangzhou, 1 June 2017 1 / 33 Outline 1 Inverse heat
More informationAbout the method of characteristics
About the method of characteristics Francis Nier, IRMAR, Univ. Rennes 1 and INRIA project-team MICMAC. Joint work with Z. Ammari about bosonic mean-field dynamics June 3, 2013 Outline The problem An example
More informationMetastability for the Ginzburg Landau equation with space time white noise
Barbara Gentz gentz@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gentz Metastability for the Ginzburg Landau equation with space time white noise Barbara Gentz University of Bielefeld, Germany
More informationMemory and hypoellipticity in neuronal models
Memory and hypoellipticity in neuronal models S. Ditlevsen R. Höpfner E. Löcherbach M. Thieullen Banff, 2017 What this talk is about : What is the effect of memory in probabilistic models for neurons?
More informationThe Truth about diffusion (in liquids)
The Truth about diffusion (in liquids) Aleksandar Donev Courant Institute, New York University & Eric Vanden-Eijnden, Courant In honor of Berni Julian Alder LLNL, August 20th 2015 A. Donev (CIMS) Diffusion
More informationA mathematical framework for Exact Milestoning
A mathematical framework for Exact Milestoning David Aristoff (joint work with Juan M. Bello-Rivas and Ron Elber) Colorado State University July 2015 D. Aristoff (Colorado State University) July 2015 1
More informationStochastic nonlinear Schrödinger equations and modulation of solitary waves
Stochastic nonlinear Schrödinger equations and modulation of solitary waves A. de Bouard CMAP, Ecole Polytechnique, France joint work with R. Fukuizumi (Sendai, Japan) Deterministic and stochastic front
More informationUniformly Uniformly-ergodic Markov chains and BSDEs
Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear time-invariant systems Katalin Hangos University of Pannonia Faculty
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 1. Linear systems Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Linear systems 1.1 Differential Equations 1.2 Linear flows 1.3 Linear maps
More informationThe incompressible Navier-Stokes equations in vacuum
The incompressible, Université Paris-Est Créteil Piotr Bogus law Mucha, Warsaw University Journées Jeunes EDPistes 218, Institut Elie Cartan, Université de Lorraine March 23th, 218 Incompressible Navier-Stokes
More informationOpen Quantum Systems and Markov Processes II
Open Quantum Systems and Markov Processes II Theory of Quantum Optics (QIC 895) Sascha Agne sascha.agne@uwaterloo.ca July 20, 2015 Outline 1 1. Introduction to open quantum systems and master equations
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationStochastic Gradient Descent in Continuous Time
Stochastic Gradient Descent in Continuous Time Justin Sirignano University of Illinois at Urbana Champaign with Konstantinos Spiliopoulos (Boston University) 1 / 27 We consider a diffusion X t X = R m
More informationDIFFUSION MAPS, REDUCTION COORDINATES AND LOW DIMENSIONAL REPRESENTATION OF STOCHASTIC SYSTEMS
DIFFUSION MAPS, REDUCTION COORDINATES AND LOW DIMENSIONAL REPRESENTATION OF STOCHASTIC SYSTEMS R.R. COIFMAN, I.G. KEVREKIDIS, S. LAFON, M. MAGGIONI, AND B. NADLER Abstract. The concise representation of
More informationExit times of diffusions with incompressible drifts
Exit times of diffusions with incompressible drifts Andrej Zlatoš University of Chicago Joint work with: Gautam Iyer (Carnegie Mellon University) Alexei Novikov (Pennylvania State University) Lenya Ryzhik
More informationStatistical mechanics of random billiard systems
Statistical mechanics of random billiard systems Renato Feres Washington University, St. Louis Banff, August 2014 1 / 39 Acknowledgements Collaborators: Timothy Chumley, U. of Iowa Scott Cook, Swarthmore
More informationLecture No 2 Degenerate Diffusion Free boundary problems
Lecture No 2 Degenerate Diffusion Free boundary problems Columbia University IAS summer program June, 2009 Outline We will discuss non-linear parabolic equations of slow diffusion. Our model is the porous
More information