https://www.tau.ac.il/%7Eklafter1/258.pdf

The random walk's guide to anomalous diffusion: a fractional dynamics approach

A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions

Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates

PREFACE. 1. A WORKED EXAMPLE. 1.1 A simple model. 1.2 Modulus version of the simple model. 1.3 Six--factor version of the simple model. 1.4 The simple model 'by groups'. 1.5 The (less) simple correlated--input model. 1.6 Conclusions. 2. GLOBAL SENSITIVITY ANALYSIS FOR IMPORTANCE ASSESSMENT. 2.1 Examples at a glance. 2.2 What is sensitivity analysis? 2.3 Properties of an ideal sensitivity analysis method. 2.4 Defensible settings for sensitivity analysis. 2.5 Caveats. 3. TEST CASES. 3.1 The jumping man. Applying variance--based methods. 3.2 Handling the risk of a financial portfolio: the problem of hedging. Applying Monte Carlo filtering and variance--based methods. 3.3 A model of fish population dynamics. Applying the method of Morris. 3.4 The Level E model. Radionuclide migration in the geosphere. Applying variance--based methods and Monte Carlo filtering. 3.5 Two spheres. Applying variance based methods in estimation/calibration problems. 3.6 A chemical experiment. Applying variance based methods in estimation/calibration problems. 3.7 An analytical example. Applying the method of Morris. 4. THE SCREENING EXERCISE. 4.1 Introduction. 4.2 The method of Morris. 4.3 Implementing the method. 4.4 Putting the method to work: an analytical example. 4.5 Putting the method to work: sensitivity analysis of a fish population model. 4.6 Conclusions. 5. METHODS BASED ON DECOMPOSING THE VARIANCE OF THE OUTPUT. 5.1 The settings. 5.2 Factors Prioritisation Setting. 5.3 First--order effects and interactions. 5.4 Application of Si to Setting 'Factors Prioritisation'. 5.5 More on variance decompositions. 5.6 Factors Fixing (FF) Setting. 5.7 Variance Cutting (VC) Setting. 5.8 Properties of the variance based methods. 5.9 How to compute the sensitivity indices: the case of orthogonal input. 5.9.1 A digression on the Fourier Amplitude Sensitivity Test (FAST). 5.10 How to compute the sensitivity indices: the case of non--orthogonal input. 5.11 Putting the method to work: the Level E model. 5.11.1 Case of orthogonal input factors. 5.11.2 Case of correlated input factors. 5.12 Putting the method to work: the bungee jumping model. 5.13 Caveats. 6. SENSITIVITY ANALYSIS IN DIAGNOSTIC MODELLING: MONTE CARLO FILTERING AND REGIONALISED SENSITIVITY ANALYSIS, BAYESIAN UNCERTAINTY ESTIMATION AND GLOBAL SENSITIVITY ANALYSIS. 6.1 Model calibration and Factors Mapping Setting. 6.2 Monte Carlo filtering and regionalised sensitivity analysis. 6.2.1 Caveats. 6.3 Putting MC filtering and RSA to work: the problem of hedging a financial portfolio. 6.4 Putting MC filtering and RSA to work: the Level E test case. 6.5 Bayesian uncertainty estimation and global sensitivity analysis. 6.5.1 Bayesian uncertainty estimation. 6.5.2 The GLUE case. 6.5.3 Using global sensitivity analysis in the Bayesian uncertainty estimation. 6.5.4 Implementation of the method. 6.6 Putting Bayesian analysis and global SA to work: two spheres. 6.7 Putting Bayesian analysis and global SA to work: a chemical experiment. 6.7.1 Bayesian uncertainty analysis (GLUE case). 6.7.2 Global sensitivity analysis. 6.7.3 Correlation analysis. 6.7.4 Further analysis by varying temperature in the data set: fewer interactions in the model. 6.8 Caveats. 7. HOW TO USE SIMLAB. 7.1 Introduction. 7.2 How to obtain and install SIMLAB. 7.3 SIMLAB main panel. 7.4 Sample generation. 7.4.1 FAST. 7.4.2 Fixed sampling. 7.4.3 Latin hypercube sampling (LHS). 7.4.4 The method of Morris. 7.4.5 Quasi--Random LpTau. 7.4.6 Random. 7.4.7 Replicated Latin Hypercube (r--LHS). 7.4.8 The method of Sobol'. 7.4.9 How to induce dependencies in the input factors. 7.5 How to execute models. 7.6 Sensitivity analysis. 8. FAMOUS QUOTES: SENSITIVITY ANALYSIS IN THE SCIENTIFIC DISCOURSE. REFERENCES. INDEX.

Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models

http://www.andreasaltelli.eu/file/repository/PUBLISHED_PAPER.pdf

Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index

We have tabulated the form of the coefficientsg1(x) andg2(x) as well as the boundary values [a, b] of the Fokker-Planck equation

$$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$

for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.

Orthogonal polynomial solutions of the Fokker-Planck equation

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration☆

We present by way of examples a comparison of the Fokker-Planck equations obtained by the Ito prescription and by the Stratonovich prescription for physical systems described by a Langevin equation with non-additive fluctuations. The particular examples we consider include Kubo line broadening and chemically reacting systems. We conclude that the Stratonovich prescription should always be used to analyze physical systems.

Stochastic processes with non-additive fluctuations: II. Some applications of Itô and Stratonovich calculus

We have investigated the relationship between Markovian master equations (m.e.) and the corresponding stochastic differential equations (s.d.e.) for closed systems, i.e., systems not subjected to external pumping. We show that the form of the fluctuations in the s.d.e., i.e., additive or multiplicative, depends upon the properties of the kernel of the m.e. and the range of the state space of the stochastic variable(s), i.e., bounded or unbounded. The knowledge of these two properties of the m.e. permits the determination of the way in which the fluctuations enter into the s.d.e. (i.e., additive or multiplicative) and the calculation of their statistics. Several examples are presented to illustrate the general theory.

Kurt E. Shuler

Papers

Orthogonal polynomial solutions of the Fokker-Planck equation

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration☆

Stochastic processes with non-additive fluctuations: II. Some applications of Itô and Stratonovich calculus

On the relations between Markovian master equations and stochastic differential equations

Studies in Non-Equilibrium Rate Processes. II. The Relaxation of Vibrational Non-Equilibrium Distributions in Chemical Reactions and Shock Waves