Nonlinearity and Randomness in Complex Systems1 Computing Waves in the Face of Uncertainty E. Bruce Pitman Department of Mathematics University at Buffalo

2 Part of a large project investigating geophysical mass flows  Interdisciplinary research project funded by NSF (ITR and EAR)  UB departments/people involved:  Mechanical engineering: A Patra, A Bauer, T Kesavadas, C Bloebaum, A. Paliwal, K. Dalbey, N. Subramaniam, P. Nair, V. Kalivarappu, A. Vaze, A. Chanda  Mathematics: E.B. Pitman, C Nichita, L. Le  Geology: M Sheridan, M Bursik, B.Yu, B. Rupp, A. Stinton, A. Webb, B. Burkett  Geography ( National Center for Geographic Information and Analysis ): C Renschler, L. Namikawa, A. Sorokine, G. Sinha  Center for Computational.Research M Jones, M. L. Green  Iowa State University E Winer

3 Mt. St. Helens, USA

4 Volcan Colima, Mexico

5 Atenquique, Mexico 1955

6

7 San Bernardino Mountain: Waterman Canyon

8 Guinsaugon. Phillipines, 02/16/06 Heavy rain sent a torrent of earth, mud and rocks down on the village of Guinsaugon. Phillipines, 02/16/06. A relief official says 1,800 people are feared dead.

9 Ruapehu, New Zealand

10 Pico de Orizaba, Mexico Ballistic particle Simulations of pyroclastic flows and hazard map at Pico de Orizaba -- hazard maps by Sheridan et. al.

11 “Hazard map” based on flow simulations and input uncertainty characterizations Regions for which probability of flow > 1m for initial volumes ranging from 5000 m 3 to 10 8 m 3 -- flow volume distribution from historical data

12 Introduction  Geophysical flows e.g. rock falls, debris flows, avalanches, volcanic lava flows may have devastating consequences for the human population  Need “what if …?” simulation tool to estimate hazards for formulating public safety measures  We have developed TITAN2D  Simulate flows on natural terrain,  Be robust, numerically accurate and run efficiently on a large variety of serial and parallel machines,  Quantify the effect of uncertain inputs  Have good visualization capabilities.

13 Goals of this talk  Basic mathematical modeling  Will not address extensions such as erosion, two phase flows, that are important in the field  Uncertainty Quantification  Hyperbolic PDE system – poses special difficulties for uncertainty computations  Ultimate aim is Hazard Maps

Nonlinearity and Randomness in Complex Systems14 Modeling Savage, Hutter, Iverson, Denlinger, Gray, Pitman, …

15 Modeling Many models – complex physics is still not perfectly represented ! Savage-Hutter Model Iverson-Denlinger mixture theory Model Pitman-Le Two-phase model Debris Flows are hazardous mixture of soil, rocks, clasts with interstitial fluid present

16 Micromechanics and Macromechanics  Characteristic length scales (from mm to Km)  e.g. for Mount St. Helens (mudflow –1985)  Runout distance  31,000 m  Descent height  2,150 m  Flow length( L )  100-2,000  Flow thickness( H )  1-10 m  Mean diameter of sediment material m (data from Iverson 1995, Iverson & Denlinger 2001)

17 Model Topography and Equations(2D) ground flowing mass Upper free surface F s (x,t) = s(x,y,t) – z = 0, Basal material surface F b (x,t) = b(x,y) – z = 0 Kinematic BC: Iverson and Denlinger JGR, 2001; Pitman et. al. Phys. Fluids, 2003; Patra et. al, JVGR, 2005

18 Model System-Basic Equations Solid Phase Only The conservation laws for a continuum incompressible medium are: stress-strain rate relationship derived from Coulomb theory [Aside: this system of equations is ill-posed (Schaeffer 1987) ] Boundary conditions for stress:  : basal friction angle

19 Model System-Scaling Scaling variables are chosen to reflect the shallowness of the geophysical mass L – characteristic length in the downstream and cross-stream directions ( Ox,Oy ) H – characteristic length in normal direction to the flow ( Oz ) Drop (most) terms of O (  )

20 Model System-Depth Average Theory Depth average where is the avalanche thickness z – dimension is removed from the problem - e.g. for the continuity equation: where are the averaged lateral velocities defined as:

21 Modeling of Granular Stresses Earth pressure coefficient is employed to relate normal stresses Shear stresses assumed proportional to normal stresses Hydraulic assumption in normal direction

22 Depth averaging and scaling: Hyperbolic System of balance laws continuity x momentum 1.Gravitational driving force 2.Resisting force due to Coulomb friction at the base 3.Intergranular Coulomb force due to velocity gradients normal to the direction of flow Model System – 2D 132

Nonlinearity and Randomness in Complex Systems23 Uncertainty Dalbey, Patra

24 Modeling and Uncertainty “Why prediction of grain behavior is difficult in geophysical granular systems” ” “…there is no universal constitutive description of this phenomenon as there is for hydraulics” the variability of granular agglomerations is so large that fundamental physics is not capable of accurately describing the system and its variations P. Haff ( Powders and Grains ’97 )

25 Uncertainty in Outputs of Simulations of Geophysical Mass Flows Model Uncertainty Model Formulation: Assumptions and Simplifications Model Evaluation: Numerical Approximation, Solution strategies – error estimation Data Uncertainty propagation of input data uncertainty

26 Modeling Uncertainty Sources of Input Data Uncertainty Initial conditions – flow volume and position Bed and internal friction parameters Terrain errors Erosion and two phase model parameters

27 INPUT UNCERTAINTY PROPAGATION Model inputs – material, loading and boundary data are always uncertain range of data and its distributions may be estimated propagate input range and distribution to an output range and distributions e.g. maximum strain, maximum excursion How does uncertain input produce a solution distribution?

28 Effect of different initial volumes Left – block and Ash flow on Colima, V =1.5 x 10 5 m 3 Right – same flow -- V = 8 x10 5 m 3

29 Effect of initial position, friction angles Figure shows output of simulation From TITAN2D – A) initial pile location, C) and D) used different friction angles,and, F) used a perturbed starting location Figure shows output of simulation From TITAN2D – A) initial pile location, C) and D) used different friction angles,and, F) used a perturbed starting location Figure shows output of simulation from TITAN2D – A) initial pile location, C) and D) used different friction angles, and, F) used a perturbed starting location

30 Comparison of Models San Bernardino Single phase model – low basal friction 4 deg! Single phase model – water with frictional dissipation term!

31 Comparison of Models 50% solid fraction 70% solid fraction

32 Quantifying Uncertainty -- Approach Methods Monte Carlo (MC) Latin Hypercube Sampling (LHS) Polynomial Chaos (PC) Non Intrusive Spectral Projection (NISP) ―Polynomial Chaos Quadrature (PCQ) Stochastic Collocation } Random sampling based } Functional Approximation

33 Quantifying Uncertainty -- MC Approach Monte Carlo (MC): random sampling of input pdf Moments can be computed by running averages e.g. mean and standard deviation is given by: Central Limit Theorem :  Computationally expensive. Estimated computational time for error in sample TITAN calculation on 64 processors ~ 217 days

34 Latin Hypercube Sampling -- MMC 1. For each random direction (random variable or input), divide that direction into N bin bins of equal probability; 2. Select one random value in each bin; 3. Divide each bin into 2 bins of equal probability; the random value chosen above lies in one of these sub-bins; 4. Select a random value in each sub-bin without one; 5. Repeat steps 3 and 4 until desired level of accuracy is obtained. McKay 1979, Stein 1987, …

35 Functional Approximations In these approaches we attempt to compute an approximation of the output pdf based on functional approximations of the input pdf Prototypical method of this is the Karhunen Loeve expansion

36 Karhunen-Loeve (KL) Expansions KL expansions are representations of the random process f(t,  ) in terms of the eigenfunctions of the covariance function R ff (t 1,t  ) s R ff (t 1,t 2 )  i (t 2 ) d t 2 = i  i (t 1 ) f(t,  ) = +  { i } 0.5  i (t)  i (  ) i=1.. 1

37 Quantifying Uncertainty -- Approach Polynomial Chaos (PC): approximate pdf as the truncated sum of infinite number of orthogonal polynomials  i Multiply by  m and integrate to use orthogonality Wiener ’34, Xiu and Karniadakis’02

38 Chaos solver

39 PC for Burger’s equation Let = k  k U= U i  I i=1..n k=1..n Multiply by ψ m and integrate Coupled across all Equations m=1..n

40 Polynomial Chaos Quadrature Instead of Galerkin projection, integrate by quadrature weights Analogy with Non-Intrusive Spectral Projection Stochastic Collocation Leads to a method that has the simplicity of MC sampling and cost of PC Can directly compute all moment integrals Efficiency degrades for large number of random variables

41 NISP Replace integration with quadrature and interchange order of integration of time and stochastic dimension

42 Polynomial Chaos Quadrature (PCQ)

43 NISP / Polynomial Chaos Quadrature Replace integration with quadrature and interchange order of integration of time and stochastic dimension

44 Quantifying Uncertainty -- Approach PCQ : a simple deterministic sampling method with sample points chosen based on an understanding of PC and quadrature rules  makes PC computationally feasible for non- linear non-polynomial forms  easy to implement and parallelize; statistics obtained directly  can use random variables from multiple distributions simultaneously  difficult to find sample points for very high orders  “curse of dimensionality” -- samples required grows exponentially as a function of number of RV

45 Quantification of Uncertainty Test Problem Application to flow at Volcan Colima Starting location, and, Initial volume are assumed to be random variables distributed according to assumption, or available data

46 Test Problem Burgers equation Figure shows statistics of time required to reach steady state for randomly positioned shock in initial condition; PCQ converges much faster than Monte Carlo

47 Volume vs. frequency for flows at Montserrat

48 Quantifying Uncertainty Starting location  Gaussian with std. deviation of 150m Mean Flow Flow from starting locations 3 std. dev away Mean Flow Flow from starting locations 3 std. dev away

49 Application to Volcan Colima Initial volume  uniformly distributed from 1.57x10 6 to 1.57x10 7 Mean and standard deviation of flow spread computed with MC and PCQ Monte Carlo PCQ

50 Mean Flow for Volcan Colima for initial volume uncertainty

51 Mean+3std. dev for Volcan Colima -- initial volume uncertainty

52 “Hazard Map” for Volcan Colima Probability of flow Exceeding 1m for Initial volume ranging From 5000 to 10 8 m 3 And basal friction from 28 to 35 deg

53 Conclusions  Developed robust simulation tool for modeling geophysical mass flows on natural terrain.  Tool uses:  Adaptive meshing  Parallel computing  Integrated with Geographical Information Systems  Quantification of uncertainty for probabilistic inputs  All developed software is available free and open source from  Software can be accessed on the Computational Grid (DOE Open Science Grid) at

54 Conclusions  PCQ is an attractive methodology for determining the solution distribution as a consequence of uncertainty  Find full pdf  Curse of dimensionality still strikes  MC, LH, NISP, Point Estimate methods, PCQ – which to use depends on the problem at hand

55 Conclusions  How to handle uncertainty in terrain? In the models?  More work to integrate PCQ into output functionals that prove valuable  All developed software is available free and open source from  Software can be accessed on the Computational Grid (DOE Open Science Grid) at

56 Current/Future Work  Better theory is needed to incorporate effect of interstitial fluid  Integrate erosion with two fluid model  Residual based error estimates in quantities of interest  Solve the full 3D system.  Moving Least Squares  Free surface/basal surface tracking with level set method.