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### Computing Waves in the Face of Uncertainty

### Modeling

### Uncertainty

E. Bruce Pitman

Department of Mathematics

University at Buffalo

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): CRenschler, L. Namikawa, A. Sorokine, G. Sinha
- Center for Computational.Research M Jones, M. L. Green
- Iowa State University E Winer

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.

Pico de Orizaba, Mexico

Ballistic particle Simulations of pyroclastic flows and hazard map

at Pico de Orizaba -- hazard maps by Sheridan et. al.

“Hazard map” based on flow simulations and input uncertainty characterizations

Regions for which probability of flow > 1m for initial volumes ranging from 5000 m3 to 108 m3 -- flow volume distribution from historical data

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.

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

Savage , Hutter, Iverson, Denlinger, Gray, Pitman, …

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

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 0.001-10 m

(data from Iverson 1995, Iverson & Denlinger 2001)

ground

Model Topography and Equations(2D)Upper free surface

Fs(x,t) = s(x,y,t) – z = 0,

Basal material surface

Fb(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

Model System-Basic EquationsSolid 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:

d: basal friction angle

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()

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:

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

Model System – 2D

Depth averaging and scaling: Hyperbolic System of balance laws

continuity

x momentum

- Gravitational driving force
- Resisting force due to Coulomb friction at the base
- Intergranular Coulomb force due to velocity gradients normal to the direction of flow

1

2

3

Dalbey, Patra

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)

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

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

- 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?

Effect of different initial volumes

Left – block and Ash flow on Colima, V =1.5 x 105 m3

Right – same flow -- V = 8 x105 m3

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

Comparison of Models San Bernardino

Single phase model – water with frictional dissipation term!

Single phase model – low basal friction 4 deg!

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

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 10-3 error in sample TITAN calculation on 64 processors ~ 217 days

Latin Hypercube Sampling -- MMC

McKay 1979, Stein 1987, …

- For each random direction (random variable or input), divide that direction into Nbin bins of equal probability;
- Select one random value in each bin;
- Divide each bin into 2 bins of equal probability; the random value chosen above lies in one of these sub-bins;
- Select a random value in each sub-bin without one;
- Repeat steps 3 and 4 until desired level of accuracy is obtained.

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

Quantifying Uncertainty -- Approach

Wiener ’34, Xiu and Karniadakis’02

- Polynomial Chaos (PC): approximate pdf as the truncated sum of infinite number of orthogonal polynomials yi

- Multiply by ym and integrate to use orthogonality

PC for Burger’s equation

Let= kk U= UiI i=1..n k=1..n

Multiply by ψm and integrate

Coupled across all

Equations m=1..n

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

NISP

Replace integration with quadrature and interchange

order of integration of time and stochastic dimension

Quantifying Uncertainty -- Approach

PCQ: a simple deterministic sampling method with sample points chosen based on an understanding of PC and quadrature rules

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

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

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

Application to Volcan Colima

Initial volume uniformly

distributed from 1.57x106

to 1.57x107

Mean and standard deviation

of flow spread computed with

MC and PCQ

Monte Carlo PCQ

“Hazard Map” for Volcan Colima

Probability of flow

Exceeding 1m for

Initial volume ranging

From 5000 to 108 m3

And basal friction from

28 to 35 deg

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

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 www.gmfg.buffalo.edu
- Software can be accessed on the Computational Grid (DOE Open Science Grid) at http://grid.ccr.buffalo.edu

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