To Intrude Or Not To Intrude?Algorithmic Challenges in Uncertainty Propagation

To Intrude Or Not To Intrude?Algorithmic Challenges in Uncertainty Propagation

Paul Constantine, David Gleich, and Gianluca Iaccarino

Thermal and Fluid Sciences Affiliates and Sponsors ConferenceFebruary 5, 2009

Supported by

DOE PSAAP Program

The Modeling ProcessThe Modeling Process

Reality

Computational Model

Mathematical Model

Valid

ati

on

Verifica

tio

n

Pre

dic

tio

n

Coding

Assimilation

Qualification

Input uncertainty

QoI

Output uncertaintyData

1

2

3

4

The Modeling ProcessThe Modeling Process

Reality

Computational Model

Mathematical Model

Valid

ati

on

Verifica

tio

n

Pre

dic

tio

n

Coding

Assimilation

Qualification

Input uncertainty

QoI

Output uncertaintyData

1

2

3

4

Redefining The ProblemRedefining The Problem

Assume you want to compute a temperature field…

“Certain”

T = T(x)

“Uncertain”

T = T(x,y)

introduce parameters y

Recall that the new parameters may represent uncertainties in measured input quantities, geometries, model parameters, boundary conditions, etc.

This introduces a new parameter space for the quantity of interest (e.g. temperature).

What New Questions Can We Ask?What New Questions Can We Ask?

You now may ask…

What is the average temperature over the range of y at a point x?

What is the variance of temperature at a point x?

€

E[T](x) = T(x,y)dP(y)∫

€

Var[T](x) = (T(x, y) − E[T](x))2 dP(y)∫

What is the probability that the temperature will remain within some critical threshold at a point x?

€

Pr(T(x) < Tcritical ) = ρ (T(x,y))dP(y)−∞

Tcritical

∫

How Can We Compute These Statistics?How Can We Compute These Statistics?

Monte Carlo Methods

• Random sampling from the parameter space of y.

• Non-intrusive, but slow convergence.

How Can We Compute These Statistics?How Can We Compute These Statistics?

Monte Carlo Methods

• Random sampling from the parameter space of y.

• Non-intrusive, but slow convergence.

Interpolation (Stochastic Collocation)

• Interpolate solution at quadrature points in y, and integrals are quadrature rules.

• Non-intrusive and fast convergence, but aliasing error and curse of dimensionality.

How Can We Compute These Statistics?How Can We Compute These Statistics?

Monte Carlo Methods

• Random sampling from the parameter space of y.

• Non-intrusive, but slow convergence.

Interpolation (Stochastic Collocation)

• Interpolate solution at quadrature points in y, and integrals are quadrature rules.

• Non-intrusive and fast convergence, but aliasing error and curse of dimensionality.

Projection (Polynomial Chaos)

• Project the solution onto a polynomial basis of the parameter space.

• Fast convergence and best approximation, but intrusive and curse of dimensionality.

How Can We Compute These Statistics?How Can We Compute These Statistics?

Monte Carlo Methods

• Random sampling from the parameter space of y.

• Non-intrusive, but slow convergence.

Interpolation (Stochastic Collocation)

• Interpolate solution at quadrature points in y, and integrals are quadrature rules.

• Non-intrusive and fast convergence, but aliasing error and curse of dimensionality.

Projection (Polynomial Chaos)

• Project the solution onto a polynomial basis of the parameter space.

• Fast convergence and best approximation, but intrusive and curse of dimensionality.There are efficient alternatives to Monte Carlo!

A Simple ExampleA Simple Example

€

5 + 2y −y

−y 4 + 3y

⎡

⎣ ⎢

⎤

⎦ ⎥t1(y)

t2(y)

⎡

⎣ ⎢

⎤

⎦ ⎥=

1

1

⎡

⎣ ⎢

⎤

⎦ ⎥

Compute:Given the equation:

€

E[t1(y)],

€

E[t2(y)]

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

•

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

• Scalability. (One large scale run for each quadrature point.)

•

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

• Scalability. (One large scale run for each quadrature point.)

• Curse of dimensionality. (Exponential increase in cost.)

•

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

• Scalability. (One large scale run for each quadrature point.)

• Curse of dimensionality. (Exponential increase in cost.)

• Global approximations. (Discontinuities and singularities in y.)

•

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

• Scalability. (One large scale run for each quadrature point.)

• Curse of dimensionality. (Exponential increase in cost.)

• Global approximations. (Discontinuities and singularities in y.)

• Biased estimates. (Hard to estimate error.)

•

Challenges for Polynomial Approximation Challenges for Polynomial Approximation MethodsMethods

There are pressing challenges that keep the polynomial approximation methods from mainstream use despite their fast convergence properties.

• Scalability. (One large scale run for each quadrature point.)

• Curse of dimensionality. (Exponential increase in cost.)

• Global approximations. (Discontinuities and singularities in y.)

• Biased estimates. (Hard to estimate error.)

• Intrusive or Non-intrusive?

Addressing the ChallengesAddressing the Challenges(What We Have Been Doing)(What We Have Been Doing)

We have developed a way to compute the (best approximation) projection with a weakly intrusive implementation.

Assume the discrete solution is computed via an appropriate matrix equation.

€

A(y)u(y) = b(y)

Addressing the ChallengesAddressing the Challenges(What We Have Been Doing)(What We Have Been Doing)

We have developed a way to compute the (best approximation) projection with a weakly intrusive implementation.

Assume the discrete solution is computed via an appropriate matrix equation.

€

A(y)u(y) = b(y)

€

A(y0)−1b(y0)

Non-intrusive

Addressing the ChallengesAddressing the Challenges(What We Have Been Doing)(What We Have Been Doing)

We have developed a way to compute the (best approximation) projection with a weakly intrusive implementation.

Assume the discrete solution is computed via an appropriate matrix equation.

€

A(y)u(y) = b(y)

€

A(y0)−1b(y0)

€

A00 L A0n

M O M

An 0 L Ann

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

u0

M

un

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

b0

M

bn

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Non-intrusive Intrusive

Addressing the ChallengesAddressing the Challenges(What We Have Been Doing)(What We Have Been Doing)

We have developed a way to compute the (best approximation) projection with a weakly intrusive implementation.

Assume the discrete solution is computed via an appropriate matrix equation.

€

A(y)u(y) = b(y)

€

A(y0)−1b(y0)

€

A00 L A0n

M O M

An 0 L Ann

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

u0

M

un

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

b0

M

bn

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

A(y0)v,

Non-intrusive Intrusive Weakly

Intrusive

€

b(y0)

AcknowledgementsAcknowledgements

We wish to acknowledge:

• Generous support from the DOE ASC/PSAAP Program.

• Valuable feedback and comments from the Stanford UQ Group.THANKS FOR YOUR ATTENTION!THANKS FOR YOUR ATTENTION!

QUESTIONS?QUESTIONS?

Take-home MessageTake-home Message

• There are efficient alternatives to Monte Carlo that are easy to implement and ready for use.

• Stay tuned for reusable software libraries.