# Presentation: 302170

## A Sparse Grid Collocation Approach to Stochastic Inverse Problems

- Sibley School of Mechanical & Aerospace Engineering
- 188 Frank H T Rhodes Hall
- Cornell University
- Ithaca, NY 14853
- Email: zabaras@cornell.edu
- URL: http://mpdc.mae.cornell.edu

- NICHOLAS ZABARAS

***A Sparse Grid Collocation Approach to Stochastic Inverse Problems***

## CASCADING UNCERTAINTIES IN REAL PHENOMENA

- Only statistical/averaged quantities available to characterize these structures experimentally.
- Eg. Volume fraction, two-point correlation and semi-variograms

- Application

- Thermal analysis of functionally graded systems

- Analysis of oil flow in geological strata

- Uncertainties across multiple length and time scales. Must incorporate them for realistic analysis

## DESIGN/ESTIMATION IN THE PRESENSE OF UNCERTAINTY

- Design deformation process, thermal processes for optimal performance, in the presence of multiple uncertainties

- Estimate location/magnitude of initiation events, eg contamination source detection, parameter identification in the presence of uncertainties

## KEY ISSUES WITH UNCERTAINTY INDUCED DESIGN

- Can ideas from _deterministic design and optimization_ be incorporated?
- Can _validated, mature, deterministic models_ be directly used?
- Can the framework be made _scalable_? how _parallelizable_ can it be?
- How can _experimental information be encoded_ for effective design?

## STATE-OF-THE-ART IN STOCHASTIC INVERSE DESIGN 

- _Ghanem et. al.,(__[2006](http://portal.acm.org/citation.cfm?id=1167055)_)  On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data, Using Bayesian to construct stochastic fields based on experimental measurements
- _Kaipio_ (_[2007](http://portal.acm.org/citation.cfm?id=1232207&dl=ACM&coll=portal&CFID=15151515&CFTOKEN=6184618)_), Analyze the effect of discretization errors in inverse problems by using a statistical approach.
- _Ferreira_(_[2005](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK9-4KWTF8V-1&_user=492137&_coverDate=04/01/2007&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=472af1693fff9ccaf6e0f21605095b2e)_) A multi-scale random field model as a prior in multiscale permeability estimation using Bayesian method.
- Sequential Monte Carlo methods (Particle filtering) has also attracted attention in dynamical inverse problems.
- _Zabaras et.al._(_[2004](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V3H-4CB057X-4&_user=492137&_coverDate=08/31/2004&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=618a0954ff0f8a12ca2a763767ae88d6)_), A Bayesian inference approach to the stochastic inverse heat conduction problem

- STATE-OF-THE-ART IN STOCHASTIC INVERSE DESIGN

- State of the art is to use Bayesian techniques for inversion. A few applications to design. Recent advances in coupling GPCE with Bayesian

## STATE-OF-THE-ART IN STOCHASTIC INVERSE DESIGN 

- Recent significant advances in modeling  uncertainties.
- Non-statistical methods involve representing the uncertainty using spectral1,2,3, polynomial4,5,6 and wavelet7 basis functions.
- Mature framework to implement a design framework8.

- P Spanos and R Ghanem, Stochastic finite elements: A spectral approach, Springer-Verlag (1991).
- W Schoutens, Stochastic processes and orthogonal polynomials, Lecture notes in statistics, Springer (1998)
- D. Xiu and G. Karniadakis, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J Sci Computing, 24 (_[2002](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJOCE3000024000002000619000001&idtype=cvips&gifs=yes)_) 619-644
- I. M. Babuska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Journal on Numerical Analysis 45 (_[2007](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000003001005000001&idtype=cvips&gifs=yes)_) 1005-1034
- F. Nobile, R. Tempone and C. G. Webster, A sparse grid stochastic collocation method for elliptic partial differential equations with random input data. (_[Preprint](http://people.scs.fsu.edu/~webster/Clayton%20Webster/CV_files/sparse_collocation.pdf)_)
- B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation methods for stochastic natural convection problems, Journal of Computational Physics 225 (_[2007](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-4MSPV7N-9&_user=492137&_coverDate=07/01/2007&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=27eb8649265bc853bf50630b21e0193e)_) 652-685
- O. P. Le Maître et. al. Multi-resolution analysis of wiener-type uncertainty propagation schemes, Journal of Computational Physics 197 (_[2004](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHY-4BT1745-2&_user=492137&_coverDate=07/01/2004&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000022719&_version=1&_urlVersion=0&_userid=492137&md5=c0f755e989752ecb401f7d38f8ada9b3)_) 502-531
- V. A. BadriNarayanan, N. Zabaras, Stochastic inverse heat conduction using a spectral approach, International Journal for Numerical Methods in Engineering 60 (_[2004](http://www3.interscience.wiley.com/cgi-bin/abstract/109075116/ABSTRACT?CRETRY=1&SRETRY=0)_) 1569-1593

## SCALABLE STOCHASTIC OPTIMIZATION FRAMEWORK

- Framework based on finite dimensional representation of input uncertainty.
- Estimation/Design posed as a functional/discrete optimization problem
- Use appropriate stochastic representation forms to convert (infinite dimensional) stochastic optimization into a finite dimensional deterministic optimization
- Optimization strategy based on repeated solutions to the direct problem (comparison with Bayesian)
- The solution of the stochastic direct problem: A scalable, embarrassingly parallel strategy that allows the use of blackbox deterministic simulators

## SOLUTION TO THE STOCHASTIC DIRECT PROBLEM

- Various sources of uncertainty.
- Represent each using a finite set of uncorrelated random variables
- The dependant variable lies in the produce space of all these variations

## COLLOCATION BASED FRAMEWORK

- The GPCE method approximates this *M* dimensional function using a spectral element discretization (which causes the coupled set of equations)

- Spectral discretization represents the function as a linear combination of sets of orthogonal polynomials.
- Instead of a spectral discretization, use   other kinds of polynomial approximations
- Use simple interpolation.
- That is, sample the function at some points and construct a polynomial  approximation

## COLLOCATION BASED FRAMEWORK

- Stochastic function in 2 dimensions

- Need to represent this function

- Sample the function at a finite set of points

- Use polynomials (Lagrange polynomials) to get a approximate representation

## COLLOCATION BASED FRAMEWORK

- Stochastic domain is approximated using *multidimensional interpolating functions*

- Once the interpolation function has been constructed, the function value at any point    is just

- One can construct the stochastic solution by solving at the *N* points specified by

- Going back to our problem of interest, we want to construct a polynomial approximation to our variables (u,p,θ)

- This is just a set of deterministic equations, with a specified value of uncertainty given by these M-tuples

## CONSEQUENCES

- Construct the complete stochastic solution by sampling the stochastic space at N distinct points.
- That is, solve the deterministic set of equations defined at the M- tuple

-  This results in a completely DECOUPLED set of equations. Have to solve N distinct sets of equations
-  Possibility of parallelization: Have to solve these equations separately, this makes the solution embarrassingly parallel.

## CONSTRUCTING MULTIDIMENSIONAL INTERPOLANTS

- For instance, if M=10 dimensions and we use k points in each direction

| Number of points in each direction, k | Total number of sampling points |
| --- | --- |
| 2 | 1024 |
| 3 | 59049 |
| 4 | 1.05x106 |
| 5 | 9.76x106 |
| 10 | 1x1010 |

*One idea is only to pick the* *more important points from the tensor product grid.*

*Simple tensor products*

## SPARSE GRID COLLOCATION

- 1D sampling

- 2D sampling

- In higher dimensions, not all the points are equally important.
- Some points contribute less to the accuracy of the solution (e.g. points where the function is very smooth, regions where the function is linear, etc.). Discard the points that contribute less: SPARSE GRID COLLOCATION

- Tensor product

## SPARSE GRID COLLOCATION

- Provides a way to construct interpolation functions based on minimal number of points
- Provides a way to progress naturally from univariate interpolation to multivariate interpolation.
- First construct one dimensional interpolation functions
- Construct the required multi-dimensional as a linear combination of products of these 1D functions
- How to choose the degree and coefficients of these products?
- Smolyak (1963) came up with a set of rules to construct such products1.

- S. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Soviet Math. Dokl., 4 (1963) 240--243.

## SPARSE GRIDS

- Number of points = 29
- Number of points full grid = 81

- Number of points = 145
- Number of points full grid = 1089

- Number of points = 69
- Number of points full grid = 729

- Number of points = 441
- Number of points full grid = 35937

## SPARSE GRIDS

- Depth of interpolation, q

- Number of sampling points, N

- The number of sampling points in M dimensions is given by the cardinality of the set:

- The plot shows the number of sampling points as a function of the depth of interpolation for d = 2, 3, 5 .

- M. Babuska, F. Nobile, R. Tempone, A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data, SIAM J Num. Anal. 45 (_[2007](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000003001005000001&idtype=cvips&gifs=yes)_) 1005-1034
- V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (_[2000](http://www.springerlink.com/content/rvjl1822m67hv3l2/)_) 273–288

- It is straightforward to come up with bounds on N1,2.

## ERROR ESTIMATES

- V. Barthelmann, E. Novak and K. Ritter, High dimensional polynomial interpolation on sparse grids, Adv. Comput. Math. 12 (_[2000](http://www.springerlink.com/content/rvjl1822m67hv3l2/)_), 273–288
- E. Novak, K. Ritter, R. Schmitt and A. Steinbauer, On an interpolatory method for high dimensional integration, J. Comp. Appl. Math. 112 (_[1999](http://www.ingentaconnect.com/content/els/03770427/1999/00000112/00000001/art00222)_), 215–228

- Depending on the order of the one-dimensional interpolant, one can come up with error bounds on the approximating quality of the interpolant1,2.

- If piecewise linear 1D interpolating functions are used to construct the sparse interpolant, the error estimate is given by:

- Where N is the number of sampling points,

- If 1D polynomial interpolation functions are used, the error bound is:

- assuming that f has k continuous derivatives

## IMPLEMENTATION

- PREPROCESSING
- Compute list of collocation points based on number of stochastic dimensions, d and level of interpolation, q

- Solve the deterministic problem defined by each set of collocated points

- POSTPROCESSING
- Compute moments and other statistics with simple operations of the deterministic data at the collocated points

- Solution Methodology

- Use any validated deterministic solution procedure.
- Completely non intrusive

- Std deviation of temperature: Natural convection

## POSTPROCESING

- Once the function values are available at all the sparse points, one can construct the stochastic solution.
- One is usually interested in the mean, standard deviation and the probability density function of the dependant variables at various points in the physical domain
- Mean and higher moments:
-      These moments are computed very efficiently as a linear combination of the function values at the sparse grid points with a set of a priori computed weights
- Probability density function  (PDF)
-      The function is approximated using the sparse grid method. Following this a large number of function evaluations are performed. A normalized histogram of the function distribution then gives the PDF

## SOLVING THE STOCHASTIC DIRECT PROBLEM: RECAP

- Given multiple sources of uncertainty affecting the system in the form of a function of a finite set of uncorrelated random variables.

- Total dimensionality of the space in which the dependant variable resides in is

- The solution to the stochastic PDE

- Is found by solving N deterministic PDE’s

- The solution is written as

## THE STOCHASTIC INVERSE PROBLEM: INTUITIVE IDEAS

- The solution to the direct problem _***assumes***_ that a neat representation of the input uncertainties in terms of a finite number of uncorrelated random variables is available

- -1(a)

- In the inverse problem, we are given the stochastic dependant variable and are asked to compute the input stochastic processes that result in the given dependant variable

- But what is actually given?

- (a)

**Notes:**
Consider a deterministic inverse problem. Given some experimental measurements/sensor data, the task is to reconstruct the best deterministic input parameter that results in the given data. 
In the context of a stochastic inverse problem, there is a big gap between what is measured and what is required by the framework
If we are given a neat stochastic representation (i.e in terms of a finite set of uncorrelated random variables) of the measured variable, the it is possible to come up with strategies of solving this inverse problem.

## THE STOCHASTIC INVERSE PROBLEM: INTUITIVE IDEAS

- _Issues:_

- The measured/desired quantity is a stochastic process that depends on multiple sources of uncertainty. How to isolate the effect of one process?
- Physically motivated issues: In most applications, one measures realizations and subsequently provides ONLY moments and/or a p.d.f of the dependant variable

- (a)

- -1(a)

- HOW?

- Source A

- Source B

- Source C

## THE STOCHASTIC INVERSE PROBLEM: INTUITIVE IDEAS

- Given limited information about the dependant stochastic variable- in terms of ONLY, say, k moments or a p.d.f – is it possible to construct a stochastic representation of the design variables?

- For clarity, consider the direct problem again.

- (a)

- Moments, pdf

_*Suppose that we are ONLY given moments or a p.d.f of the input stochastic process, what can be said about the dependant stochastic process*_

- Results in ambiguity in the exact nature of the input stochastic process

## POSING THE STOCHASTIC INVERSE PROBLEM

- Given this ambiguity, can anything be said about the dependant stochastic process?

- Ch. Schwab, R. A. Todur, Sparse finite elements for Stochastic elliptic problems - Higher moments, Computing 71 (_[2003](http://www.springerlink.com/content/buartcrfdjaemuyd/)_) 43-63
- I. Babuska, R. Tempone, G. E. Zouraris, Galerkin finite elements approximation of stochastic finite elements, SIAM J. Numer. Anal. 42 (_[2004](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000042000002000800000001&idtype=cvips&gifs=yes)_) 800–825.
- I. M. Babuska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Journal on Numerical Analysis 45 (_[2007](http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=SJNAAM000045000003001005000001&idtype=cvips&gifs=yes)_) 1005-1034

- Limit ourselves to the case of a stochastic elliptic equation
- Pose the direct problem in the variational (weak) form.
- Under certain smoothness conditions of the coefficients and load1,2,3:

_**Theorem 1**_: The stochastic elliptic problem has a unique solution (in the expectation sense)

_**Theorem 2**_:  Under certain smoothness conditions for the input stochastic process, the k-th order moment of solution exists

_**Theorem 3**_: Given the k-order moment of the input stochastic process, the k-th order moment of the solution exists and is unique

## POSING THE STOCHASTIC INVERSE PROBLEM

- Physically meaningful designed quantity would be moments and/or a p.d.f of the corresponding input stochastic process

- -1(a)

- -1(a)

- Given measurements in terms of p.d.f/moments => Construct the optimal p.d.f/moments of input quantities

## PROBLEM STATEMENT

- For the sake of generality, we consider the stochastic parabolic problem (transient heat conduction), though some of our proofs are for the stochastic elliptic problem (heat conduction).

- _Input uncertainties_: Thermal conductivity (known), heat flux (unknown)
- _Given_: p.d.f or moments of temperature at some sensor locations

_*TYPE I*_: Find the _moments of the stochastic heat flux_, such that the solution of the system defined by the above SPDE results in a temperature profile whose k moments match the given experimental data at s sensor loactions

_*TYPE II*_: Find the _p.d.f of the stochastic heat flux_, such that the solution of the system defined by the above SPDE results in a temperature profile whose p.d.f matches the given experimental data at s sensor loactions

- ∂θ

- 

- sensors

- ∂h

## THE DESIGN PROBLEM

- The unknown stochastic heat flux can be written in terms of a finite set of uncorrelated random variables.
- Following the usual procedure in collocation based solution techniques, the unknown stochastic heat flux is written in terms of nq deterministic realizations of the heat flux

- Have to design these nq deterministic functions. Pose problems in terms of designing these functions

_*TYPE I*_: _Designing Moments_:  Find q(x,t,ξi), i=1,...,nq such that the solution of the system defined by the above SPDE results in a temperature profile whose k moments match the given experimental data at s sensor locations

_*TYPE II*_: _Designing p.d.f_: Find q(x,t,ξi), i=1,...,nq, such that the solution of the system defined by the above SPDE results in a temperature profile whose p.d.f matches the given experimental data at s sensor locations

## THE OPTIMIZATION PROBLEM

- Assuming that a solution exists in the sense of Tikhonov, look for a solution

-             is a cost functional that quantifies how well the designed heat flux performs.
- Convert the design problem into an optimization problem.

_*Problem Type I*_:  Define cost functional in terms of difference in moments

_*Problem Type II*_:  Define cost functional in terms of difference in pdf

- Computed moments

- Measured/desired moments

-           is the inverse c.d.f (chosen for computational efficiency and known support1. [0 1] )

- N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Computational Physics, submitted (_[Preprint](http://mpdc.mae.cornell.edu/Publications/PDFiles/StochasticSensitivityAnalysis.pdf)_)

## THE OPTIMIZATION PROBLEM

- Represent the cost functional in terms of the nq deterministic functions.
- Utilize the sparse grid representation to simplify the cost functional.
- Using the sparse representation of the dependant variable ...

- Represent the moments as ...

- And the inverse cumulative distribution function as ...

## GRADIENT OF THE COST FUNCTIONAL

- Gradient based optimization utilized.
- Need to define the gradient of the cost functional.

- Define the notion of the directional derivative

- After some manipulations, can write the gradient as

_*Problem Type I*_:

_*Problem Type II*_:

## STOCHASTIC SENSITIVITY EQUATIONS

- Directional derivative term inside the gradient can be simplified to:

- Where Θ is the linear in ∆qv part of the temperature

- Thus, the calculation of the gradient of the cost functional requires the computation of the stochastic sensitivity of the temperature.
- But stochastic sensitivity definition very simple using sparse grid collocation

## STOCHASTIC SENSITIVITY EQUATIONS

- The stochastic sensitivity equations are simply directional derivatives of the N deterministic equation w.r.t each of the design parameters qi.

- The stochastic sensitivity can then be written in terms of solution to these N*n equations as:

- The stochastic sensitivity equations are

## THE OPTIMIZATION FRAMEWORK

- The stochastic optimization framework uses:
- A minimal set of decoupled, deterministic direct problems
- A minimal set of decoupled, deterministic sensitivity problems
- The only serial part of this framework is the computation of the cost functional and the gradient => highly scalable

## NUMERICAL TESTS

## 1D IHCP: Test 1: Problem Type I- Reconstructing moments

- 1D rod of length L=1.0

- The thermal conductivity and the heat flux are stochastic.
- The thermal conductivity, k varies as:

- The moments of the temperature at the sensor location are given.
- These ‘experimental’ moments are computed by running a direct problem using a predefined distribution for the stochastic heat flux. A standard Beta distribution with p=1.5 and q = 2 is used
- Run direct problem using higher discretization in physical space and stochastic space than corresponding inverse problem- prevent inverse crimes

## 1D IHCP: Test 1: Reconstructing moments

- Convergence with increasing depth of interpolation

| depth | Design variables |
| --- | --- |
| 2 | 5 |
| 4 | 17 |
| 6 | 65 |
| 8 | 257 |

- Use one r.v to represent q. ξ = U[0,1]
- First few (~7) moments captured with very coarse representation of the space

- Accuracy increases with depth

## Hierarchical stochastic optimization: A rudimentary example

- The optimization with lower depth of interpolation is faster
- Utilize this to construct a coarse stochastic solution.
- Use this solution as an initial guess to run a refined optimization problem

- Similar in idea to accelerated convergence in multi-grid methods
- We call this ‘hierarchical stochastic optimization method’. Results in significant reduction in computational effort

## 1D IHCP: Test 1b: Reconstructing moments

- Invariance with choice of the representation of q

- Use one random variable used to represent q.
- 4 runs with different r.v and associated probability measures were performed

- Almost identical results using all four different representations

| N | Exp | Beta | Gamma | Normal | Uniform |
| --- | --- | --- | --- | --- | --- |
| 1 | 0.42858 | 0.42855 | 0.42864 | 0.42854 | 0.42856 |
| 2 | 0.23811 | 0.23802 | 0.23779 | 0.23835 | 0.23882 |
| 3 | 0.15155 | 0.15193 | 0.15180 | 0.15177 | 0.14919 |
| 4 | 0.10494 | 0.10533 | 0.10538 | 0.10503 | 0.10572 |
| 5 | 0.07698 | 0.07698 | 0.07724 | 0.0768 | 0.07764 |
| 6 | 0.05889 | 0.05831 | 0.05879 | 0.05834 | 0.06095 |

## 1D IHCP: Test 2: Problem Type II- Reconstructing pdf’s

- 1D rod of length L=1.0

- The thermal conductivity and the heat flux are stochastic.
- The thermal conductivity, k varies as:

- The pdf of the temperature at the sensor location is given.
- These ‘experimental’ moments are computed by running a direct problem using a predefined distribution for the stochastic heat flux.
- CASE 1: Finite support: A standard Beta distribution with p=1.5 and q = 2 is used
- CASE 2: Unbounded support: A standard Normal distribution

## 1D IHCP: Test 2: Reconstructing pdf’s

- Slightly pixilated representation for medium depths of interpolation
- Almost perfect representation for higher depths of interpolation

- Use one random variable to represent q. ξ = U[0,1]

- Convergence with increasing depth of interpolation: Finite support

## 1D IHCP: Test 2: Reconstructing pdf’s

- For lower depths of interpolation, the tails are not captured well.
- Slightly pixilated representation for medium depths of interpolation
- Almost perfect representation for higher depths of interpolation

- Use one random variable to represent q. ξ = U[0,1]

- Convergence with increasing depth of interpolation: Infinite support

## 1D IHCP: Test 2b: Reconstructing pdf’s

- Invariance with choice of the representation of q

- Use one r.v to represent q.
- 4 runs with different r.v and associated probability measures were performed.
- Used:
- Uniform random variable  U[0,1]
- Normal random variable  N(0,1)
- Gamma random variable G(2)
- Beta random variable B(1.5,2)

## 1D IHCP: Test 3: Reconstructing pdf’s

- Reconstructing high dimensional input stochastic processes

- The thermal conductivity and the heat flux are stochastic.
- The thermal conductivity, k varies as:

- The pdf of the temperature at the sensor location is given.
- These ‘experimental’ moments are computed by running a direct problem using a multidimensional stochastic representation of q

## 1D IHCP: Test 3: Reconstructing pdf’s

- Reconstructing high dimensional input stochastic processes

- Convergence w.r.t depth of interpolation.
- Use one random variable to represent q.

- Comparison of reconstructed and original stochastic heat flux.
- Circles represent experimental pdf, line is the reconstructed pdf

## 2D IHCP: Transient Analysis

- T= 0.5

- Variable conductivity

- See if framework is applicable to solve transient problems
- Consider a 2D domain [-0.5 0.5]^2
- The temperature on the right wall is held constant
- The thermal conductivity is stochastic.

- Design q

- The thermal conductivity has a mean value of 20 and is correlated with a correlation kernel that is exponential. The correlation length is 0.1.
- Represent the conductivity using a finite set of uncorrelated random variables- Karhunen Loeve expansion

- Use first three random variables to represent (known) variability in k

## 2D IHCP: Problem definition and ‘experimental’ data

- T= 0.5

- Variable conductivity

- Design q

*Given the pdf of temperature at sensor locations over the time interval [0, T] , reconstruct the pdf of heat flux applied on the left boundary in [0,T]*

- Sensors are located at a distance 0.1L into the domain

_*Constructing the pdf of temperatures at the sensors*_: A known decaying stochastic heat flux profile  is applied on the left boundary.
- The heat flux is assumed to have an exponential correlation with b = 0.5
- Represent the ‘experimental’ heat flux using a finite set of uncorrelated random variables. Use the first three eigen values

## 2D IHCP: Problem definition and ‘experimental’ data

- Input random process, flux

- Output random process, Temp

- Making sure ‘inverse crimes’ are not committed. Very high discretization spatially, temporally and in stochastic space.

## 2D IHCP: Optimization

- _Spatial discretization_:  40x40 quad elements
- _Time discretization_:  50 time steps, dt = 0.001
- _Representation of k_: Level 5 sparse grid ~ 441 points in 3 dim stochastic space

*Design each nodal dof on the boundary for each time: 41*50 stochastic quantities *

- Each q(y,t,ξ) represented using one uniform random variable. Use sparse grid interpolants to represent these

- Set N = 65. Corresponds to level 6 interpolation
- Total number of design variables = N*41*50 = 133,250
- Each direct stochastic problem requires solution of 65*441 = 28665 deterministic problems
- Each direct problem corresponds to 41*41*50 = 84,050 variables
- Total dof/iteration = 2.41 *billion*

## 2D IHCP: Results

- Use 40 nodes of local linux cluster ~ 160 processors
- Each iteration took about 74 minutes.
- Movies compare the time evolution of reconstructed pdf and the experimental pdf at two locations

- Reconstructed flux pdf

- Experimental flux pdf

- Midpoint, y = 0.0

- Endpoint, y = -0.5

## 2D IHCP: Scalability/Speedup

- Run the problem using different number of processes
- Estimate the parallelizability of the framework by looking at time taken to complete one iteration in the optimization problem

- Number of processor almost inversely proportional to the time taken.  Sequential part is negligibly small.
- Very highly scalable framework

## Robust design: A 3D example

- Heat sinks in microelectronic devices

- Thermal tiles in aerospace structures

- Most of the critical components in many applications/devices are usually fabricated from polycrystalline/functionally graded/heterogeneous materials

- The properties at the device-scale (thermal and electrical conductivity, elastic moduli and failure mechanisms) depend on the microstructure and material distribution at the meso-scale

- But it is not possible to completely characterize the microstructure distribution of a device.

- Have to design the operating conditions/parameters taking into account this limited information about the microstructure.

- Robust design in the presence of topological uncertainty

## Robust design: Problem definition

- Given _limited topological information_ about the microstructure, _design the optimal (stochastic)  heat flux_ to be applied on one end of the device such that a _required (stochastic) temperature_ is maintained at the other end.

- Design flux here

- Maintain temperature here

- Topology/material distribution unknown

- Statistical descriptors available

- Limited information in the form of a single experimental image of the microstructure. A Silver-Tungsten metal-metal composite

- Desired temperature profile:
-  Mean temperature is a linearly varying profile
-  Standard deviation of 0.05
-  Choose a normal distribution N(Tmean. Tstd)

- 2a

## Robust design: Problem definition

-  Maintain a specified thermal profile on the right wall
-  This thermal profile is given in terms of a pdf

_***Additional uncertainties:***_
-  Do not know the exact microstructure that the device is made up of.
-  Only know certain statistical correlations that the microstructure satisfies.
-  Consider the microstructure to be a random field .

_***Design variables:***_
-  Must design the pdf of the optimal heat flux

## Robust design: Problem definition

- One realization of the microstructure

- Dimension reduction methodology applied to the problem of representing the microstructure variability.
- Results in a 9 dimensional stochastic space.
- This space represents  the set of all microstructures satisfying the experimentally determined statistics (volume fraction and two point correlation)

_*Robust design to account for this uncertainty*_

- Physical domain: 20 μm x 20μm x 20μm
- Computational domain: 65x65x65
- Microstructure stochastic space: 1177 collocation points
- The design stochastic heat flux represented as a Bezier surface: number of parameters: 25
- The design heat flux is stochastic- belongs to a stochastic space
- The temperature depends on the uncertainty in the heat flux and the microstructural uncertainty

- Experimental statistics

## Robust design: Results

- Each iteration ~ 9 hours
- Solve 1177*65 ~ 76505 deterministic problems in each iteration
- Each deterministic problem has 65x65x65 = 275625 dof’s
- Total number of design variables = 65*5*5 ~ 1625
- Total dof/iteration= 21.25 billion

## Robust design: First/second order statistics

- Mean heat flux surface

- Bounds on the flux

- The mean value of the optimal heat flux
- The figure on the right shows the bounds on the heat flux. (Mean +/- one standard deviation)

## Robust design: PDF’s of heat flux and temperature

- Left, PDF of designed optimal heat flux at three points
- Right, PDF of resulting temperatures at three points

## Temperature statistics on the boundary

- Check to see if the designed optimal stochastic heat flux produces a temperature that satisfies the design requirements

- Left: The standard deviation of the temperature. The standard deviation is less then 0.05 in the complete region
- Right: The mean temperature on the right boundary. There is a decrease in the temperature along the z axis

## Convergence

- Convergence with increasing depth of interpolation

- Level 6

- Level 7

- Level 8

- Top: Mean heat flux surface

- Bottom: Bounds on the flux

## Convergence

- Convergence with  refining the Bezier representation

- Mean heat flux surface

-  Bounds on the flux

- 5x5

- 9x9

- Solve optimization problem with two different Bezier representations

- (5,5) representation:
- 3175  Design variables

- (9,9) representation:
- 10287  Design variables

- Less than 0.1% variation in solutions

## Conclusions

- Developed a scalable framework for estimation/design problems in the presence of uncertainty
- Framework provides an attractive alternative to Bayesian based methods. Does not require an assumption of the prior
- Framework is based on ideas from stochastic sparse grid representation of stochastic processes and deterministic optimization  principles
- Seamlessly allows the linking of ***any*** validated deterministic blackbox simulator. Significant ramifications in processes that are affected by multiple sources of uncertainty.
- Future prospects: Application to deformation process design, multiscale estimation problems

## Related publications

- B. Ganapathysubramanian and N. Zabaras, "*Sparse grid collocation methods for stochastic natural convection problems*", Journal of Computational Physics, Vol. 225, pp. 652-685, 2007
- B. Ganapathysubramanian and N. Zabaras, "*Modelling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multi-scale method*", Journal of Computational Physics, in press (preprint)
- B. Ganapathysubramanian and N. Zabaras, "*A non-linear dimension reduction methodology for generating data-driven stochastic input models*", Journal of Computational Physics, submitted (preprint).
- N. Zabaras, B. Ganapathysubramanian, "*A sparse grid collocation approach for the solution of stochastic inverse problems*", J. Computational Physics, submitted (preprint).