Radioactive waste disposal ◮ Assessing the safety of a potential repository is a major scientific undertaking. ◮ Groundwater pathway - route for radionuclides to get to surface environment. ◮ Long timescales (many thousands of years) make modelling essential. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 8/30
Radioactive waste disposal ◮ Assessing the safety of a potential repository is a major scientific undertaking. ◮ Groundwater pathway - route for radionuclides to get to surface environment. ◮ Long timescales (many thousands of years) make modelling essential. ◮ Uncertainty is a major issue. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 8/30
Radioactive waste disposal ◮ Assessing the safety of a potential repository is a major scientific undertaking. ◮ Groundwater pathway - route for radionuclides to get to surface environment. ◮ Long timescales (many thousands of years) make modelling essential. ◮ Uncertainty is a major issue. ◮ Objective is to make a good decision. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 8/30
Radioactive waste disposal ◮ Assessing the safety of a potential repository is a major scientific undertaking. ◮ Groundwater pathway - route for radionuclides to get to surface environment. ◮ Long timescales (many thousands of years) make modelling essential. ◮ Uncertainty is a major issue. ◮ Objective is to make a good decision. ◮ NB: Not necessarily interested in the worse possible case. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 8/30
Treatment of uncertainty As we know, there are known knowns. There are things we know we know. We also know there are known unknowns. That is to say we know there are some things we do not know. But there are also unknown unknowns, the ones we don’t know we don’t know. Donald Rumsfeld university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 9/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty ◮ eg. meteorite impact university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty ◮ eg. meteorite impact 4. Explicit quantification university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty ◮ eg. meteorite impact 4. Explicit quantification ◮ eg. travel times in the groundwater pathway university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty ◮ eg. meteorite impact 4. Explicit quantification ◮ eg. travel times in the groundwater pathway 5. Stylised approach university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Treatment of uncertainty Five-fold strategy for dealing with uncertainty (Bailey 2005): 1. No impact on safety ◮ eg. sorption coefficient for short lived radionuclides 2. Bounding uncertainty ◮ eg. solubility limit 3. Excluding uncertainty ◮ eg. meteorite impact 4. Explicit quantification ◮ eg. travel times in the groundwater pathway 5. Stylised approach ◮ eg. biosphere (ICRP and IAEA projects) university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 10/30
Case study - WIPP ◮ WIPP - Waste Isolation Pilot Plant. Figure: Cross section through the rock at the WIPP site university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 11/30
Case study - WIPP ◮ WIPP - Waste Isolation Pilot Plant. ◮ US DOE repository for radioactive waste situated in New Mexico. Figure: Cross section through the rock at the WIPP site university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 11/30
Case study - WIPP ◮ WIPP - Waste Isolation Pilot Plant. ◮ US DOE repository for radioactive waste situated in New Mexico. ◮ Located at a depth of 655m within bedded evaporites, primarily halite. Figure: Cross section through the rock at the WIPP site university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 11/30
Case study - WIPP ◮ WIPP - Waste Isolation Pilot Plant. ◮ US DOE repository for radioactive waste situated in New Mexico. ◮ Located at a depth of 655m within bedded evaporites, primarily halite. ◮ The most transmissive rock in the region is the Culebra dolomite. Figure: Cross section through the rock at the WIPP site university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 11/30
Case study - WIPP ◮ WIPP - Waste Isolation Pilot Plant. ◮ US DOE repository for radioactive waste situated in New Mexico. ◮ Located at a depth of 655m within bedded evaporites, primarily halite. ◮ The most transmissive rock in the region is the Culebra dolomite. ◮ Culebra would be the principal pathway for transport of Figure: Cross section through radionuclides away from the the rock at the WIPP site repository in the event of an accidental breach. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 11/30
Case study - WIPP ◮ Region containing data is measurement point 20km by 30km with the WIPP 30000 0 repository in the centre. 25000 20000 y (m) 15000 10000 5000 0 0 5000 10000 15000 20000 x (m) university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 12/30
Case study - WIPP ◮ Region containing data is measurement point 20km by 30km with the WIPP 30000 0 repository in the centre. 25000 ◮ Measurements of transmissivity, T , and 20000 freshwater head, h , are available at 39 locations. y (m) 15000 10000 5000 0 0 5000 10000 15000 20000 x (m) university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 12/30
Case study - WIPP ◮ Region containing data is measurement point 20km by 30km with the WIPP 30000 0 repository in the centre. 25000 ◮ Measurements of transmissivity, T , and 20000 freshwater head, h , are available at 39 locations. y (m) 15000 ◮ Measurements are irregularly 10000 spaced, most concentrated around the repository in centre 5000 of region. 0 0 5000 10000 15000 20000 x (m) university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 12/30
Case study - WIPP ◮ One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 13/30
Case study - WIPP ◮ One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. ◮ Radionuclides are released into the Culebra dolomite and then transported by groundwater. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 13/30
Case study - WIPP ◮ One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. ◮ Radionuclides are released into the Culebra dolomite and then transported by groundwater. ◮ Flow is two-dimensional to a good approximation. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 13/30
Case study - WIPP ◮ One scenario at WIPP is a release of radionuclides by means of a borehole drilled into the repository. ◮ Radionuclides are released into the Culebra dolomite and then transported by groundwater. ◮ Flow is two-dimensional to a good approximation. ◮ The time taken to travel from the repository to the boundary of the WIPP site is an important quantity. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 13/30
Case study - WIPP ◮ Find h ( x ) such that: university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 14/30
Case study - WIPP ◮ Find h ( x ) such that: ∇ T ∇ h = 0 , x ∈ D , h = h 0 , x ∈ ∂ D , T ( x i ) = T i , x i ∈ D , i = 1 , ..., 39 . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 14/30
Case study - WIPP ◮ Find h ( x ) such that: ∇ T ∇ h = 0 , x ∈ D , h = h 0 , x ∈ ∂ D , T ( x i ) = T i , x i ∈ D , i = 1 , ..., 39 . ◮ Then solve ζ = − T ( ζ ) ˙ e φ ∇ h ( ζ ) , ζ ( 0 ) = ζ 0 , and compute the time, t f , at which ζ ( t ) ∈ ∂ D W , where e is the thickness of the layer, φ the porosity and ∂ D W is the boundary of the WIPP site. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 14/30
Case study - WIPP ◮ Find h ( x ) such that: ∇ T ∇ h = 0 , x ∈ D , h = h 0 , x ∈ ∂ D , T ( x i ) = T i , x i ∈ D , i = 1 , ..., 39 . ◮ Then solve ζ = − T ( ζ ) ˙ e φ ∇ h ( ζ ) , ζ ( 0 ) = ζ 0 , and compute the time, t f , at which ζ ( t ) ∈ ∂ D W , where e is the thickness of the layer, φ the porosity and ∂ D W is the boundary of the WIPP site. ◮ Let s f = log ( t f ) . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 14/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . ◮ Limited information on the transmissivity leads to uncertainty. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . ◮ Limited information on the transmissivity leads to uncertainty. ◮ It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . ◮ Limited information on the transmissivity leads to uncertainty. ◮ It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. ◮ One approach to uncertainty quantification is to use probabilistic techniques. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . ◮ Limited information on the transmissivity leads to uncertainty. ◮ It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. ◮ One approach to uncertainty quantification is to use probabilistic techniques. ◮ Probabilistic solution - distribution function for s f , F S f ( s ) . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Case study - WIPP ◮ Problem on previous slide is ill-posed since we don’t know T for all x ∈ D . ◮ Limited information on the transmissivity leads to uncertainty. ◮ It is important to understand and quantify the effect of this uncertainty on the results - in particular on the travel time. ◮ One approach to uncertainty quantification is to use probabilistic techniques. ◮ Probabilistic solution - distribution function for s f , F S f ( s ) . ◮ Interpretation of probability is important - a subjective Bayesian approach is used here. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 15/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: ◮ Step 0 - collect data. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: ◮ Step 0 - collect data. ◮ Step 1 - build stochastic model. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: ◮ Step 0 - collect data. ◮ Step 1 - build stochastic model. ◮ Step 2 - solve the stochastic model. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: ◮ Step 0 - collect data. ◮ Step 1 - build stochastic model. ◮ Step 2 - solve the stochastic model. ◮ Step 3 - post process - calculate the quantities of interest - travel times. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Probabilistic Uncertainty Quantification ◮ Goal is to compute F S f ( s ) . The following steps are required: ◮ Step 0 - collect data. ◮ Step 1 - build stochastic model. ◮ Step 2 - solve the stochastic model. ◮ Step 3 - post process - calculate the quantities of interest - travel times. ◮ Step 4 - interpret the results. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 16/30
Stochastic model for WIPP ◮ Represent log transmissivity field, Z = log ( T ) , as a Gaussian random field with mean and covariance: E [ Z ( x )] = β 0 + β 1 x 1 + β 2 x 2 Cov [ Z ( x ) , Z ( x ′ )] = ω 2 C ( x , x ′ ) = ω 2 exp −� x − x ′ � /λ � � . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 17/30
Stochastic model for WIPP ◮ Represent log transmissivity field, Z = log ( T ) , as a Gaussian random field with mean and covariance: E [ Z ( x )] = β 0 + β 1 x 1 + β 2 x 2 Cov [ Z ( x ) , Z ( x ′ )] = ω 2 C ( x , x ′ ) = ω 2 exp −� x − x ′ � /λ � � . ◮ The joint probability distribution, f Θ ( θ ) , for the hyperparameters θ = ( β 0 , β 1 , β 2 , ω 2 , λ ) is estimated using a standard Bayesian analysis and MCMC to generate samples of θ drawn from f Θ ( θ ) . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 17/30
Stochastic model for WIPP 8e−05 6e−05 Density 4e−05 2e−05 0e+00 0 10000 20000 30000 40000 lambda Figure: Probability density function for λ from MCMC calculation. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 18/30
Stochastic model for WIPP ◮ The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 19/30
Stochastic model for WIPP ◮ The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). ◮ This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 19/30
Stochastic model for WIPP ◮ The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). ◮ This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). ◮ Note there is uncertainty in the hyperparameters as well as that inherent in the random field model for T . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 19/30
Stochastic model for WIPP ◮ The measurements of transmissivity are also used to condition the random field model (so that each realisation agrees with the measured values at the measurement points). ◮ This leads to a low-rank modification of the covariance operator (so that the variance is zero at the measurement points). ◮ Note there is uncertainty in the hyperparameters as well as that inherent in the random field model for T . ◮ This uncertainty must be taken into account in the analysis. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 19/30
Stochastic model for WIPP ◮ Gaussian RF Z ( x , ω ) can be written in the form ∞ � � Z ( x , ω ) = z ( x ) + λ i ψ i ( x ) ξ i ( ω ) . i = 1 where the ξ i are iid standard normal random variables. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 20/30
Stochastic model for WIPP ◮ Gaussian RF Z ( x , ω ) can be written in the form ∞ � � Z ( x , ω ) = z ( x ) + λ i ψ i ( x ) ξ i ( ω ) . i = 1 where the ξ i are iid standard normal random variables. ◮ We have E [ Z ] = z ( x ) , ∞ � � Cov [ Z ( x ) , Z ( y )] = λ i λ j ψ i ( x ) ψ j ( y ) . i = 1 [Karhunen, 1947], [Loève, 1948] university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 20/30
Stochastic model for WIPP ◮ Gaussian RF Z ( x , ω ) can be written in the form ∞ � � Z ( x , ω ) = z ( x ) + λ i ψ i ( x ) ξ i ( ω ) . i = 1 where the ξ i are iid standard normal random variables. ◮ We have E [ Z ] = z ( x ) , ∞ � � Cov [ Z ( x ) , Z ( y )] = λ i λ j ψ i ( x ) ψ j ( y ) . i = 1 [Karhunen, 1947], [Loève, 1948] ◮ Here Cov [ Z ( x ) , Z ( y )] is the covariance function of the conditioned field. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 20/30
Stochastic model for WIPP ◮ Solve ∇ · [ T ( x , ξ ) ∇ h ( x , ξ )] = 0 x ∈ D , a . s . h ( x , ξ ) = h 0 ( x ) x ∈ ∂ D , a . s . where ∞ � � log T ( x , ξ ) = z ( x ) + λ i ψ i ( x ) ξ i . i = 1 university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 21/30
Stochastic model for WIPP ◮ Solve ∇ · [ T ( x , ξ ) ∇ h ( x , ξ )] = 0 x ∈ D , a . s . h ( x , ξ ) = h 0 ( x ) x ∈ ∂ D , a . s . where ∞ � � log T ( x , ξ ) = z ( x ) + λ i ψ i ( x ) ξ i . i = 1 ◮ Compute the log of travel time, s f , from h and T . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 21/30
Stochastic model for WIPP ◮ Solve ∇ · [ T ( x , ξ ) ∇ h ( x , ξ )] = 0 x ∈ D , a . s . h ( x , ξ ) = h 0 ( x ) x ∈ ∂ D , a . s . where ∞ � � log T ( x , ξ ) = z ( x ) + λ i ψ i ( x ) ξ i . i = 1 ◮ Compute the log of travel time, s f , from h and T . ◮ Thus s f depends on ξ and θ , and the problem is to compute the stochastic law of s f from the joint law of ξ and θ . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 21/30
Solution of stochastic model ◮ Taking account of uncertainty in hyperparameters: university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 22/30
Solution of stochastic model ◮ Taking account of uncertainty in hyperparameters: ◮ Now � F S f ( s ) = f Ξ , Θ ( ξ, θ ) d ξ d θ, s f ( ξ,θ ) ≤ s university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 22/30
Solution of stochastic model ◮ Taking account of uncertainty in hyperparameters: ◮ Now � F S f ( s ) = f Ξ , Θ ( ξ, θ ) d ξ d θ, s f ( ξ,θ ) ≤ s � �� � = f Ξ | Θ ( ξ | θ ) d ξ f Θ ( θ ) d θ, s f ( ξ,θ ) ≤ s university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 22/30
Solution of stochastic model ◮ Taking account of uncertainty in hyperparameters: ◮ Now � F S f ( s ) = f Ξ , Θ ( ξ, θ ) d ξ d θ, s f ( ξ,θ ) ≤ s � �� � = f Ξ | Θ ( ξ | θ ) d ξ f Θ ( θ ) d θ, s f ( ξ,θ ) ≤ s � = F S f | Θ ( s | θ ) f Θ ( θ ) d θ. where � F S f | Θ ( s | θ ) = f Ξ | Θ ( ξ | θ ) d ξ. s f ( ξ,θ ) ≤ s university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 22/30
Solution of stochastic model ◮ F S f | Θ ( s | θ ) is obtained from the solution of the stochastic PDE for each value of θ . university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 23/30
Solution of stochastic model ◮ F S f | Θ ( s | θ ) is obtained from the solution of the stochastic PDE for each value of θ . ◮ We do not have an explicit expression for f Θ ( θ ) , just a (large) sample, { θ i } N θ i = 1 , of equally probable values of θ generated by the MCMC calculation. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 23/30
Solution of stochastic model ◮ F S f | Θ ( s | θ ) is obtained from the solution of the stochastic PDE for each value of θ . ◮ We do not have an explicit expression for f Θ ( θ ) , just a (large) sample, { θ i } N θ i = 1 , of equally probable values of θ generated by the MCMC calculation. ◮ Use Monte-Carlo to compute F S f ( s ) as N θ F S f ( s ) = 1 � F S f | Θ ( s | θ i ) N θ i = 1 where { θ i } N θ i = 1 is the MCMC sample. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 23/30
Solution of the stochastic PDE ◮ Probably the most expensive part of the calculation university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 24/30
Solution of the stochastic PDE ◮ Probably the most expensive part of the calculation ◮ Currently a very active area of research university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 24/30
Solution of the stochastic PDE ◮ Probably the most expensive part of the calculation ◮ Currently a very active area of research ◮ Curse of dimension is the main issue (see later slide) university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 24/30
Solution of the stochastic PDE ◮ Probably the most expensive part of the calculation ◮ Currently a very active area of research ◮ Curse of dimension is the main issue (see later slide) ◮ Options include: ◮ Brute force Monte-Carlo ◮ Quasi Monte-Carlo methods ◮ Generalised polynomial chaos methods ◮ Multi-level Monte-Carlo methods university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 24/30
Solution of the stochastic PDE ◮ Probably the most expensive part of the calculation ◮ Currently a very active area of research ◮ Curse of dimension is the main issue (see later slide) ◮ Options include: ◮ Brute force Monte-Carlo ◮ Quasi Monte-Carlo methods ◮ Generalised polynomial chaos methods ◮ Multi-level Monte-Carlo methods ◮ Many challenges remain in this area, with several promising lines of research. university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 24/30
Solution of the stochastic PDE ◮ Truncated KL expansion N � � Z N ( x , ω ) = z ( x ) + λ i ψ i ( x ) ξ i ( ω ) . i = 1 university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 25/30
Solution of the stochastic PDE ◮ Truncated KL expansion N � � Z N ( x , ω ) = z ( x ) + λ i ψ i ( x ) ξ i ( ω ) . i = 1 ◮ Solve ∇ · [ T ( x , ξ ) ∇ h ( x , ξ )] = 0 x ∈ D , a . s . h ( x , ξ ) = h 0 ( x ) x ∈ ∂ D , a . s . where N � � log T ( x , ξ ) = z ( x ) + λ i ψ i ( x ) ξ i . i = 1 university-logo Andrew Cliffe What Should We Do with Radioactive Waste? 25/30
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