Scalable Hierarchical Sampling of Gaussian Random Fields for Large-Scale Multilevel Monte Carlo Simulations Quantification of Uncertainty: Improving Efficiency and Technology Sarah Osborn 1 Panayot Vassilevski 1 LLNL-PRES-734783 Umberto Villa 2 1 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory 2 Institute for Computational Engineering and Sciences (ICES), University of Texas at Austin July 19, 2017 LLNL-PRES-XXXXXX This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC
Forward Propagation Uncertainty Quantification § When modeling some physical phenomena, inputs are often subject to uncertainty. § Example : Uncertainty in Groundwater Flow with Darcy's Law — Permeability coefficient k is subject to uncertainty. — Model k as a spatially correlated log-normal random field. — 𝑙 𝑦, 𝜕 = exp 𝜄 𝑦, 𝜕 where 𝜄 is a Gaussian random field with known mean and covariance. § Goal : Given prior assumptions about uncertainty in input data, quantify uncertainty in the solution for large-scale simulations using Monte Carlo sampling methods. 2 LLNL-PRES-xxxxxx
Key Computational Challenges for Large-Scale Monte Carlo Sampling Methods Multilevel Monte Carlo Many samples are necessary with a fine • Use specialized element-based agglomeration technique to spatial discretization. construct hierarchy. SPDE Sampling Technique Scalable generation of •Solve a stochastic PDE (SPDE) with random input coefficient mixed finite element method. realizations •Requires solution of saddle point problem with random right hand side. Specialized preconditioners Efficient solution of •Discretization leads to matrices with saddle point structure. forward problem •Employ methods from element-based multigrid. 3 LLNL-PRES-xxxxxx
Key Computational Challenges for Large-Scale Monte Carlo Sampling Methods Multilevel Monte Carlo Many samples are necessary with a fine • Use specialized element-based agglomeration technique to spatial discretization. construct hierarchy. SPDE Sampling Technique Scalable generation of •Solve a stochastic PDE (SPDE) with random input coefficient mixed finite element method. realizations •Requires solution of saddle point problem with random right hand side. Specialized preconditioners Efficient solution of •Discretization leads to matrices with saddle point structure. forward problem •Employ methods from element-based multigrid. 4 LLNL-PRES-xxxxxx
Monte Carlo Method § Goal : Estimate 𝔽[𝑅] , the expected value of a quantity of interest 𝑅(𝒀(𝑦, 𝜕) ) where 𝒀(𝑦, 𝜕) is the solution of a PDE with random field coefficient. N X = 1 E [ Q ] ≈ b Q MC Q h ( ω i ) h N i =0 where 𝑅 2 (𝜕 3 ) is the 𝑗 -th sample of 𝑅 approximated with spatial discretization ℎ. Mean Square Error (MSE) of method: 1 + ( E [ Q − Q h ]) 2 N V [ Q h ] � �� � � �� � Bias: Discre�za�on Error Es�mator Variance 5 LLNL-PRES-xxxxxx
Multilevel Monte Carlo Method § This variance reduction technique uses a sequence of spatial approximations 𝑅 ℓ , ℓ = 𝑀, … , 1 which approximate 𝑅 ; = 𝑅 2 with increasing accuracy (and cost). § Linearity of expectation implies E [ Q ] ≈ E [ Q h ] = E [ Q L ] + P L − 1 ` =0 E [ Q ` − Q ` +1 ] . The multilevel MC estimator is h i P N L P L − 1 P N ` b Q MLMC 1 1 = i =0 Q L ( ω i ) + i =0 ( Q ` ( ω i ) − Q ` +1 ( ω i )) . h i =0 N L N ` M. Giles . Oper. Res. (2008) K. Cliffe , M. Giles , R. Scheichl , and A. Teckentrup . Comput. Vis. Sci., (2011) 6 LLNL-PRES-xxxxxx
Multilevel Acceleration of Monte Carlo Method § The MSE of the MLMC Estimator is L − 1 1 1 � + ( E [ Q − Q h ]) 2 V [ Q L ] + V [ Q � − Q � +1 ] N L N � � �� � � =1 � �� � Discre�za�on error � �� � Fixed cost independent of h V [ Q � − Q � +1 ] � V [ Q � ] For a desired tolerance, the number of samples on each level is chosen to minimize the total computational cost. 7 LLNL-PRES-xxxxxx
Generation of Hierarchy of Spatial Discretizations with Element-Based Multigrid (AMGe) Recall pros/cons of multigrid (MG) methods: § Geometric Multigrid (GMG) § Algebraic Multigrid (AMG) — Scalable for many regular/semi- — Optimal and effective solver for many structured grid problem PDEs on arbitrary grids — Requires a nested hierarchy of grids — Requires only the fine-grid matrix; no — Uses information from discretization spatial mesh needed — Infeasible to implement for arbitrary — Closer to a black-box method unstructured-grid problems 8 LLNL-PRES-xxxxxx
Element-based Multigrid (AMGe) AMGe methods aim to leverage the advantages of the two approaches and to mitigate their shortcomings. GMG with nonstandard elements • (agglomerates of fine-grid ones) and operator-dependent coarse finite element spaces. By using some “extra” information, • AMGe can handle effectively a broader class of problems than classical AMG. Hierarchy of agglomerated meshes Coarse spaces have guaranteed • approximation properties. 9 LLNL-PRES-xxxxxx
Coarsening de Rham Complexes on Agglomerated Elements The de Rham complex plays an important role in analysis and discretization of PDEs. r⇥ r r · H 1 L 2 H (curl) H (div) r⇥ r r · S h Q h R h W h Π S Π Q Π R Π W r⇥ r r · S H Q H R H W H § Generate a coarse sequence such that — The sequence is exact. — The commutativity property is preserved. — The spaces are conforming. — The approximation properties of the original spaces are preserved . J. Pasciak , P. Vassilevski . SISC. (2008) I. Lashuk , P. Vassilevski . CMAM. (2011) 10 LLNL-PRES-xxxxxx
One hierarchy, many uses….. The hierarchy of de Rham sequences with operator-dependent coarse spaces with approximation properties can be used for — Robust multilevel preconditioners — Discretization on a hierarchy of levels • Numerical upscaling • Multilevel Monte Carlo simulations • Scalable Generation of Gaussian Random Fields A parallel distributed memory C++ library for an AMGe framework to coarsen a wide class of PDEs on general unstructured meshes developed at LLNL. https://github.com/LLNL/parelag 11 LLNL-PRES-xxxxxx
Key Computational Challenges for Large-Scale Monte Carlo Sampling Methods Multilevel Monte Carlo Many samples are necessary with a fine • Use specialized element-based agglomeration technique to spatial discretization. construct hierarchy. SPDE Sampling Technique Scalable generation of •Solve a stochastic PDE (SPDE) with random input coefficient mixed finite element method. realizations •Requires solution of saddle point problem with random right hand side. Specialized preconditioners Efficient solution of •Discretization leads to matrices with saddle point structure. forward problem •Employ methods from element-based multigrid. 12 LLNL-PRES-xxxxxx
Scalable Sampling of a Gaussian Random Field Challenge : How to generate realizations of a Gaussian field (GF) on a hierarchy of spatial discretizations?? We consider a stationary isotrophic field with Mat é rn covariance function σ 2 cov ( x, y ) = 2 ν − 1 Γ ( ν )( κ ∥ y − x ∥ ) ν K ν ( κ ∥ y − x ∥ ) where x, y ∈ R d . Karhunen-Lo 𝐟̀ ve Expansion : — Dense eigenvalue computation Bottleneck — State of the art methods exploit Fast Multipole Methods and randomized eigensolvers to alleviate this issue. 13 LLNL-PRES-xxxxxx
Scalable Sampling of a Gaussian Random Field Gaussian Markov random field representation: GFs with Mat é rn covariance functions are solutions of the stochastic PDE ( κ 2 − ∆ ) α / 2 θ ( x, ω ) = g W ( x, ω ) , x ∈ R d , α = ν + d 2 𝒳 𝑦, 𝜕 : spatial Gaussian white noise with unit variance • g: scaling factor to impose unit marginal variance • 𝜆 ∈ ℝ: inversely proportional to correlation length • Special case: In 3D, realizations of a Gaussian random field with exponential covariance function are solutions of the stochastic reaction diffusion problem . F. Lindgren , H. Rue, J. Lindstrom. J R Stat Soc Series B Stat Methodol . (2011) 14 LLNL-PRES-xxxxxx
Stochastic PDE (SPDE) Sampler Let 𝜉 = 1 ( 2D ) or 𝜉 = E F (3D) , then the realizations of the GF solve ( κ 2 − ∆ ) θ ( x, ω ) = g W ( x, ω ) , x ∈ R d Using the mixed finite element method, let piecewise constant func�ons Θ h ⊂ L 2 ( D ) lowest-order Raviart-Thomas elements R h ⊂ H (div , D ) Find ( u h , θ h ) ∈ ( R h , Θ h ) such that � ( u h , v h ) + ( θ h , div v h ) = 0 ∀ v h ∈ R h (div u h , q h ) − κ 2 ( θ h , q h ) = g ( W ( ω ) , q h ) ∀ q h ∈ Θ h with boundary condi�ons u h · n = 0 . 15 LLNL-PRES-xxxxxx
Stochastic PDE (SPDE) Sampler R D i W ( ω ) ∼ N (0 , | D i | ) we obtain Noting that " # 0 B T � u h � M h h ξ ∼ N (0 , I ) = , 1 − κ 2 W h B h θ h − gW h ξ 2 where • M h is the mass matrix for the space R h • W h is the (diagonal) mass matrix for space Θ h • B h stems from the divergence constraint. Able to leverage existing scalable solvers and preconditioners! 16 LLNL-PRES-xxxxxx
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