W HAT IS GOING TO HAPPEN TO THE POLAR ICE CAPS ? They affect - PowerPoint PPT Presentation

S CALABLE S OLVERS FOR F ORWARD AND I NVERSE P ROBLEMS IN G EOPHYSICS Toby Isaac tisaac@uchicago.edu University of Chicago January 11, 2016 CAAM Department Rice University T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 1 / 58

M OTIVATION 1 L ARGE - SCALE B AYESIAN I NFERENCE FOR P ARAMETER F IELDS 2 S CALABLE S OLVERS FOR N ONLINEAR S TOKES E QUATIONS 3 T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 2 / 58

Motivation Bayesian Inference Stokes Solvers References M Y I NTERESTS 1 Adaptive mesh refinement data structures and algorithms for large-scale scientific computing (slides on my website) 2 Scientific software development: p4est (including hybrid extension p6est ), PETSc (including stand-alone plugins like DofColumns ), Deal.II , mangll , . . . 2016 Scientific Software Days, February 25-26, Austin! scisoftdays.org 3 Uncertainty quantification, Bayesian inversion, PDE-constrained optimization 4 Scalable, efficient solvers for implicit PDEs (3) and (4) today. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 3 / 58

Motivation Bayesian Inference Stokes Solvers References W HAT IS GOING TO HAPPEN TO THE POLAR ICE CAPS ? They affect temperature, sea level (70 m equivalent), ocean currents, etc. Quantities of Interest (QoIs) : q visibleearth.nasa.gov T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 4 / 58

Motivation Bayesian Inference Stokes Solvers References W HAT IS GOING TO HAPPEN TO THE POLAR ICE CAPS ? They affect temperature, sea level (70 m equivalent), ocean currents, etc. Quantities of Interest (QoIs) : q earthobservatory.nasa.gov T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 4 / 58

Motivation Bayesian Inference Stokes Solvers References W HAT IS GOING TO HAPPEN TO THE POLAR ICE CAPS ? They affect temperature, sea level (70 m equivalent), ocean currents, etc. Quantities of Interest (QoIs) : q earthobservatory.nasa.gov T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 4 / 58

Motivation Bayesian Inference Stokes Solvers References W HAT IS GOING TO HAPPEN TO THE POLAR ICE CAPS ? They affect temperature, sea level (70 m equivalent), ocean currents, etc. Quantities of Interest (QoIs) : q earthobservatory.nasa.gov T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 4 / 58

Motivation Bayesian Inference Stokes Solvers References S CIENTISTS GATHER LOTS OF DATA Observations/Data : d obs Ice cores / boreholes, Ice-penetrating radar (+stratigraphy), GRACE gravity field measurements, Interferometric synthetic aperture radar (InSAR) / lidar, etc. (NASA Earth Observatory) T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 5 / 58

Motivation Bayesian Inference Stokes Solvers References S CIENTISTS GATHER LOTS OF DATA Observations/Data : d obs Ice cores / boreholes, Ice-penetrating radar (+stratigraphy), GRACE gravity field measurements, Interferometric synthetic aperture radar (InSAR) / lidar, etc. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 5 / 58

Motivation Bayesian Inference Stokes Solvers References S CIENTISTS GATHER LOTS OF DATA Observations/Data : d obs Ice cores / boreholes, Ice-penetrating radar (+stratigraphy), GRACE gravity field measurements, Interferometric synthetic aperture radar (InSAR) / (UT Center for Space Research) lidar, etc. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 5 / 58

Motivation Bayesian Inference Stokes Solvers References S CIENTISTS GATHER LOTS OF DATA Observations/Data : d obs Ice cores / boreholes, Ice-penetrating radar (+stratigraphy), GRACE gravity field measurements, Interferometric synthetic aperture radar (InSAR) / lidar, etc. (NASA/JPL-Caltech) T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 5 / 58

Motivation Bayesian Inference Stokes Solvers References H OW CAN WE USE DATA TO GET Q O I S ? Pure data analysis Pure deterministic inversion Use statistical tools and known Use unified models of processes values of q to understand/project explaining d obs and q (united by the relationship between d obs and parameter set m ) to invert for m , predicted values of q : then project: E [ q | d obs ]; F obs : m → d ; Var( q | d obs ); F QoI : m → q ; q := F QoI ( “ F − 1 etc. obs ( d obs ) ” ) . Model-based Uncertainty Quantification Use models to better inform statistical relationships between d obs and q : E [ q = F QoI ( m ) | d obs ]; Var[ . . . ]; etc. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 6 / 58

Motivation Bayesian Inference Stokes Solvers References PDE-C ONSTRAINED F OBS The observations d obs belong to a vector space R N d . The observations are made on the state w ∈ X via b : X → R N d w is implicitly defined by the forward (PDE) model A parameterized by m : A ( w ; m ) = 0 . F obs : m �→ b ( w ) s.t. A ( w ; m ) = 0 . T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 7 / 58

Motivation Bayesian Inference Stokes Solvers References PDE-C ONSTRAINED F OBS : ICE SHEET PROBLEM [Rignot et al., 2011a] [Creyts and Schoof, 2009] Our d obs is the MEaSUREs surface velocity dataset for Antarctica [Rignot et al., 2011b]. Our model parameter m is log β , where β is the sliding coefficient between ice and substrate (units stress velocity − 1 ): phenomenological; highly spatially variable. Our state variables w are the velocity u and pressure p in the ice sheet. Our model A is a nonlinear Stokes boundary value problem, where m is a coefficient field for a Robin-type boundary condition. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 8 / 58

Motivation Bayesian Inference Stokes Solvers References I CE SHEET DYNAMICS ( INSTANTANEOUS ) Balance of linear momentum and mass 2 ( ∇ u + ∇ u T )] [ D ( u ) = 1 − ∇ · [2 µ ( T, u ) D ( u ) − I p ] = ρ g , ∇ · u = 0 , Constitutive relations 1 − n µ ( T, u ) = 1 [ D ( u ) II = 1 2 tr( D ( u ) 2 )] 2 A ( T )( D ( u ) II + ǫ ) 2 n Boundary conditions σ n | Γ top = 0 , u · n | Γ base = 0 , T � ( σ n + β ( m ) u ) | Γ base = 0 T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 9 / 58

Motivation Bayesian Inference Stokes Solvers References I CE SHEET DYNAMICS ( INSTANTANEOUS ) Balance of linear momentum and mass 2 ( ∇ u + ∇ u T )] [ D ( u ) = 1 − ∇ · [2 µ ( T, u ) D ( u ) − I p ] = ρ g , ∇ · u = 0 , Constitutive relations shear thinning with second invariant 1 − n µ ( T, u ) = 1 [ D ( u ) II = 1 2 tr( D ( u ) 2 )] 2 A ( T )( D ( u ) II + ǫ ) 2 n Boundary conditions σ n | Γ top = 0 , u · n | Γ base = 0 , T � ( σ n + β ( m ) u ) | Γ base = 0 T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 9 / 58

Motivation Bayesian Inference Stokes Solvers References I CE SHEET DYNAMICS ( INSTANTANEOUS ) Balance of linear momentum and mass 2 ( ∇ u + ∇ u T )] [ D ( u ) = 1 − ∇ · [2 µ ( T, u ) D ( u ) − I p ] = ρ g , ∇ · u = 0 , Constitutive relations 1 − n µ ( T, u ) = 1 [ D ( u ) II = 1 2 tr( D ( u ) 2 )] 2 A ( T )( D ( u ) II + ǫ ) 2 n Boundary conditions stress-free surface σ n | Γ top = 0 , u · n | Γ base = 0 , T � ( σ n + β ( m ) u ) | Γ base = 0 T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 9 / 58

Motivation Bayesian Inference Stokes Solvers References I CE SHEET DYNAMICS ( INSTANTANEOUS ) Balance of linear momentum and mass 2 ( ∇ u + ∇ u T )] [ D ( u ) = 1 − ∇ · [2 µ ( T, u ) D ( u ) − I p ] = ρ g , ∇ · u = 0 , Constitutive relations 1 − n µ ( T, u ) = 1 [ D ( u ) II = 1 2 tr( D ( u ) 2 )] 2 A ( T )( D ( u ) II + ǫ ) 2 n Boundary conditions basal sliding σ n | Γ top = 0 , u · n | Γ base = 0 , T � ( σ n + β ( m ) u ) | Γ base = 0 T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 9 / 58

Motivation Bayesian Inference Stokes Solvers References G ENERAL B AYESIAN I NVERSION F RAMEWORK Assume for the moment a finite-dimensional parameter space M h . The likelihood of d obs given m is π like ( d obs | F obs ( m )) . Bayes’ law: the posterior probability of the parameter given the observations is π prior ( m ) π like ( d obs | F obs ( m )) π post ( m | d obs ) = M h π prior ( m ) π like ( d obs | F obs ( m )) d m . � T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 10 / 58

Motivation Bayesian Inference Stokes Solvers References G ENERAL B AYESIAN I NVERSION F RAMEWORK Thus, e.g., � M h F QoI ( m ) π prior ( m ) π like ( d obs | F obs ( m )) d m E [ F QoI ( m ) | d obs ] = . � M h π prior ( m ) π like ( d obs | F obs ( m )) d m Expectations require integration over M h (high-dimensional) . . . . . . w.r.t. π post ( m | d obs ) (implicitly-defined, cannot be sampled directly). General solution: Metropolis-Hasting. Constructs a Markov chain of samples { m i } such that � N i =1 F QoI ( m i ) lim = E [ F QoI ( m ) | d obs ] a.s. N N →∞ Good performance requires drawing samples from a proposal distribution similar to π post ( m ) to generate a sample chain. T. Isaac (U. Chicago) Scalable Forward & Inverse Solvers January 11, 2016 11 / 58