Reduced-order models for uncertainty quantification and parameter - PowerPoint PPT Presentation

Reduced-order models for uncertainty quantification and parameter estimation in cardiac models ∆ Stefano Pagani ∆ ∫ ∆ Alfio Quarteroni Andrea Manzoni MOX-Dipartimento di Matematica Politecnico di Milano (Italy) ∫ MATH-CMCS Modelling and Scientific Computing EPFL (Switzerland) ∆

Integrating data within mathematical models potential recordings imaging 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 Clinical -0.6 -0.6 -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 data -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] Pre-processing Forward model Parameter selection Personalization Prediction Pipeline segmentation electrophysiology sensitivity analysis parameter estimation evaluation of +meshing electromechanics forward UQ backward UQ new scenarios Challenging issues: I computational complexity of full-order models (e.g. finite element method); I noisy clinical data; I uncertainties related to geometry, (partially known) physical coefficients, boundary/ initial conditions. Stefano Pagani

Integrating data within mathematical models potential recordings imaging 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 Clinical -0.6 -0.6 -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 data -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 0 100 200 300 400 0 100 200 300 400 t [ms] t [ms] Pre-processing Forward model Parameter selection Personalization Prediction Pipeline segmentation electrophysiology sensitivity analysis parameter estimation evaluation of +meshing electromechanics forward UQ backward UQ new scenarios Many-query problems: I parameter selection for reducing the uncertainty space dimension (sensitivity analysis); I uncertainty propagation on outputs of clinical interest (forward UQ); I parameter estimation for model personalization (backward UQ). Stefano Pagani

Methods Variance-based sensitivity analysis LOCAL Reduced Order Models for parameter selection local approximation of both nonlinear term and solution RB-MCMC sampling procedure SURROGATE MODELs kriging and GP-based ROM error surrogate (ROMES) Reduced basis Ensemble Kalman filter ROMES for time-dependent outputs for sequential state/parameter estimation Forward model Parameter selection Personalization electrophysiology sensitivity analysis parameter estimation electromechanics forward UQ backward UQ - S. Pagani, A. Manzoni, A. Quarteroni . “Numerical approximation of - S. Pagani . “Reduced-order models for inverse problems and uncertainty parametrized problems in cardiac electrophysiology by a local reduced basis quantification in cardiac electrophysiology”. PhD Thesis (2017) . method”. In preparation , 2017 . - S. Pagani, A. Manzoni and A. Quarteroni . “Efficient state/parameter - D. Bonomi . “Reduced order models for the parametrized cardiac estimation in nonlinear unsteady PDEs by a reduced basis ensemble Kaman filter”. electromechanical problem”. PhD Thesis (2017) . SIAM/ASA Journal on Uncertainty Quantification , 5(1): 890–921, 2017 . - M. Drohmann and K. Carlberg . “The ROMES method for statistical - A. Manzoni, S. Pagani and T. Lassila . “Accurate solution of Bayesian modeling of reduced-order-model error”. SIAM/ASA Journal on Uncertainty inverse uncertainty quantification problems using model and error reduction Quantification , 3(1):116–145, 2015. methods”. SIAM/ASA Journal on Uncertainty Quantification , 4(1):380–412, 2016. Stefano Pagani

Reduced basis method in a nutshell Goal : compute efficiently the solution of a problem when a set of parameters vary V h M h u h ( µ 1 ) µ 1 µ N µ 2 µ 2 µ d u h ( µ N ) u h ( µ d ) µ P µ 1 u h ( µ 2 ) • parameter-dependent PDEs 
 (e.g. cardiac electrophysiology, 
 nonlinear mechanics, 
 coupled electro-mechanics,…) • (un)steady (non)linear PDEs • physical/geometrical parameters { u h ( µ 1 ) , . . . , u h ( µ N ) } u h ( µ n ) ✓ material coefficients ✓ electrical conductivities ✓ initial/boundary data ✓ geometrical configuration … Stefano Pagani

Reduced basis method in a nutshell Test case: forward problem I Idea: Galerkin approximation on a low dimensional subspace V n ⇢ V h (reduced basis space) of dimension n ⌧ N h = dim( V h ). Linear steady case: Finite Elements method Reduced Basis method u h u n ϕ i φ i N h n X X u n u h u n ( x ; µ ) = i φ i ( x ) u h ( x ; µ ) = i ϕ i ( x ) N h ≫ n i =1 i =1 A h f h A n f n u h u n = = Stefano Pagani

Reduced basis method in a nutshell A numerical example RB Approximation 
 { u n ( µ ) ) : µ ∈ P} (new parameter value) V h M h u h ( µ 1 ) φ 2 µ 1 µ N µ 2 µ 2 µ d u h ( µ d ) µ P µ 1 φ 1 u h ( µ 2 ) φ n • parameter-dependent PDEs 
 (e.g. cardiac electrophysiology, 
 nonlinear mechanics, 
 coupled electro-mechanics,…) • (un)steady (non)linear PDEs • physical/geometrical parameters V n = span { φ 1 , . . . , φ n } ✓ material coefficients ✓ electrical conductivities P n ✓ initial/boundary data ✓ geometrical configuration … A n f n u n = Stefano Pagani

Galerkin projection Test case: forward problem I Construction of the subspace: proper orthogonal decomposition (POD) on the { u h ( µ 1 ) , . . . , u h ( µ N ) } set of high-fidelity snapshots { u ( ` ) h ( µ ) } and { w ( ` ) h ( µ ) } . I ROM: projection of the full-order arrays on the reduced subspace V n through an orthogonal projection. Given a training set P train ⊂ P of N train parameter vectors, we compute the so-called Finite Elements method Reduced Basis method V = [ φ 1 , . . . , φ n ] snapshots matrix by solving the full-order system for each µ ∈ P train : u h u n S u = [ u h ( t (0) ; µ 1 ) , u h ( t (1) ; µ 1 ) ,..., u h ( t (0) ; µ 2 ) , u h ( t (1) ; µ 2 ) ,..., ] ∈ R N h × N s , ϕ i φ i of dimensions N h × N s , with N s = N train N t . A n V T A h V = The POD technique selects as basis functions { φ i } of the reduced-space the first n left N h n singular vectors of the snapshots matrix S u . X X u n u h u n ( x ; µ ) = i φ i ( x ) u h ( x ; µ ) = i ϕ i ( x ) N h ≫ n i =1 i =1 ζ T     σ 1   1 . ...     . S u =  φ 1 ... φ n φ N  . ...      .      f n V T f h = ζ T σ N N A n f n u n = A h f h u h = Stefano Pagani Stefano Pagani

Discrete empirical interpolation method Test case: forward problem I To deal efficiently with the nonlinear terms at the reduced order level we employ the discrete empirical interpolation method (DEIM) N ( u n ; µ ) ≈ V T U ( P T U ) − 1 N ( P T Vu n ; µ ) . | {z } | {z } n × m D m D × 1 Procedure : I compute the snapshots matrix of the nonlinear term N : S N = [ N ( u (1) h ; µ 1 ) , N ( u (2) h ; µ 1 ) ,..., N ( u (1) h ; µ 2 ) , N ( u (2) h ; µ 2 ) ... ] ∈ R N h × N s ; I compute the matrix of basis functions U = [ φ 1 ,..., φ m D ] by applying the POD technique on S N ; I select m D degrees of freedom { i 1 ,..., i m D } and construct the index matrix P = [ e i 1 ,..., e i m D ] ( e i ) j = δ ij . Reduced-mesh : we need to assemble the nonlinear operator on the elements related to the degrees of freedom { i 1 ,..., i m D } selected by the DEIM algorithm. S. Chaturantabut and D. C. Sorensen . “Nonlinear model reduction via discrete empirical interpolation”. SIAM J. Sci. Comp. , 32(5):2737–2764, 2010 Stefano Pagani March 1, 2017

Local Reduced basis method A numerical example I Warning: for advection dominant or traveling front problem it is difficult to ensure that n ⌧ N h . V h M h φ 2 2 u h ( µ 1 ) φ 1 µ 1 µ N µ 2 µ 2 µ d 2 u h ( µ d ) φ 1 φ 2 µ n P µ 1 n u h ( µ 2 ) φ 1 1 φ 2 1 Tested clustering techniques for offline snapshots subdivision: • time based • parameter based • state based: -projection error based n = span { φ i 1 , . . . , φ i V i n i } i = 1 , . . . , N c -k-means P i n S. Pagani, A. Manzoni, A. Quarteroni . “Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced A n f n basis method”. In preparation (2017) . u n = Stefano Pagani