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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)

∫ ∆ ∆ ∫ ∆ ∆

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.

Clinical data

segmentation +meshing sensitivity analysis forward UQ

Pipeline

potential recordings Forward model parameter estimation backward UQ evaluation of new scenarios Parameter selection Personalization Prediction imaging Pre-processing

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Integrating data within mathematical models

Stefano Pagani

electrophysiology electromechanics

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).

Clinical data

segmentation +meshing sensitivity analysis forward UQ

Pipeline

potential recordings Forward model electrophysiology electromechanics parameter estimation backward UQ evaluation of new scenarios Parameter selection Personalization Prediction imaging Pre-processing

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Integrating data within mathematical models

Stefano Pagani

sensitivity analysis forward UQ Forward model parameter estimation backward UQ Parameter selection Personalization

Methods

Stefano Pagani

SURROGATE MODELs LOCAL Reduced Order Models

local approximation of both nonlinear term and solution kriging and GP-based ROM error surrogate (ROMES)

Variance-based sensitivity analysis for parameter selection RB-MCMC sampling procedure Reduced basis Ensemble Kalman filter for sequential state/parameter estimation

ROMES for time-dependent outputs

electrophysiology electromechanics

• S. Pagani, A. Manzoni, A. Quarteroni. “Numerical approximation of

parametrized problems in cardiac electrophysiology by a local reduced basis method”. In preparation, 2017.

• D. Bonomi. “Reduced order models for the parametrized cardiac

electromechanical problem”. PhD Thesis (2017).

• M. Drohmann and K. Carlberg. “The ROMES method for statistical

modeling of reduced-order-model error”. SIAM/ASA Journal on Uncertainty Quantification, 3(1):116–145, 2015.

• S. Pagani. “Reduced-order models for inverse problems and uncertainty

quantification in cardiac electrophysiology”. PhD Thesis (2017).

• S. Pagani, A. Manzoni and A. Quarteroni. “Efficient state/parameter

estimation in nonlinear unsteady PDEs by a reduced basis ensemble Kaman filter”. SIAM/ASA Journal on Uncertainty Quantification, 5(1): 890–921, 2017.

• A. Manzoni, S. Pagani and T. Lassila. “Accurate solution of Bayesian

inverse uncertainty quantification problems using model and error reduction methods”. SIAM/ASA Journal on Uncertainty Quantification, 4(1):380–412, 2016.

Vh Mh

uh(µ1)

uh(µN)

uh(µ2)

{uh(µ1), . . . , uh(µN)}

uh(µn)

Goal: compute efficiently the solution of a problem when a set of parameters vary

Reduced basis method in a nutshell

Stefano Pagani

µ

µN

µ1 µ2

µ1

P µ2 µd

• parameter-dependent PDEs


(e.g. cardiac electrophysiology,
 nonlinear mechanics, 
 coupled electro-mechanics,…)

• (un)steady (non)linear PDEs
• physical/geometrical parameters

✓ material coefficients ✓ electrical conductivities ✓ initial/boundary data ✓ geometrical configuration …

uh(µd)

I Idea: Galerkin approximation on a low dimensional subspace Vn ⇢ Vh (reduced

basis space) of dimension n ⌧ Nh = dim(Vh).

Test case: forward problem

= Ah fh uh = un fn An

Finite Elements method Reduced Basis method

Nh ≫ n

uh ϕi φi un un(x; µ) =

n

X

i=1

un

i φi(x)

uh(x; µ) =

Nh

X

i=1

uh

i ϕi(x)

Linear steady case:

Stefano Pagani

Reduced basis method in a nutshell

µ

µN

µ1 µ2

µ1

P µ2

Vh Mh

uh(µ1)

uh(µ2)

{

RB Approximation 
 (new parameter value)

un(µ) ) : µ ∈ P}

A numerical example

Stefano Pagani

Reduced basis method in a nutshell

Pn

• parameter-dependent PDEs


(e.g. cardiac electrophysiology,
 nonlinear mechanics, 
 coupled electro-mechanics,…)

• (un)steady (non)linear PDEs
• physical/geometrical parameters

✓ material coefficients ✓ electrical conductivities ✓ initial/boundary data ✓ geometrical configuration …

φ1 φ2 φn

µd = un fn An

Vn = span{φ1, . . . , φn} uh(µd)

I Construction of the subspace: proper orthogonal decomposition (POD) on the

set of high-fidelity snapshots {u(`)

h (µ)} and {w(`) h (µ)}.

Given a training set Ptrain ⊂ P of Ntrain parameter vectors, we compute the so-called snapshots matrix by solving the full-order system for each µ ∈ Ptrain: Su = [uh(t(0);µ1),uh(t(1);µ1),...,uh(t(0);µ2),uh(t(1);µ2),...,] ∈ RNh×Ns ,

• f dimensions Nh ×Ns, with Ns = NtrainNt.

The POD technique selects as basis functions {φi} of the reduced-space the first n left singular vectors of the snapshots matrix Su. Su =   φ1 ...φn ... φN        σ1 ... σN         ζ T

1

. . . ζ T

N

   .

I ROM: projection of the full-order arrays on the reduced subspace Vn through an

• rthogonal projection.

Test case: forward problem

Stefano Pagani

= Ah fh uh = un fn An

Finite Elements method Reduced Basis method

Nh ≫ n

uh ϕi φi un un(x; µ) =

n

X

i=1

un

i φi(x)

uh(x; µ) =

Nh

X

i=1

uh

i ϕi(x)

= = An fn Ah fh V VT VT V = [φ1, . . . , φn]

Stefano Pagani

Galerkin projection

{uh(µ1), . . . , uh(µN)}

I To deal efficiently with the nonlinear terms at the reduced order level we employ

the discrete empirical interpolation method (DEIM) N(un;µ) ≈ VT U(PT U)−1 | {z }

n×mD

N(PT Vun;µ) | {z }

mD×1

. Procedure:

I compute the snapshots matrix of the nonlinear term N:

SN = [N(u(1)

h ;µ1),N(u(2) h ;µ1),...,N(u(1) h ;µ2),N(u(2) h ;µ2)...] ∈ RNh×Ns ; I compute the matrix of basis functions U = [φ1,...,φmD] by applying the POD

technique on SN;

I select mD degrees of freedom {i1,...,imD} and construct the index matrix

P = [ei1,...,eimD ] (ei)j = δij. Reduced-mesh: we need to assemble the nonlinear operator on the elements related to the degrees of freedom {i1,...,imD} selected by the DEIM algorithm.

Test case: forward problem

• S. Chaturantabut and D. C. Sorensen. “Nonlinear model reduction via discrete empirical

interpolation”. SIAM J. Sci. Comp., 32(5):2737–2764, 2010

Discrete empirical interpolation method

March 1, 2017 Stefano Pagani

I Warning: for advection dominant or traveling front problem it is difficult to

ensure that n ⌧ Nh. µ

µN

µ1 µ2

µ1

P µ2

Vh Mh

uh(µ1)

uh(µ2)

φ1

n

A numerical example

Stefano Pagani

Local Reduced basis method

µd = un fn An

uh(µd)

V i

n = span{φi 1, . . . , φi ni} i = 1, . . . , Nc

φ1

1

φ1

2

φ2

n

φ2

2

φ2

1

P i

n

Tested clustering techniques for offline snapshots subdivision:

• time based
• parameter based
• state based: -projection error based
• k-means
• S. Pagani, A. Manzoni, A. Quarteroni. “Numerical approximation
• f parametrized problems in cardiac electrophysiology by a local reduced

basis method”. In preparation (2017).

We consider the Monodomain equation (tissue ΩH level) coupled with the Aliev-Panfilov model (cell level): find u(x,t) and w(x,t) such that 8 > > > > > > > < > > > > > > > : Am ✓ Cm ∂u ∂t +Ku(u −a)(u −1)+wu ◆ −div(D(x)∇u) = AmIapp(t) in ΩH,t ∈ (0,T) ∂w ∂t = ✓ ε0 + c1w c2 +u ◆ (−w −Ku(u −a−1)) in ΩH,t ∈ (0,T) ∇u(t)·n = ∇w(t)·n = 0

• n ∂ΩH,t ∈ (0,T)

u(0) = u0 w(0) = w0 in ΩH,t = 0, Under the assumption that the left-ventricle tissue is an axisymmetric anisotropic media the conductivity tensor is given by: D(x) = σt I+(σl −σt) f0(x)⊗f0(x), where f0(x) is a vector parallel to the fiber direction at any point x ∈ ΩH. Finally f0 is rotated with from an angle θepi on the epicardium to an angle θendo on the endocardium with the following relationship: θ = (θepi −θendo) r −r1 r2 −r1 +θendo.

Stefano Pagani

Test case

Stefano Pagani

Influence of the parameters on the solution

Fig: activation times on varying the epicardial and the endocardial angle of the fibers Fig: activation times on varying the longitudinal and the traversal conductivity

Ingredient: a criterium for the subdivision of the snapshots matrix. Su = [...,u(30)

h

(µ1),...,u(100)

h

(µ1),...,u(180)

h

(µ1),...,u(10)

h

(µ2),...,u(150)

h

(µ2),...]. Algorithm:

• 1. Su is partitioned into Nc submatrices Sk

u, k = 1,...,Nc;

• 2. SI (matrix of nonlinear term snapshots) is partitioned into Nc submatrices Sk

I ,

k = 1,...,Nc;

• 3. the localized basis functions are constructed through the POD technique applied

to each Sk

u and Sk I , k = 1,...,Nc;

• 4. the reduced arrays forming the reduced system are computed by means of the

Galerkin projection.

Stefano Pagani

Local ROM method

{S1

u, . . . , Sk u} = arg min Su Nc

X

k=1

X

uh∈Sk

u

kuh − ck

hk

Nc

Stefano Pagani

K-means clustering

c1

h

c2

h

c3

h

c4

h

c5

h

c6

h

c7

h

c8

h

c9

h

c10

h

2

2

Stefano Pagani

Results

Localized reduced-order model based on k-means clustering Finite elements DOFs: 31764 Number of Elements: 140271 POD/DEIM ROM speedup: 4.6x

Parameters μ: - epicardial and endocardial angles

• longitudinal and traversal conductivities

max # basis functions: 192 min # basis functions: 11 Localized ROM speedup 25.2x

log10(|uh − un|) un local ROM reduction error FOM uh