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 Modelslocal 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(µ
N )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
µ1P
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 nNh = dim(Vh).
Test case: forward problem
=Ah fhuh =un fnAn
Finite Elements method Reduced Basis method
Nh ! n
uh
ϕi
φi
un
un(x;µ) =
nX
i=1
un
iφi(x)uh(x;µ) =
NhX
i=1
uh
iϕi(x)
Linear steady case:
Stefano Pagani
Reduced basis method in a nutshell
µ
µN
µ1
µ2
µ1P
µ2
Vh
Mh
uh(µ1)
uh(µ2)
uRB 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 fnAn
Vn = spanφ1, . . . ,φ
n
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 ,
of 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
ζT1
...
ζTN
.
I ROM: projection of the full-order arrays on the reduced subspace Vn through an
orthogonal projection.
Test case: forward problem
Stefano Pagani
=Ah fhuh
=un fnAn
Finite Elements method Reduced Basis method
Nh ! n
uh
ϕi
φi
un
un(x;µ) =
nX
i=1
un
iφi(x)uh(x;µ) =
NhX
i=1
uh
iϕi(x)
=
=
An
fn
Ah
fh
VVT
VT
V = [φ1, . . . , φn]
Stefano Pagani
Galerkin projection
∈ ∈
uh(µ1), . . . , uh(µ
N )
I To deal efficiently with the nonlinear terms at the reduced order level we employ
the discrete empirical interpolation method (DEIM)
N(un;µ)≈ VTU(PT
U)−1
| z
n×mD
N(PTVun;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) . . .] ∈ R
Nh×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, 2017Stefano Pagani
I Warning: for advection dominant or traveling front problem it is difficult to
ensure that nNh.
µ
µN
µ1
µ2
µ1P
µ2
Vh
Mh
uh(µ1)
uh(µ2)
φ1
n
A numerical example
Stefano Pagani
Local Reduced basis method
µd
=un fnAn
uh(µd)
V i
n= spanφi
1, . . . ,φi
ni i = 1, . . . , Nc
φ1
1
φ1
2
φ2
n
φ2
2
φ2
1
Pi
n
Tested clustering techniquesfor offline snapshots subdivision:• time based• parameter based• state based: -projection error based
-k-means
S. Pagani, A. Manzoni, A. Quarteroni. “Numerical approximation of 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 that8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
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 on ∂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 Sku , k = 1, . . . ,Nc ;
2. SI (matrix of nonlinear term snapshots) is partitioned into Nc submatrices SkI ,
k = 1, . . . ,Nc ;
3. the localized basis functions are constructed through the POD technique applied
to each Sku 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, . . . ,S
k
u = argmin
Su
NcX
k=1
X
uh∈Sku
kuh − ck
hk
Nc
Stefano Pagani
K-means clustering
c1
hc2
hc3
hc4
h c5
h
c6
hc7
h c8
hc9
hc10
h
2
2
Stefano Pagani
Results
Localized reduced-order model based on k-means clustering
Finite elements DOFs: 31764Number of Elements: 140271
POD/DEIM ROM speedup: 4.6x
Parameters μ: - epicardial and endocardial angles
-longitudinal and traversal conductivities
max # basis functions: 192min # basis functions: 11
Localized ROM speedup 25.2x
log10(|uh − un|)un
local ROM reduction errorFOM
uh
