simulation du système cardiovasculaire et modélisation réduite
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Jean-Frédéric GerbeauProject-team REO
INRIA Paris-Rocquencourt & Laboratoire J-L. LionsFrance
Simulation du système cardiovasculaire
et modélisation réduite6 février 2013
Université de Rouen
2
Cardiovascular system
Aneurisms
Atherosclerosis
Biomedical devices
Electrophysiology(Arythmia, drug effect,...)
Vascular diseases
Outline
• Computational Hemodynamics- Motivations- Inverse problems in fluid-structure
• Electrocardiograms (ECGs)• Reduced order modeling - Motivations- The Approximated Lax Pair method
3
Outline
• Computational Hemodynamics- Motivations- Inverse problems in fluid-structure
• Electrocardiograms (ECGs)• Reduced order modeling - Motivations- The Approximated Lax Pair method
4
Computational hemodynamics:Hierarchy of models
• 3D Model (Navier-Stokes + Nonlinear elasticity)
Variables : velocity, pressure, displacement
• 1D Model (Euler equation, 1775)
Variables : flow rate, mean pressure, section area
• 0D Model (Ordinary Differential Equations)
Variables : flow rate, pressure drop
Γw0
Γ
ΩS S
w
1 2
z
A(z,t)Q(z,t)
Rp
Rd
C
(compartment modeling or boundary conditions)
Three-dimensional modelsBefore 2000 : • Pioneer works: Navier-Stokes equations, simple boundary
conditions, rarely on real anatomies2000 - 2010 :• Patient-specific anatomies: homemade solutions; free (VMTK) or
commercial software (Amira, Mimics,...)• Fluid-solid interaction: tremendous progress, now present in
commercial software (not very efficient though!)• Boundary conditions: many workarounds... still an open topic!• Main applications: stents, aneurysms, ...
Today (and tomorrow) :• Diagnosis: assimilate medical data in simulations• Prognosis & planning: long term evolution, adaption, remodeling,...• Find problems where all these works can be really useful!
6
Progress in FSI
7
1999 2007
2003
Computational time2001 2003 2007 2012100 20 4 1
Only due to progress in algorithms !
1st example: surgical planning
8
Glenn-Fontan surgery• congenital heart disease• multi-step complex
procedure
Glenn FontanNumerical simulations• Patient-specific geometries• Forecast pressure drop, flow split, wall-shear stress
Troianowski, Taylor, Feinstein, Vignon-ClementelStanford/INRIA
http://www.americanheart.org
I. Vignon-Clementel, A. Birolleau, G. Troianowski (INRIA & Stanford)
Geometry from 5 patients MRI data
10
Y-graft in general improves:• Energy efficiency• Flow distribution• SVC pressure under rest & exercise
Troianowski, Taylor, Feinstein, Vignon-Clementel
Challenge:
• About 20 “outlets”• 3 parameters per
outlets....
Rp
Rd
C
2d example: avoid clinical exams ?
11
Source: O. Peruta
• Example: aortic coarctation • After surgical repair, patients must
be followed on a regular basis• Exercise test is often necessary to
assess the patient condition• Question: With computer simulations, can we extrapolate
the rest test to avoid the stress test ?• Maybe... if we are able to evaluate the artery wall stiffness
Collaboration with R. Hose, I. Valverde, P. Beerbaum (euHeart project)
Aortic coarctation
12
Data: I. Valverde, P. Beerbaum (KCL). Segmentation and processing: R. Hose et al. (Sheffield), C. Bertoglio (INRIA)
(euHeart project)
Medical data assimilation
13
Estimate parameters- artery wall stiffness- boundary condition- ...
Measurements- medical imaging (CT, MRI, ...)- blood flow (US, PC-MRI,...)- pressure (catheter)- ...
Access to hidden quantities
- pressure- wall stress
Models- Navier-Stokes equations- Solid mechanics- ...
Improve measures- regularization- interpolation- ...
Data assimilation
14
• Partial observations of X: Z = H(X)
• Uncertainties on the initial condition X0 and the parameters ✓
• FSI dynamical system:⇢
BX = A(X, �) +RX(0) = X0
• Time discretization: Xn+1 = Fn+1(Xn, �)
• State variable: X=[u, p,df ,d,v]
• Parameters: ✓ = [Young modulus, viscosity, boundary conditions, . . . ]
Data assimilation
15
⇢X0 = X0 + ⇣X✓ = ✓ + ⇣✓
• Variational approach: - Optimization algorithms- Usually based on gradient (adjoint equations)
• Filtering approach:- Sequential correction of the state and the parameters
• Uncertainties: ⇣ = [⇣X , ⇣✓]
• Minimize
J(�) =1
2
Z T
0⇥Z �H(X)⇥2W dt+
1
2⇥�⇥2P
where X = X(�) is solution to the problem.
16Simulation : C.Bertoglio
Rp
Rd
• Parameter estimation: Young modulus Ei = 2�i in 5 regions
• Observations: wall displacement on fluid-solid interface
Example: Compliance estimationParameter estimation
17
0 0.2 0.4 0.6 0.8−0.5
0
0.5
1
1.5
2
Time [s]
Dis
plac
emen
t [m
m]
Region 1Region 2Region 3Region 4Region 5Noise
Region SNR
1 0.43
2 1.25
3 8.8
4 1.8
5 0.75
• Gaussian noise (10%) and time resampling �tobs
= 10�t
• Synthetic data
• Two tuning coeffecients:
– A priori covariance on parameters: ↵I– Gain: �
Example: Compliance estimationParameter estimation
18
0 0.2 0.4 0.6 0.80
0.5
1
1.5
2
2.5
3
3.5
Time [s]
Youn
gs m
odul
us [
MPa
]
� = 9,⇥ = 1
Example: Compliance estimationParameter estimation
Simulation : C.Bertoglio
19
0 0.2 0.4 0.6 0.8−4
−2
0
2
4
Time [s]
θ
α = 4 − β = 100
0 0.2 0.4 0.6 0.8−4
−2
0
2
4
Time [s]
θ
α = 9 − β = 20
• We can plot �n ±p
diag(P �n)
• The filtering algorithm provides an a posteriori covariance P �n
Example: Compliance estimationParameter estimation
Simulation : C.Bertoglio
Experimental validation
0 0.1 0.2 0.3 0.4 0.5 0.6−1.5
−1
−0.5
0
0.5
1
Time [s]
log 2(E
)
FSI + data assmilationMechanical test
• Silicon rubber aortic phantom
Data: Gaddum, Rutten, Beerbaum (KCL). Segmentation and processing: Hose, Barber (Sheffield)
Simulation: Bertoglio, JFG (INRIA)(euHeart project)
• Many works about fluid boundary conditions... but not that many about wall !
C. Taylor et al. , StanfordAbdominal aorta
Blood flow in aortaExternal tissue support
• Typical b.c. on the external part of the vessel :
�sn = p0n
• A simple and affordable way to model the external tissues :
�sn = �ksd� cs�d
�t
Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)
DataContours without supportContour with support
Legend:Solid
Fluid Pressure with supportPressure without supportFlow with supportFlow without support
DiastoleSystole
spinespine
spinesp
ine
spinesp
ine
π
π
π
1
2
3
π1
π2
π3
60
80
100
120
140
0.0 0.3 0.90.6
60
80
100
120
140
0.0 0.3 0.90.60
10
20
30
40
0
40
80
120
160
60
80
100
120
140
0.0 0.3 0.90.60
90
180
270
360
π0
π0
(mmHg)
(cc/s)
(mmHg)
(cc/s)
Example 3: External tissue supportModeling
Moireau, Xiao, Astorino, Figueroa, Chapelle, Taylor, JFG, (Biomech Model Mech 2011)
Outline
• Computational Hemodynamics- Motivations- Inverse problems in fluid-structure
• Electrocardiograms (ECGs)• Reduced order modeling - Motivations- The Approximated Lax Pair method
23
e.g.: Pullan et al. 05, Sundes et al. 06,...
8<
:
div(�TruT) = 0+ transmission condition
on the epicardium
Torso:Poisson equation
Electrocardiograms
24
ui
ue
APD�⌅⇤
⌅⇥
CmdVm
dt+ Iion(Vm, g) = 0
dg
dt+ G(Vm, g) = 0
Cell scale: Hodgkin-Huxley-like models
8>>>><
>>>>:
Am
✓C
m
�Vm
�t+ I
ion
(Vm
, g)
◆� div(�
i
rui
) = Am
Iapp
,
div(�e
rue
) + div(�i
rui
) = 0
�g
�t+G(V
m
, g) = 0,
Myocardium: bidomain models
12-lead ECG
25
Boulakia, Cazeau, Fernández, JFG, Zemzemi, Annals Biomed Engng 2010
• Simulated ECG:
• Real ECG:
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
-2
-1
0
1
2
0 200 400 600 800
I
-2
-1
0
1
2
0 200 400 600 800
II
-2
-1
0
1
2
0 200 400 600 800
III
-2
-1
0
1
2
0 200 400 600 800
aVR
-2
-1
0
1
2
0 200 400 600 800
aVL
-2
-1
0
1
2
0 200 400 600 800
aVF
-2
-1
0
1
2
0 200 400 600 800
V1
-2
-1
0
1
2
0 200 400 600 800
V2
-2
-1
0
1
2
0 200 400 600 800
V3
-2
-1
0
1
2
0 200 400 600 800
V4
-2
-1
0
1
2
0 200 400 600 800
V5
-2
-1
0
1
2
0 200 400 600 800
V6
V5
V6
V1
V2
V3
V4
D1
aVR
aVL
aVFD2
D3
(from www.wikipedia.org)
Bundle branch blocks
Healthy case Right bundle branch block
Chapelle, Fernández, JFG, Moireau, Sainte-Marie, Zemzemi, FIMH 2009
26
Outline
• Computational Hemodynamics- Motivations- Inverse problems in fluid-structure
• Electrocardiograms (ECGs)• Reduced order modeling
- Motivations- The Approximated Lax Pair method
27
Motivation: Blood flows
• Output of interest Ex: Pressure drop
28
• Rapid prototypingBoundary conditions (Windkessel), constitutive law coefficients,...
• Moderate variabilityExamples: pulmonary arteries
Astorino, JFG, Pantz, Traoré, CMAME 2009
• OptimizationEx: arterial by-pass optimization (Quarteroni-Rozza, Sankaran-Marsden)
Aortic coarctation
Motivation: electrophysiology
29
• OptimizationEx: optimize multi-sites pacing
Forward
Inverse
• Inverse problemsElectrocardiology
Potential in heart and the torso Electrocardiogram(ECG)
Reduced Order Modeling
Many options:• Simplify the geometry and/or the physics
Examples:- 1D model for blood flows (Formaggia, Peiró, ...)
- Eikonal model in electrophysiology (Franzone, Sermesant, Frangi,...)
• Keep the equations and the geometry, but reduce the approximation spaceExamples:- Reduced Basis Method (Patera, Maday, Quarteroni, Rozza,...)
- Proper Orthogonal Decomposition (POD), or Karhunen-Loève expansion (Iollo, Farhat, Karniadakis, Kunisch, Gunzburger, Volkwein,...)
30
POD in a nutshell
31
• Let ('i)i=1..n be a finite element basis
• Singular Value Decomposition: S = ⇥�⇤T, with � = diag(�1, . . . ,�p)
• Reduced Order Model (ROM): search for Uh =PN
j=1 Uj�j such that:d
dt(Uh,�i) + a(Uh,�i) = (f,�i), �i = 1..N
• (�1, . . . ,�N ): N columns of � corresponding to the N largest �i, N � n
• Let S be the matrix whose columns are the Si, i = 1..p.
• Full Order Model (FOM): search for uh =Pn
j=1 uj⇥j such that:d
dt(uh,⇥i) + a(�;uh,⇥i) = (f,⇥i), �i = 1..n
• Compute p “snapshots” (i.e. solutions at p time instants, or parameters �):
S1(u11, . . . , u
1n), . . . , S
p(up1, . . . , u
pn)
32
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
u
t=0exact POD(10)
0 5 10 15 20 25 30 35 40 45 5010−18
10−16
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
n
eigenvalues
POD with 10 modes
Simulation : D. Lombardi
Example 1:
@u
@t
� @
2u
@x
2= 0
Eigenvalues
33
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u
t=0 t=1
0 5 10 15 20 25 30 35 40 45 5010−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
n
eigenvalues
Simulation : D. Lombardi
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
u
exactPOD(10)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
u
exactPOD(50)
POD with 10 modes POD with 50 modes
Example 2:
@u
@t
+ c
@u
@x
= 0
Eigenvalues
Basic idea
34
Uh =NX
j=1
Uj�j(x, t)
Uh =NX
j=1
Uj(t)�j(x)
• Instead of
• Try to work with something like :
• Question : how to propagate the modes ?
Step 1: Semi Classical Signal Analysis (SCSA)
• Let u(x) be a non-negative signal and the “scattering” operator
35
L�(u)� = ���� �u�
• And approximate u(x) by the Deift-Trubowitz formula
u(x) = �
�1
N�X
m=1
m�
2m
• Solve the eigenvalue problem
L�(u)� = ��
where �m are the negative eigenvalueswith m =p��m
Laleg, Crépeau, Sorine (2007 & 2012)
• Choose � > 0 to achieve a given accuracy
• Remark: can be exact for “reflectionless” potentials
36
Reconstruction of aortic pressureFrom: Laleg, Médigue, Papelier, Crépeau, et al. (2010)
Lax pair in a nutshell
37
• If an operator M(t) is such that
• If (�k(t),�k(t)) is an eigenpair of L(t) then
- @t�k = 0
- @t�k(t) = M(t)�k(t)
@tL+ LM�ML = 0
• Let L(t) be a self-adjoint operator in a Hilbert space
Lax pair in a nutshell
• Example: Korteweg de Vries equation:
38
@t
u+ 6u@x
u+ @3x
u = 0
• Lax pair:
L(u)v = �@2x
v � uv
M(u)v = 4@3x
v + 6u@x
v + 3v@x
u
• Example: 1-soliton: if
then:
with: �1(x) = sech(x)
u0(x) =
21
2�
21(1x)
u(x, t) =
21
2�
21(1(x�
21t))
Goal: • Consider a general nonlinear evolution PDE :
39
⇢@tu = F (u)u(0) = u0
• For example: F (u) = �u+ u(1� u)
• Difficulty: the Lax pair are not known in general
• Use the SCSA and the Lax pair to define a reduced order model
40
@t�m = ��hF (u) m, mi
• The evolution of �m is governed by
• The evolution of m satifies, for p 2 {1, . . . , NM},
h@t m, pi = Mmp(u)
(Mmp(u) =
�
�p � �mhF (u) m, pi, if p 6= m and �p 6= �m,
Mmp(u) = 0, if p = m or �p = �m.with
⇢@tu = F (u)u(0) = u0
• For m 2 {1, . . . , NM}, let �m(t) be an eigenvalue of L�(u(x, t))and m(x, t) an associated eigenfunction, normalized in L
2(⌦)
• Assume there exists an operator M(u) such that
@t m = M(u) m.
Proposition: let u be solution to Step 2: Approximation of Lax Pair (ALP)
Step 3: Approximated “inverse scattering”
41
• (�m,�m)m=1..N� : eigenfunctions/values of L�(u)
• We look for an approximation of u:
u =
N�X
k=1
↵k�2k
• Inserting this expression in
• We findN�X
k=1
↵kh�2k,�
2mi = � 1
�(�m + h��m,�mi) .
which gives the ↵k.
hL�(u)�m,�pi = �mh�m,�pi
42
Summary: the ALP algorithmInitialisation (SCSA of u0): compute NM eigenmodes m.
The N� first are denoted �m and
u0(x) ⇡ �
�1
N�X
m=0
m�m(x)
1. Compute the matrix M(u
n) approximating the operator M(u(t
n))
2. Compute the eigenvalues �
n+1m : �
n+1m = �
nm � �thF (u
n)
nm,
nmi
3. Compute the eigenfunctions
n+1m :
n+1m =
PNM
p=1
hI + �tM(u
n) +
�t2
2 M(u
n)
2i
mp
np
4. Solve for ↵
n+1p :
PN�p=1 ↵
n+1p h�2p,�2mi = � 1
�
��
n+1p + h��n+1
m ,�
n+1m i
�.
5. Compute u
n+1with u
n+1=
PN�p=1 ↵
n+1p
��
n+1p
�2
Numerical test cases• Korteweg-de Vries:
★ one-soliton (analytically known)★ only one negative eigenvalue
43
−15 −10 −5 0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x
u
t=0 t=T
−15 −10 −5 0 5 10 15−0.5
0
0.5
1
1.5
2
x
u
exactROM
−15 −10 −5 0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x
u
exactROM
10 modes 25 modes
Simulation : D. Lombardi
44
• Korteweg-de Vries: ★ three-soliton★ three negative eigenvalues
−15 −10 −5 0 5 10 150
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x
u
t=0
−15 −10 −5 0 5 10 150
0.5
1
1.5
2
2.5
3
3.5
4
x
u
t=T
Simulation : D. Lombardi
• Comparison of:★ projection of the analytic solution on 20 POD
modes built of snapshots obtained from the half first time history
★ ALP integration with 20 modes
−15 −10 −5 0 5 10 15−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x
u
exactSTMPOD
• Fisher-Kolmogorov-Petrovski-Piskunov:
45
@u
@t��u = u(1� u)
46
• P1 Finite-element:★ 9000 dof ★ Time step 2.5e-4
• Challenges:★ Sharp front, splitting in two directions★ No precomputed solutions : POD cannot work
Simulation : D. Lombardi
47
• ALP★ 6 modes for the solution, 25 modes for the operator★ Time step 2.5e-3 (10 times larger than FEM)
Simulation : D. Lombardi
48
55 main arteries
Γw0
Γ
ΩS S
w
1 2
z
A(z,t)Q(z,t)
velocity and pressure
flow rate and mean pressure(D. Lombardi)
In progress....
Perspectives
• Many things to do★ Adaptation of the number of modes★ Different operators for the scattering step★ Role of positive eigenvalues★ Error estimator★ Analysis★ ...
• Applications (in progress)★ systemic network of arteries (Damiano Lombardi)★ cardiac electrophysiology (Elisa Schenone)
49
Inria Research Report 8137 available on HALhttp://hal.inria.fr/hal-00752810
• Remark:
50
Z
⌦(@t m)2 ⇡
NMX
n,l=1
Mml(u)Mmn(u)h n, li =NMX
n=1
Mmn(u)2
• Thus, the Frobenius norm of M(u) can be used as an estimator of the approximation of the dynamics
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