Parallel preconditioners for problems arising from whole-microwave - PowerPoint PPT Presentation

Parallel preconditioners for problems arising from whole-microwave system modeling for brain imaging Pierre-Henri Tournier 1 Pierre Jolivet 2 Marcella Bonazzoli 3 Victorita Dolean 3 , 4 eric Hecht 1 eric Nataf 1 Fr´ ed´ Fr´ ed´ 1 LJLL, Universit´ e Pierre et Marie Curie, INRIA ´ equipe ALPINES, Paris 2 IRIT-CNRS, Toulouse 3 LJAD, Universit´ e de Nice Sophia Antipolis, Nice 4 Dept. of Mathematics and Statistics, University of Strathclyde, Glasgow, UK Numerical methods for wave propagation and applications August 31, 2017 Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 1/ 26

Motivation 2 types of cerebro vascular accidents (strokes): ischemic (85 %) hemorrhagic (15 %) The correct treatment depends on the type of stroke: ⇒ restore blood flow ⇒ lower blood pressure = = Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 2/ 26

Motivation In order to differentiate between ischemic and hemorrhagic stroke, CT scan or MRI is typically used. Microwave tomography is a novel and promising imaging technique, especially for medical and brain imaging. CT scan MRI microwave tomography resolution excellent excellent good fast ✗ ✗ ✓ mobile ∼ ✗ ✓ cost ∼ 300 000 e ∼ 1 000 000 e < 100 000 e safe ✗ ✓ ✓ monitoring ✗ ✗ ✓ Diagnosing a stroke at the earliest possible stage is crucial for all following therapeutic decisions. Monitoring: Clinicians wish to have an image every fifteen minutes. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 3/ 26

Motivation EMTensor GmbH, Vienna, Austria. First-generation prototype: cylindrical chamber composed of 5 rings of 32 antennas (ceramic-loaded waveguides). Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 4/ 26

The direct problem We consider in Ω a linear, isotropic, non-magnetic, dispersive, dissipative dielectric material. The direct problem consists in finding the electromagnetic field distribution in the whole chamber, given a known material and transmitted signal. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 5/ 26

The direct problem For each of the 5 × 32 antennas, the as- sociated electric field E j is the solution of Maxwell’s equations: � �  ω 2 ε + i ωσ ∇ × ( ∇ × E j ) − µ 0 E j = 0 in Ω ,      E j × n = 0 on Γ metal ,  (1) ( ∇ × E j ) × n + i β ( E j × n ) × n = g j on Γ j ,      ( ∇ × E j ) × n + i β ( E j × n ) × n = 0 on Γ i , i � = j ,  where µ 0 is the permeability of free space, ω is the incident angular frequency , β is the wavenumber of the waveguide, ε > 0 is the dielectric permittivity, σ > 0 is the conductivity and g j corresponds to the excitation. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 6/ 26

The direct problem Spatial discretization using N´ ed´ elec edge finite elements yields a large sparse linear system Au = f j for each transmitting antenna j . We need a robust and efficient solver for second order time-harmonic Maxwell’s equations with heterogeneous coefficients. = ⇒ Use domain decomposition methods to produce parallel preconditioners for the GMRES algorithm. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 7/ 26

Overlapping domain decomposition methods Consider the linear system: Au = f ∈ C n . Ω Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 8/ 26

Overlapping domain decomposition methods Consider the linear system: Au = f ∈ C n . Given a decomposition of � 1; n � , ( N 1 , N 2 ), define: ◮ the restriction operator R i from � 1; n � into N i , ◮ R T as the extension by 0 from N i into � 1; n � . i Then solve concurrently: u m +1 1 + A − 1 u m +1 2 + A − 1 = u m 11 R 1 ( f − Au m ) = u m 22 R 2 ( f − Au m ) 1 2 where u i = R i u and A ij := R i AR T j . Ω 1 [Schwarz 1870] Ω 2 Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 8/ 26

Overlapping domain decomposition methods Duplicated unknowns coupled via a partition of unity : 1 N 1 1 � R T I = 1 i D i R i . 2 2 i =1 To solve Au = f Schwarz methods can be viewed as preconditioners for a fixed point algorithm: u n +1 = u n + M − 1 ( f − Au n ) . N ◮ M − 1 � R T i D i A − 1 R i with A i = R i AR T RAS := i i i =1 N ◮ M − 1 i D i B − 1 � R T ORAS := R i Optimized transmission conditions i i =1 [B. Despr´ es 1991] for Helmholtz Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 9/ 26

HPDDM HPDDM is an efficient parallel implementation of domain decomposition methods by Pierre Jolivet and Fr´ ed´ eric Nataf ◮ header-only library written in C++11 with MPI and OpenMP ◮ interfaced with the open source finite element software FreeFem++ (Fr´ ed´ eric Hecht) Time to solution (in seconds) Setup Solve Ideal (54) 500 (61) 200 (73) (94) 50 10 512 1 , 024 2 , 048 4 , 096 # of subdomains Strong scalability test for Maxwell 3D with edge ele- ments of degree 2 - 119M d.o.f. - Curie (TGCC, CEA) Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 10/ 26

Comparison with experiments The experimental measurements obtained from the antennas are the reflection and transmission coefficients. Their numerical counterparts are computed as Γ i E j · E 0 � i d γ S ij = i | 2 d γ , for i , j = 1 , ..., 160 , Γ i | E 0 � where E 0 i is the fundamental mode of the waveguide i . ring 3 - empty ring 3 - empty 70 200 simulation simulation experiment experiment 60 150 50 100 40 50 magnitude (dB) phase (degree) 30 0 20 -50 10 -100 0 -150 -10 -200 -200 -150 -100 -50 0 50 100 150 200 -200 -150 -100 -50 0 50 100 150 200 angle (degree) angle (degree) Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 11/ 26

The inverse problem The inverse problem consists in recovering ε and σ such that for each transmitting antenna j , the solution E j to the associated Maxwell’s problem matches the measurements: Γ i E j · E 0 � i d γ i | 2 d γ = S mes for each receiving antenna i . Γ i | E 0 ij � Difficulties: ◮ inverse problems are ill-posed ◮ noise in the experimental data ◮ solving the inverse problem means solving the direct problem multiple times = ⇒ time-consuming Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 12/ 26

The inverse problem Let κ := µ 0 ( ω 2 ε + i ωσ ) be the unknown parameter of our inverse problem. Solving the inverse problem corresponds to minimizing the following cost functional: 160 160 J ( κ ) =1 2 � � � � � S ij ( κ ) − S mes � � ij 2 � j =1 i =1 2 160 160 � Γ i E j ( κ ) · E 0 � � i d γ =1 � � � � − S mes . � � ij Γ i | E 0 2 � i | 2 d γ � � j =1 i =1 � � S ij ( κ ) depends on the solution E j ( κ ) to ∇ × ( ∇ × E j ) − κ E j = 0  in Ω ,    E j × n = 0 on Γ metal ,   ( ∇ × E j ) × n + i β ( E j × n ) × n = g j on Γ j ,     ( ∇ × E j ) × n + i β ( E j × n ) × n = 0 on Γ i , i � = j .  Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 13/ 26

The inverse problem For j = 1 , ..., 160, we introduce the adjoint problem  ∇ × ( ∇ × F j ) − κ F j = 0 in Ω ,    F j × n = 0 on Γ metal ,    ( ∇ × F j ) × n + i β ( F j × n ) × n = ( S ij ( κ ) − S mes ) ij E 0  on Γ i ,  i  Γ i | E 0 i | 2 d γ �   i = 1 , ..., 160 .  We have 160 �� � � ℜ δκ E j · F j dx DJ ( κ, δκ ) = . Ω j =1 We can then compute the gradient to use in a gradient-based optimization algorithm. Here we use a limited-memory BFGS algorithm. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 14/ 26

Numerical experiment - hemorrhagic stroke ◮ Brain model from X-ray and MRI data (362 × 434 × 362) ◮ simulated hemorrhagic stroke of ellipsoidal shape ◮ f = 1 GHz ◮ waveguides (ceramic) : ǫ r = 59 ◮ matching liquid : ǫ r = 44 + 20 i ◮ 10% multiplicative white Gaussian noise on synthetic data Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 15/ 26

Numerical experiment - hemorrhagic stroke Idea: reconstruct the permittivity slice by slice, by taking into account the transmitting antennas corresponding to only one ring and truncating the computational domain. Pierre-Henri Tournier Parallel preconditioners for problems arising from microwave system modeling for brain imaging 16/ 26