,
Reduced basis methods and high performance computing. Applications to non-linear multi-physics problems Christophe Prud’homme
[email protected] Cemosis - http://www.cemosis.fr IRMA Université de Strasbourg
23 octobre 2014
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Collaborators on the framework • Cecile Daversin (IRMA - UdS,
Strasbourg)
Non-intrusive RB • Rachida Chakir (LJLL
- UPMC, Paris)
• Christophe Prud’homme (IRMA -
UdS, Strasbourg)
• Yvon Maday (LJLL -
UPMC, Paris)
• Alexandre Ancel (IRMA - UdS, Current Funding
Strasbourg) • Christophe Trophime
(CNRS/LNCMI, Grenoble) • Stephane Veys (LJK - UJF,
Grenoble) • Jean-Baptiste Wahl
ANR/CHORUS, ANR/HAMM, FRAE/RB4FASTSIM, ANR/Vivabrain, Labex IRMIA, IDEX
(IRMA,Strasbourg) and former collaborators : S. Vallaghe, E. Schenone.
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1 Motivations and Framework 2 Reduced Basis for Non-Linear Problems and Extension to Multiphysics 3 Applications 4 Conclusions
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Motivations and Framework
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Motivations and Context
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An open reduced basis framework
Objectives • Provide an open framework for certified reduced basis methods
and low rank methods • Provide a rapid prototyping framework using the Feel++
language for the standard finite/spectral element methods • Provide interfaces to various open "mathematical" programming
environments such as Python/OpenTURNS(UQ) or Octave Where to get it ? • sources are available at http://www.feelpp.org • available as Debian/Ubuntu packages
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Feel++ Features • Galerkin methods (fem,sem, cG, dG) in 1D, 2D and 3D on simplices and hypercubes • Interfaces to PETSc/SLEPc • Language embedded in C++ close to variational formulation language that shortens tremendously the “time to results” // Th = {elements} auto mesh = loadMesh ( new Mesh < Simplex > ); // Xh = {v ∈ C 0 (Ω)|vK ∈ P2 (K), ∀K ∈ Th } auto Xh = Pch ( mesh ); auto u = Xh - > element () , v = Xh - > element (); auto a = form2 ( _test = Xh , _trial = Xh , ); R // a(u, v) = Ω ∇u · ∇v a = integrate ( elements ( mesh ) , gradt ( u )* trans ( grad ( v ))); auto b = form2 R ( _test = Xh , _trial = Xh ); // a(u, v) = Ω u v b = integrate ( elements ( mesh ) , idt ( u )* id ( v ) );
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Feel++ Reduced Basis/Low Rank Methods Features
Lower bound for coercivity/inf-sup
OK
90%
CRB Linear Elliptic case
OK
100% (coercive) 90% (non-coercive)
CRB Linear Parabolic case
OK
100%
Automatic differentiation
OK
90%
EIM
OK
100%
CRB Nonlinear case
Ok
80%
Not OK
50%
CRB for multiphysics
OK
80%
Low Rank Tensor
OK
Demonstrator
CRB automated affine decomposition
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Feel++ Framework : the User point of view Parametric FEM
Cmd line Python
Specifications Geometry, PDE,... µ0 , µ1 , ..., µP , t
Affine decomposition to be automated
CRB ...
Code generator SCM
Octave EIM Offline/Online Database handling
Gmsh/OneLab
Auto. diff.
class MyModel : public ModelBase < ParameterSpace , decltype ( Pch ( Mesh < Simplex { ... init (){ auto mesh = loadMesh ( _mesh = new Mesh < Simplex setFunctionSpaces ( Pch ( mesh ) ); // define bounds for $D ^\ mu$ ... // affine decomposition here ... }
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Wrappers : Python/OpenTURNS Parametric FEM
Cmd line Python
Specifications Geometry, PDE,... µ0 , µ1 , ..., µP , t
Affine decomposition to be automated
CRB ...
Code generator SCM
Octave EIM Offline/Online Database handling
Gmsh/OneLab
Auto. diff.
Wrappers are automatically generated by the framework Python Code (OpenTURNS wrapper for UQ) # fem code modelfem = Numer icalM athFun ction ( " modelfem " ) # crb code modelrb = Num erica lMathF uncti on ( " modelrb " ) mu [0] = 10 mu [1] = 7e -3 outputfem = modelfem ( mu ) outputrb = modelrb ( mu ) 10/70
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Wrapper : Octave Parametric FEM
Cmd line Python
Specifications Geometry, PDE,... µ0 , µ1 , ..., µP , t
Affine decomposition to be automated
CRB ...
Code generator SCM
Octave EIM Offline/Online Database handling
Gmsh/OneLab
Auto. diff.
Wrappers are automatically generated by the framework Octave code # setup parameters # kIC : thermal conductivity ( default : 2) inP (1) = 1.0 e +1; # D : fluid flow rate ( default : 5e -3) inP (2) = 7.0 e -3; .... for i =1: N inP (1)= 0.2+( i -1)*(150 -0.2)/ N ; D =[ D inP (1)]; s = [ s opuseadscrb ( inP )]; end 11/70
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High Performance Computing
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HPC Strategies MPI
Computing ressources
HARTS processors
cores
mesh partition
Objectives : GPU
• Realistic geometries • Accurate
computing
GPU
Consequences : • Memory
GPU
• Performance Technologies • MPI : Data partitioning • MT : Global assembly • GPU : Local assembly
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High Performance Computing and RB
• Training parameter sampling • Offline : identical on all cores • Online : distributed (greedy) • Parallel in spatial domain • RB construction • Residual parameter independent terms • SCM • Collective communications • Scalar products • Outputs • Opportunities : Robust preconditioners reuse • Memory hungry RB : matrix free operators
Reconstruction of reduced basis fields : for visualisation or usage in other context might be tricky due to spatial partitioning.
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HPC Strategies MPI
Computing ressources
HARTS processors
cores
mesh partition
Objectives : GPU
• Realistic geometries • Accurate
computing
GPU
Consequences : • Memory
GPU
• Performance Technologies • MPI : (Spatial,Parameter)Data
partitioning • (MT,)GPU : Online computations
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“Truth” FEM Approximation
Let I = (0, Tf ) , with Tf is the final time. Let µ ∈ Dµ and t ∈ I, evaluate sN (t; µ) = `(t; uN (µ)) , where uN (t; µ) ∈ X N ⊂ X satisfies N ∂u (µ) m , v; µ + a(uN (µ), v; µ) = f (t; v), ∂t
∀ v ∈ XN
∀t∈I .
• a(·, ·; µ) and m(·, ·; µ) are bilinear, f (·; µ) and `(·; µ) are linear • f and ` are bounded
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“Truth” FEM Approximation : Inner products and Norms T
Let ∆t the time step defined by ∆t = Kf . ∀w, v ∈ X and 1 ≤ k ≤ K we next define the • energy inner product and associated norm (parameter dependent)
w(tk ), v(tk )
= m(w(tk ), v(tk ); µ)+
µ
k X
0
0
aS (w(tk ), v(tk ); µ)∆t
k0 =1
q k 0 0 v(t ) = m(v(tk ), v(tk ); µ) + aS (v(tk ), v(tk ); µ)∆t µ
• X-inner product and associated norm (parameter independent)
w(tk ), v(tk ) = w(tk ), v(tk ) X µ ¯ k k v(t ) = v(t ) X
µ ¯
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∀w, v ∈ X, 1 ≤ k ≤ K ∀v ∈ X, 1 ≤ k ≤ K
Spaces ¯ a postivie number such that 1 ≤ N ¯ ≤ Nmax . Let N Parameter Samples : ¯, Sample : SN¯ = {µ1 ∈ Dµ , . . . , µN¯ ∈ Dµ } 1 ≤ N ≤ N with S1 ⊂ S2 . . . SN¯ ⊂ Dµ n o Let Snap(µ) = uN (tk , µ), 1 ≤ k ≤ K a snapshot set with µ ∈ SN¯ . ¯ = Nmax . Let Nm such that 1 ≤ Nm ≤ K, So we have N Nm N Let ρi (µ) ∈ X ,1 ≤ i ≤ Nm , eigen modes (associated to µ) selected a Proper Orthogonal Decomposition algorithm n o ρi (µ) ∈ X N , 1 ≤ i ≤ Nm = P OD Snap(µ), Nm n o WN = span ζn ≡ ρi (µ) such that bn/Nm cNm +i ≤ n, µ ∈ SN¯ , n = 1, . . . , N with W1 ⊂ W2 . . . WNmax ⊂ X N ⊂ X 18/70
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Algorithm
POD algorithm : Algorithm 1 {ρi (µ) ∈ X N , 1 ≤ i ≤ Nm } = P OD(Snap(µ), Nm ) Build matrix M such that M = uN (ti , µ), uN (tj , µ) 1 ≤ i, j ≤ K ij
X
Solve M ϕi = λi ϕi for (ϕi ∈ RK , λi ∈ R)1≤i≤Nm associated to Nm larger eigen values of M PK Build ρi (µ) = k=1 ϕk uN (tk , µ) with 1 ≤ i ≤ Nm .
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A posteriori error estimation
Given µ ∈ Dµ we define εN (tk ; µ) = r(uN (tk ; µ), v; µ)
X0
, 1 ≤ k ≤ K.
Let e(tk , µ) = uN (tk , µ) − uN (tk , µ) X , we can write v u u k k e(t , µ) ≤ ∆N (t , µ) ≡ t
k ∆t X εN (tk0 ; µ)2 , 1 ≤ k ≤ K. αLB (µ) 0 k =1
∀µ ∈ Dµ and 1 ≤ k ≤ K the output error bound is given by N k k s (t , µ) − sN (tk , µ) ≤ ∆sN (tk , µ) ≡ ∆N (tk , µ) ∆du N (t , µ).
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Example Heat Shield : Problem statement
−∆u + ∂u ∂t = 0 boundaries conditions
in Ω, on ∂Ω.
Boundaries conditions
∂Ωext : heat transfert with Tair = 1 • −∇u · n = Biotext (u − Tair );
∂Ωint : heat transfert, Tair = 0 ∂Ωiso (10,4)
∂Ωext (1,1) (0,0)
• −∇u · n = Biotint (u − Tair );
∂Ωiso : Insulated.
(3,3)
(4,1)
∂Ωint
2 parameters
Ω
• Biotext and Biotint .
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Example Heat Shield • final time : 20 s ; • time step : 0.2 s ; • Minimum parameters values.
• Maximum parameters values.
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Example Heat Shield • Configuration : • 33 000 dofs ; • Preconditioner : LU(MUMPS) – Solver : KSP • Ξ : parameter sampling of dimension 430. • Let es (Tf ; µ) =
|sN (Tf ; µ) − sN (Tf ; µ)| , plot max es (Tf ; µ) µ∈Ξ sN (Tf ; µ) 101
es (Tf ; µ) ∆sN (Tf ; µ)/sN (Tf ; µ)
Relative error
10−1 10−3 10−5 10−7 10−9 0
5
10
15 N
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20
25
30
Example Heat Shield : Scalability • Time to build the first reduced basis • Configuration : 292 000 dofs for FEM approximation. • Solver : KSP. Preconditioner : LU(MUMPS), GASM(1 or 2),
Multigrid Multigrid Theoretical LU GASM(overlap=1) GASM(overlap=2)
Speed up
10
5
20
40
60
Number of processors
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Example Heat Shield : Basis construction • Time to build the full reduced basis on 32 procs • Configuration : 292 000 dofs for FEM approximation. • Solver : KSP Preconditioner : LU(MUMPS, GASM(1 or 2),
Time (s)
Multigrid Multigrid GASM(overlap=1) GASM(overlap=2) LU
104
103
0
5
10
Number of elements in the RB
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Reduced Basis for Non-Linear Problems and Extension to Multiphysics
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Notations
NonLinear µ-parametrized PDE • Set of parameters : µ ∈ D µ ⊂ Rp , • Solution of the nonlinear µ-PDE : u(µ) ∈ X ≡ H 1 (Ω ⊂ Rd ), • PDE weak formulation : We look for u(µ) ∈ X such that
g(v; µ, u(µ)) = 0, ∀ v ∈ X .
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Bootstrap Truth approximation
• X N ⊂ X : finite element approximation of dimension N >> 1. • uN (µ) ∈ X N is solution of g(v; µ, uN (µ)) = 0, ∀ v ∈ X N . • Solution strategies such as Newton or Picard iterations, e.g. given 0 N
u , build the nonlinear iterates 1 uN , . . . , k uN , . . . j(δ k uN (µ), v; k uN (µ), µ) = −g(v; µ, k uN (µ))
where δ k uN (µ) = k+1 uN (µ) − k uN (µ). We recover the linear case. • Equate u(µ) and uN (µ), i.e. ku(µ) − uN (µ)kX ≤ tol, ∀ µ ∈ D µ . • Bootstrapping the reduced basis method
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Decomposition of the jacobian and residual
Non affine decomposition of g and j
We suppose that we have ∀v ∈ X and ∀µ ∈ Dµ , j reads j(u(µ), v; µ, w(µ)) =
Qj Z X
σqj (x, µ; w(µ)) jq (u(µ), v)
ωq
q=1
and g reads g(v; µ, w(µ)) =
Qg Z X q=1
σqg (x, µ; w(µ))gq (v).
ωq
where jq and gq no longer depend on µ, however σqj and σqg depend on x, µ and u. ωq denotes a part of the computational domain.
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Recovering affine decomposition (Nonlinear-case) Apply EIM on all σqg and build gAD (u(µ), v; µ) such that Mg
g (v; µ, w(µ), ) ≈ gAD (v; µ, w(µ)) =
Qg q X X
βgqm (µ; w(µ))
Z
qm (x)gq (v) ,
ωq
q=1 m=1
|
{z
g qm (v)
and similarly build EIM expansion for σqj and build jAD (u(µ), v; µ, w(µ)) such that j (u(µ), v; µ; w(µ)) ≈ jAD (u(µ), v; µ; w(µ)) = Mj
Qj q X X
βjqm (µ; w(µ))
Z
q=1 m=1
|
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j qqm (x)jq (u(µ), v) ,
ωq
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{z
j qm (u(µ),v)
}
(1)
}
Truth Finite element approximations
• X N ⊂ X : finite element approximation of dimension N >> 1. N N • uN is solution of gAD (uN . AD (µ) ∈ X AD (µ), v; µ) = 0, ∀ v ∈ X
• Solution strategies such as Newton or Picard iterations, e.g. given 0 N uAD ,
k N build the nonlinear iterates 1 uN AD , . . . , uAD , . . . k N k N jAD δ k uN AD (µ), v; µ; uAD (µ) = −gAD v; µ, uAD (µ) ,
for the increment δ k uN (µ) defined by δ k uN (µ) =
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u (µ) − k uN (µ) .
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(2)
Empirical Interpolation Method
[Barrault et al., 2004]
Ingredients µ • Training set Ξµ train ⊂ D • Offline step µ • Sample SM = {µ1 ∈ Ξµ train , ..., µM ∈ Ξtrain }, Interpolation points
t1 , . . . , t M ∈ Ω • Approximation space WM = span{q1 (x), . . . , qM (x)} • Residual rm (x) = σ(x, µm ; uN (µm )) − σm (x, µm ; uN (µm )) • qm+1 (x) = rrm(t(x)) (matrix (Bi,j ) = qj (ti ) lower triangular) m
m
• Online step : Compute approximation coefficients βm (µ; uN AD (µ))
σM (ti ; µ; uN AD (ti , µ)) =
M X
N βm (µ; uN AD (ti , µ)) qm (ti ) = σ(ti ; µ; uAD (ti , µ))
m=1
∀i = 1, . . . , M
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A language embedded in C++ : EIM expansion Let u be the solution of g(u, v; µ) = 0 ∀v ∈ X N and σ(u) the non linear expression involving a field. p a r a m e te rsp ac e_p tr typ e D ; parameter_type mu ; //µ ∈ Dµ auto TrainSet = Dmu - > sampling (); int eim_sampling_size = 1000; TrainSet - > randomize ( eim_sampling_size ); // expression we want EIM expansion auto sigma = ref ( mu (0))/(1+ ref ( mu (2))*( idv ( u ) - u0 )); // call Feel ++ function eim auto eim_sigma = eim ( _model = solve ( g(u, v; µ; x) = 0 ) , _element =u , // uN (µ) _parameter = mu , // µ _expr = sigma , // σ(u) _space =XN , _name = " eim_sigma " , _sampling = TrainSet ); // then we can have access to β coefficients PM // of EIM expansion m=1 β(µ, uN (µ)) qm (x) std :: vector < double > beta_sigma = eim_sigma - > beta ( mu );
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Reduced Basis Approximation • Build SN = {µi , i = 1, ..., N } : a parameter sampling • Build XN = {uN AD (µi ), i = 1, ...N } : reduced basis approximation
space of dimension N γ 99.0% : 343K = 70◦ C 80.0% : 336.5K = 63.5◦ C
L2 relative error FEM/RB max(relative L2 error)
Helix magnet - Convergence study Random
10−1 10−3 10−5 5
10
Number of basis (RB)
max(relative L2 error)
Output error Random 10−2 10−4 10−6 5 Number of basis (RB)
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10
Helix magnet - Performances
PFEM time
Simulation characteristics
• FEM approx. using affine
• Number of dofs ≈ 2.4 × 106 • 16 processors • Dofs / proc ≈ 155000
decomposition • 16 processors
• Number of inputs : 6 • σ0 , α, L, U , h, Tw
Mean time ≈ 35min
• Reduced basis approximation : • EIM basis space : 10 basis (sampling size = 100) • RB space : 25 basis (sampling size = 1000)
RB time • RB approximation • For any number of procs
Gain factor ≈ 17
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Mean time ≈ 2min
Perspectives
Ongoing work on electro-thermal model • Continue investigations with large simulations • Increase number of basis • Analyse convergence (EIM, RB) • ... • Work on error estimation for such a model • Error estimation for EIM approximation • Dealing with non-linearity
Towards full reduced model • Add Linear Elasticity model • Add Magnetostatic model
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Conclusions
• Framework in place, need to add rigor in the nonlinear case if
possible • HPC is definitely required for (non-linear multiphysics) RB but it
should be as seamless as possible • Still some ingredients are missing • hp framework for eim and rb ; • Lego simulation ( with domain decomposition ) ; • Another application : Aerothermal problems
(Navier-Stokes+Heat Transfer) ; • Embed advanced analysis tools (automatic differentiation,
polynomial chaos, ...)
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References
Barrault, M., Maday, Y., Nguyen, N. C., and Patera, A. (2004). An empirical interpolation method : application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique, 339(9) :667–672.
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OPUS Heat Transfer Benchmark
Back
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Thermal Testcase Description y Hot air outflow
Overview
Γ3
• Heat-Transfer with conduction
Ω43 Ω4 (Air) = ∪4i=1 Ω4i eIC
Γ32 = Γ23 = Ω 2 ∩ Ω3
hPCB Ω42
Ω2 (IC2 )
hIC
x2
Γ1
Γ2 Ω44 eIC
and convection possibly coupled with Navier-Stokes • Simple but complex enough to contain all difficulties to test the certified reduced basis • non symmetric, non
compliant
Γ1 Γ31 = Γ13 = Ω 1 ∩ Ω3
Ω1 (IC1 )
• steady/unsteady • physical and geometrical
hIC
parameters
x1
Ω41
Ω3 (PCB) Γ4 ePCB
x ea
Cooling air inflow (fan)
• coupled models
• Testcase can be easily
complexified Back
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Thermal Testcase Description y Hot air outflow
Heat transfer equation Γ3 Ω43 Ω4 (Air) = ∪4i=1 Ω4i eIC
∂T ∂t
+v·∇T −∇·(ki ∇T ) = Qi , i = 1, 2, 3, 4
Γ32 = Γ23 = Ω 2 ∩ Ω3
hPCB Ω42
ρCi
Ω2 (IC2 )
hIC
Inputs
x2
Γ1
Γ2 Ω44
µ = {ea ; kIc ; D; Q; r}.
eIC Γ1 Γ31 = Γ13 = Ω 1 ∩ Ω3
Ω1 (IC1 )
Outputs
hIC
x1
Ω41
Ω3 (PCB) Γ4 ePCB
x ea
Cooling air inflow (fan)
Z 1 s1 (µ) = T eIC hIC Ω2 Z 1 s2 (µ) = T ea Ω4 ∩Γ3 Back
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Thermal Testcase Description y Hot air outflow
Fluid model Γ3
Poiseuille flow or Navier-Stokes flow
Ω43 Ω4 (Air) = ∪4i=1 Ω4i eIC
Γ32 = Γ23 = Ω 2 ∩ Ω3
hPCB Ω42
Ω2 (IC2 )
• on Γ3 ∩ Ω3 , a zero flux
hIC
• on Γ3 ∩ Ω4 , outflow
x2
Γ1
Boundary conditions
Γ2 Ω44 eIC
• on Γ4 , (0 ≤ x ≤ eP cb + ea , y = 0)
temperature is set
Γ1 Γ31 = Γ13 = Ω 1 ∩ Ω3
Ω1 (IC1 )
• Γ1 and Γ2 periodic
hIC
• at interfaces between the ICs and
x1
Ω41
PCB, thermal discontinuity (conductance)
Ω3 (PCB) Γ4 ePCB
x ea
• on other internal boundaries, the
Cooling air inflow (fan)
continuity of the heat flux and temperature Back
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Thermal Testcase Description
Finite element method • Pk , k = 1, ..., 4 Lagrange elements
y Hot air outflow
• Weak treatment of Dirichlet Γ3
conditions
Ω43 Ω4 (Air) = ∪4i=1 Ω4i eIC
• CIP Stabilisation • Locally Discontinuous FEM
Γ32 = Γ23 = Ω 2 ∩ Ω3
hPCB Ω42
Ω2 (IC2 )
Validation
x2
Γ1
functions
hIC
Γ2 Ω44 eIC
• Comparison between
Comsol(EADS) and Feel++
Γ1 Γ31 = Γ13 = Ω 1 ∩ Ω3
Ω1 (IC1 )
• Extensive testing and
hIC
comparisons
x1
Ω41
Ω3 (PCB) Γ4 ePCB
• Implementation validated, ref.
config. max rel error < 1%
x ea
Cooling air inflow (fan)
• Diff. : mesh, stabilisation,
Dirichlet Back
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Thermal Testcase Description y Hot air outflow
Γ3 Ω43 Ω4 (Air) = ∪4i=1 Ω4i eIC
Γ32 = Γ23 = Ω 2 ∩ Ω3
hPCB Ω42
Ω2 (IC2 )
hIC
x2
Γ1
Γ2 Ω44 eIC
Γ1 Γ31 = Γ13 = Ω 1 ∩ Ω3
Ω1 (IC1 )
hIC
x1
Ω41
Ω3 (PCB) Γ4 ePCB
x ea
Cooling air inflow (fan)
Figure : Temperature plot for ea ∈ {2.5e − 3, 8e − 3, 5e − 2} Back
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Aerothermic : air conditioning/environment control systems y 1
• Steady Navier-Stokes/Heat
Γ4 Γf
transfer problem
T0
Γ1
Γ3
• 2 parameters : Grashof and
Heat flux
Ω(Fluid) Γ2 0
Prandtl number
W
• Study the average
x 0
temperature on heated surface with respect to parameters • Application to aerothermal
studies (buildings, cars, airplane cabins,...)
W
Figure : Geometry of the 2D model. Consider an extrusion of this geometry in 3D case is the extrusion in z axis of length 1.
100
10
−0.2
10−0.4
10−0.6
FEM RB 101
(a) temperature isosur60/70
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(b) Stream lines
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102
103 Gr
104
105
106
(c) Tav versus Gr
Aerothermic : air conditioning/environment control systems
• Steady Navier-Stokes/Heat
transfer problem • Coupling 2D/3D mesh based
aerothermal model with ECS modelica model • ECS and A/C design
parameters to be optimized
(a) Airplane Cabin
Figure : Design and sizing of thermal control of avionic bay and cabin
(b) Airplane Bay
(c) Temperature Field
High performance computing is required ! 61/70
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Back
/
HiFiMagnet project High Field Magnet Modeling
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Laboratoire National des Champs Magnétiques Intenses Large scale user facility in France • High magnetic field : from 24 T • Grenoble : continuous magnetic field (36 T) • Toulouse : pulsed magnetic field (90 T) Magnetic Field
Application domains
• Earth : 5.8 · 10−4 T
• Magnetoscience • Solide state physic • Chemistry
• Supraconductors : 24T • Continuous field : 36T
• Biochemistry
• Pulsed field : 90T
Access • Call for Magnet Time : 2 × per year • ≈ 140 projects per year Back
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High Field Magnet Modeling
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Why use Reduced Basis Methods ? Challenges • Modeling : multi-physics non-linear models, complex geometries,
genericity • Account for uncertainties : uncertainty quantification, sensitivity
analysis • Optimization : shape of magnets, robustness of design Objective 1 : Fast
Objective 2 : Reliable
• Complex geometries • Large number of dofs • Uncertainty quantification • Large number of runs
• Field quality • Design optimization • Certified bounds • Reach material limits Back
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Non-Intrusive Low Rank Approximation
Context and Notations • (D, B, P) a probability space, • D = D 1 × . . . × D d ⊂ Rd , • B a σ-algebra on D, N • P = dk=1 Pk a probability measure, • Parametric Linear Problem : find u(µ) ∈ RN such that ˆ A(µ)u(µ) = ˆb(µ),
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∀µ ∈ D,
(3)
Non-Intrusive Low Rank Approximation
Principle
L2P (D; RN )
N
'R ⊗
L2P (D; R)
N
'R ⊗
d O
! L2Pk (Dk ; R)
,
k=1
Goal : compute a low-rank approximation of the solution u ur ∈ W = RN ⊗ S 1 ⊗ . . . ⊗ S d k With S k = span{ϕki }ni=1 ⊂ L2Pk (Dk ; R) defined on the orthonormal k nk family {ϕi }i=1
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Optimization Framework Looking for an approximation of u by solving Z min JD (v) = J(µ; v(µ))P(dµ). v∈W
D
where the quadratic functional J : D × RN → R defined by J(µ; v) =
1 T v A(µ)v − v T b. 2
Discrete formulation JD (v) ≈ JZ (v) =
X
ωz J(µz ; v(µz ))
z∈Z
Alternating Minimization Algorithm
Resolution alternatively until convergence : min JZ (q⊗β 1 ⊗. . .⊗β d ), min JZ (q⊗β 1 ⊗. . .⊗β d ), β 1 ∈S 1 q∈RN 68/70 C. Prud’homme Aristote ROC & ROM
. . . , min JZ (q⊗β 1 ⊗. . .⊗ β d ∈S d
Algorithms
Algorithm 2 Alternating minimization algorithm. Initialize β k , k = 1, . . . , d. ` ← 0. (` ≤ `max ) and (q ⊗ β 1 ⊗ . . . ⊗ β d has converged) Compute q by solving Equation (28) k = 1, . . . , d Compute β k by solving Equation (??) q ← kβ k kk q β k ← β k /kβ k kk ` ← ` + 1
Algorithm 3 Greedy rank-one algorithm. u0 ← 0 r ← 1. (r < rmax ) and (ur has converged) Compute vr+1 with previous Algorithm with b = br ur+1 ← ur + vr+1 r ← r + 1
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First Implementation In Feel++ Application to a 3-Paremeters Heat Transfert Problem
Collaboration • A. Nouy (ECN) • L. Giraldi (ECN) • C. Prud’homme (UDS/IRMA) • JB Wahl (UDS/IRMA)
Figure : Geometry of the thermal fin Problem Setting • Steady Heat
Transfert • 3 Parameters (heat
transfert coefficients) • Study of Average Figure : First modes obtained with rank-one greedy algorithm
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Temperature on ΓRoot
