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Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Reduced basis method for the reliable model reduction of Navier-Stokes equations in cardiovascular modelling Toni Lassila, Andrea Manzoni, Gianluigi Rozza CMCS - MATHICSE - ´ Ecole Polytechnique F´ ed´ erale de Lausanne In collaboration with Alfio Quarteroni (EPFL & Politecnico di Milano) Supported by the ERC-Mathcard Project (ERC-2008-AdG 227058), the Swiss National Science Foundation (Project 200021-122136), and the Emil Aaltonen Foundation Model Reduction for Complex Dynamical Systems, TU Berlin, December 2-4, 2010

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Introduction 1 Navier-Stokes equations 2 Reduced basis approximation 3 Application in cardiovascular modelling 4

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Challenges in modelling the human cardiovascular system Human cardiovascular system is a complex flow network of different spatial and temporal scales. When investigating fluid flow processes the flow geometries are changing over time. The geometric variation causes a strong nonlinearity in the equations. Medical professionals are interested in accurate simulation of spatial quantities, such as wall shear stresses at the location of a possible pathology. Computational costs can become unacceptably high, especially if the objective is to model the entire network, and strategies to reduce numerical efforts and model order are being developed.

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ ∈ D ⊂ R P , find U ( µ ) ∈ X s.t. a ( U ( µ ) , V ; µ ) = f ( V ; µ ) ∀ V ∈ X , ∀ µ ∈ D where U := ( u , p ) and V := ( v , q ) consist of the velocity field and the pressure, the product space X = V × Q ⊂ [ H 1 (Ω)] 2 × L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U ) part a 1 : a ( U , V ; µ ) := a 0 ( U , V ; µ )+ a 1 ( U , U , V ; µ ) ∀ U , V ∈ X , ∀ µ ∈ D

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ ∈ D ⊂ R P , find U ( µ ) ∈ X s.t. a ( U ( µ ) , V ; µ ) = f ( V ; µ ) ∀ V ∈ X , ∀ µ ∈ D where U := ( u , p ) and V := ( v , q ) consist of the velocity field and the pressure, the product space X = V × Q ⊂ [ H 1 (Ω)] 2 × L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U ) part a 1 : a ( U , V ; µ ) := a 0 ( U , V ; µ )+ a 1 ( U , U , V ; µ ) ∀ U , V ∈ X , ∀ µ ∈ D For example, if the parameter is simply µ = ν (fluid viscosity), we have � a 0 ( U , V ; µ ) = Ω [ µ ∇ u : ∇ v − p div ( v ) − q div ( u )] d Ω � a 1 ( U , W , V ) = Ω v · ( u · ∇ ) w d Ω + appropriate boundary conditions.

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Parametric incompressible Navier-Stokes equations (steady case) We consider the following model problem: For a given parameter vector µ ∈ D ⊂ R P , find U ( µ ) ∈ X s.t. a ( U ( µ ) , V ; µ ) = f ( V ; µ ) ∀ V ∈ X , ∀ µ ∈ D where U := ( u , p ) and V := ( v , q ) consist of the velocity field and the pressure, the product space X = V × Q ⊂ [ H 1 (Ω)] 2 × L 2 (Ω), and the problem consists of a linear part a 0 and a nonlinear (quadratic in U ) part a 1 : a ( U , V ; µ ) := a 0 ( U , V ; µ )+ a 1 ( U , U , V ; µ ) ∀ U , V ∈ X , ∀ µ ∈ D For example, if the parameter is simply µ = ν (fluid viscosity), we have � a 0 ( U , V ; µ ) = Ω [ µ ∇ u : ∇ v − p div ( v ) − q div ( u )] d Ω � a 1 ( U , W , V ) = Ω v · ( u · ∇ ) w d Ω + appropriate boundary conditions. Typically we are interested in linear functionals of the field solutions (outputs) s ( µ ) := ℓ ( U ( µ )), i.e. need to find a reduced model � s ( µ ) that has is within certified tolerance of the actual outputs: | s ( µ ) − � s ( µ ) | < TOL for all µ ∈ D .

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Finite element approximation to the Navier-Stokes solution Starting from initial guess U 0 , solve at each step k of a FP iteration for U k s.t. a 0 ( U k , V ; µ )+ a 1 ( U k − 1 , U k , V ) = f ( V ) ∀ V ∈ V × Q until convergence. Stable discretization with P 2 / P 1 FE spaces for velocity and pressure V h := { v ∈ C (Ω , R d ) : v | K ∈ [ P 2 ( K )] 2 , ∀ K ∈ T h } ⊂ V Q h := { q ∈ C (Ω , R ) : q | K ∈ P 1 ( K ) , ∀ K ∈ T h } ⊂ Q . Galerkin projection in FE space: solve at each step k for U k h s.t. h , V h ; µ )+ a 1 ( U k − 1 a 0 ( U k , U k h , V h ) = f ( V h ) ∀ V h ∈ V h × Q h h until convergence. Similar approach for the Newton’s method...

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Reduced basis approximation of the finite element solution Assumption: parametric manifold of FE solutions M h ⊂ X h 1 is 1) low dimensional and 2) depends smoothly on µ (valid for small Reynolds number) Choose a representative set of parameter values µ 1 ,..., µ N 2 Snapshot solutions u h ( µ 1 ) ,..., u h ( µ N ) span a subspace V N 3 h for the velocity and p h ( µ 1 ) ,..., p h ( µ N ) span a subspace Q N h for the pressure Galerkin reduced basis: given µ ∈ D , find U N h ( µ ) ∈ X N h s.t. 4 M = { U ( µ ) ∈ X ; µ ∈ D } a 0 ( U k , N , V N h ; µ )+ a 1 ( U k − 1 , N , U k h , V N h ) = f ( V N ∀ V N h ∈ X N h ) h h h M h = { U h ( µ ) ∈ X h ; µ ∈ D } Adaptive sampling procedure (greedy algorithm) for the 5 choice of µ 1 ,..., µ N

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling Reduced basis approximation of the finite element solution Assumption: parametric manifold of FE solutions M h ⊂ X h 1 is 1) low dimensional and 2) depends smoothly on µ (valid for small Reynolds number) Choose a representative set of parameter values µ 1 ,..., µ N 2 Snapshot solutions u h ( µ 1 ) ,..., u h ( µ N ) span a subspace V N 3 h for the velocity and p h ( µ 1 ) ,..., p h ( µ N ) span a subspace Q N h for the pressure Galerkin reduced basis: given µ ∈ D , find U N h ( µ ) ∈ X N h s.t. 4 M = { U ( µ ) ∈ X ; µ ∈ D } a 0 ( U k , N , V N h ; µ )+ a 1 ( U k − 1 , N , U k h , V N h ) = f ( V N ∀ V N h ∈ X N h ) h h h M h = { U h ( µ ) ∈ X h ; µ ∈ D } X N h = span { U h ( µ i ) , Adaptive sampling procedure (greedy algorithm) for the i = 1 ,..., N } 5 choice of µ 1 ,..., µ N Reliability / accuracy ? is based on the quality of the sampling 1 relies on computable and rigorous a posteriori error estimator ∆ N ( µ ): 2 � U h ( µ ) − U N | s ( µ ) − s N ( µ ) | ≤ ∆ s h ( µ ) � X ≤ ∆ N ( µ ) , N ( µ ) = � ℓ � X ′ h ∆ N ( µ )

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling A posteriori error estimation of the reduced basis approximation (Veroy-Patera 2005) If τ N ( µ ) < 1 and β h ( µ ) > 0 there exists a unique solution U h ( µ ) s.t. � � � h ( µ ) || X ≤ ∆ N ( µ ) =: β h ( µ ) || U h ( µ ) − U N 1 − τ N ( µ ) 1 − ρ ( µ ) Here: β h ( µ ) is the Babuska inf-sup constant that needs to be estimated da ( U h ( µ ); µ )( W , V ) inf sup = β h ( µ ) > β 0 > 0 || W |||| V || W ∈ X h V ∈ X h for the Fr´ echet derivative of a ( U , W , V ) w.r.t first argument at U h ρ ( µ ) is a Sobolev embedding constant that needs to be estimated τ N ( µ ) := 2 ρ ( µ ) ε N ( µ ) , where ε N ( µ ) := || f ( · ; µ ) − a ( U N h , · ; µ ) || X ′ h is the RB residual β h ( µ ) 2

Introduction Navier-Stokes equations Reduced basis approximation Application in cardiovascular modelling A posteriori error estimation of the reduced basis approximation (Veroy-Patera 2005) If τ N ( µ ) < 1 and β h ( µ ) > 0 there exists a unique solution U h ( µ ) s.t. � � � h ( µ ) || X ≤ ∆ N ( µ ) =: β h ( µ ) || U h ( µ ) − U N 1 − τ N ( µ ) 1 − ρ ( µ ) Here: β h ( µ ) is the Babuska inf-sup constant that needs to be estimated da ( U h ( µ ); µ )( W , V ) inf sup = β h ( µ ) > β 0 > 0 || W |||| V || W ∈ X h V ∈ X h for the Fr´ echet derivative of a ( U , W , V ) w.r.t first argument at U h ρ ( µ ) is a Sobolev embedding constant that needs to be estimated τ N ( µ ) := 2 ρ ( µ ) ε N ( µ ) , where ε N ( µ ) := || f ( · ; µ ) − a ( U N h , · ; µ ) || X ′ h is the RB residual β h ( µ ) 2 Note: for large viscosity we obtain the Stokes equations and the estimator simplifies to � h ( µ ) || ≤ ∆ N ( µ ) = ε N ( µ ) || U h ( µ ) − U N key ingredients β h ( µ )