Distributed parameter identification for the Navier–Stokes equations for obstacle detection

We present a parameter identification problem for a scalar permeability field and the maximum velocity in an inflow, following a reference profile. We utilize a modified version of the Navier–Stokes equations, incorporating a permeability term described by the Brinkman’s Law into the momentum equation. This modification takes into account the presence of obstacles on some parts of the boundary. For the outflow, we implement a directional do-nothing condition as a means of stabilizing the backflow. This work extends our previous research published in (Aguayo et al 2021 Inverse Problems 37 025010), where we considered a similar inverse problem for a linear Oseen model with do-nothing boundary conditions on the outlet and numerical simulations in 2D. Here we consider the more realistic case of Navier–Stokes equations with a backflow correction on the outflow and 3D simulations of the identification of a more realistic tricuspid cardiac valve. From a reference velocity that could have some noise or be obtained in low resolution, we define a suitable quadratic cost functional with some regularization terms. Existence of minimizers and first and second order optimality conditions are derived through the differentiability of the solutions of the Navier–Stokes equations with respect to the permeability and maximum velocity in the inflow. Finally, we present some synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.

synthetic numerical test based of recovering a 2D and 3D shape of a cardiac valve from total and local velocity measurements, inspired from 2D and 3D MRI.

Introduction
One of the most important challenges for cardiology is to design safe ways to diagnose heart valve diseases.On the one hand, since the valves are thin structures, the main non-invasive radiological examinations do not allow to obtain clear images of the valves when they are open.On the other hand, invasive procedures are only performed in the case of valve replacement surgery, requiring catheterization from the esophagus and compromising the integrity of the patient.
Previously, we proposed in [1] a technique that allows to recover obstacles and domain deformations by means of a virtual domain and a permeability parameter that follows Brinkman's Law [2,3] for the Oseen equations.In [4], the authors showed that there is an asymptotic relationship between the solutions of the Navier-Stokes equations with domains that consider obstacles and virtual domains when the obstacles are replaced by a permeability parameter validating the optimal control problem defined in [1] to detect obstacles.Compared to our previous work, the novelties of this new work are: (i) the extension of the theory to Navier-Stokes equations, presenting techniques that relax the regularity assumptions with a simplified proof, (ii) the inclusion of the maximum velocity on the inflow as a new constant parameter to be determined for a known inflow profile that complies with a Poiseuille's Law (see [5]) or a similar regime, (iii) the inclusion of the directional do-nothing (DDN) condition as an outflow boundary condition (see [6]), and (iv) the simulation of the identification algorithm in 3D of a more realistic tricuspid valve shape type geometry.
The paper is structured as follows.In section 2, a parameter identification problem is defined for the Navier-Stokes equations with a permeability term given by the Brinkman's Law with the form γu, where u is the velocity field and γ ⩾ 0 is a function that takes large values in zones where is the obstacle should be located.In section 3, we present some results of existence of solution for our direct problem and our minimization problem, considering the inclusion of the DDN condition.In sections 4 and 5, we establish the first and second order optimality conditions for our suitable cost functional improving the proof techniques presented in [1] emphasizing the details of the Fréchet-differentiability of the DDN (see remark 3.1 in [6]).Section 6 presents numerical tests that consist of recovering the shape of a heart valve from global or partial measurements of the velocity of the blood passing through the valves.For the 2D experiments, we emulate a cardiac valve and we recover its shape via estimating the permeability parameter from perturbed measurements as in [1].The 3D examples are based on a tricuspid aortic valve.In both cases, 2D and 3D, we use total or partial velocity from a numerical reference test and we use that velocity to generate a synthetic 2D or 3D MRI.Finally, conclusions and future work are presented in section 7.
The term γu, with γ ⩾ 0, represents the distributed resistance that Ω imposes to the fluid.Bounded values of γ model porous media, where porosity decreases as γ increases.According to [4], as γ values increases in a certain area, the magnitude of u tends to decrease to 0, and a zero value of γ represents that the fluid follows the Navier-Stokes equations.Then, we can interpreter 1/γ is an indicator of permeability complying with Brinkman's Law [2,3], where the media is completely permeable when γ = 0, while γ tends to +∞ in zones where there are obstacles.Thus, it is important to identify the regions where γ assumes the highest values in Ω to recover the shape of immerse obstacles.
In comparison with the traditional do-nothing condition (DDN) given by −ν ∂u ∂n + pn, the directional do nothing condition imposed on Γ N adds a correction term for backflow (see [6]).
The boundary data u D represents a default profile that the fluid inlet can follow, generally normalized with a maximum magnitude equal to 1, and that is compatible with the Navier-Stokes equations.A particular case is constituted by Poiseuille's Law that is also satisfies the DDN condition.

Existence of solution
In what follows, we suppose that u D fulfills some hypotheses so that problem (2) has an unique solution.For that, if g ∈ H 1 (Ω) verifies div g = 0 and its trace is u D (see section IX.4 in [7]), it is enough to consider that M 1 must be small enough to verify the sufficient conditions of existence and uniqueness of solution given by 2M 1 κ u D 1/2,ΓD ⩽ ν and 3κC 1 2ν 2 g 2 1,Ω < 1, where κ and C 1 are positive constants independent on u D . Taking Some notations and results are required to simplify the results in this paper.
Let Ω ⊆ R N be a bounded domain with locally Lipschitz boundary ∂Ω.There exists a linear operator T : H 1 (Ω) → H 1/2 (∂Ω) such that there exists a constant c T > 0 only depending of Ω such that and T (u) = u on ∂Ω when u is also a continuous function in Ω.
Lemma 4. The application a : is continuous and there exists a constant κ 1 > 0, only depending of Ω, such that Proof.See lemma IX.1.1 in [7].
Lemma 5.The application b : is continuous and there exists a constant κ 2 > 0, only depending of Ω, such that Proof.Direct consequence of theorems 2, 3 and lemma 4.
Definition 6.The subsets Γ + N ⊆ Γ N and Γ − N ⊆ Γ N are defined by Let s ⩾ 0. The set of admissible parameter functions is defined by where u ∈ H 1 (Ω) and p ∈ L 2 0 (Ω) are the unique variational solutions of the problem (2).

First order necessary optimality condition
In order to implement a descent method to numerically solve this problem, it is necessary to establish the differentiability of functional J.However, this directly depends on the differentiability of map A. From this result, it is possible to establish necessary optimality conditions.First, it is necessary to verify a technical result.
Theorem 12.The map A is Fréchet-differentiable for every γ ∈ A where A is well defined.
For each ] can be described by the weak solution of the following problem where Proof.Let (γ, β) , (h, ε) ∈ A, it will be proved that there is a linear application D : A → H such that where Thus, testing the first equation with δu and the second equation with δp, we obtain where, applying lemmas 2-5, we can deduce that Since sufficient condition of uniqueness of solution and uniformly boundedness of |u| 1,Ω (see remark 9) imply that there exists a constant Applying Hölder inequality, theorem 2, Sobolev Embedding theorem, lemmas 3-5, there exist constants c 3 > 0 and c 4 > 0, independents on u, u * , (γ, β), (h, ε) and and Applying lemma 11, there exist constant c 5 > 0, independent on u, u * , (γ, β), (h, ε) and (γ 1 , ε 1 ) such that Thus, there exists a constant C 1 > 0, independent on u, u * , (γ, β), (h, ε) and (γ 1 , ε 1 ) such that Repeating the same arguments, we obtain that there exists a constant C 2 > 0, independent on u, u * , (γ, β), (h, ε) and (γ 1 , ε 1 ) such that Then, there exists C > 0, independent on u, u * , (γ, β), (h, ε) and Therefore, proving the theorem. Defining ,Ω , an expression for Frechét derivatives of B and C is given by Applying chain rule, it is obtained that In order to reduce this expression, the following definition is introduced similarly to [9].
where χ ω is the indicator function of ω and Using this definition, it is possible to rewrite J ′ (γ, β) depending of the adjoint state.That expression is simpler to analyze, since it depends on the adjoint state, allowing a simple form of a first order optimality condition using a variational inequality.Theorem 14.Let γ, γ 1 ∈ A and s ⩾ 0.Then, Proof.First, using integration by parts with the adjoint states [φ, ξ] as tests functions, it is obtained that and also Then, Thus, Later, if γ * ∈ A is optimal for the problem, then Finally, it is obtained that proving this theorem.

Second order sufficient optimality condition
The stability results of the optimization algorithms depend, for the most part, on the existence of the second derivative of J. Likewise, it is possible to establish second order sufficient optimality conditions.First, it is necessary to introduce some technical results.
Testing the equations of A ′ (γ, β) with v ′ and p ′ , respectively, we obtain where Since sufficient condition of uniqueness of solution and uniformly boundedness of |u| 1,Ω (see remark 9) imply that there exists a constant c 1 > 0 such that 3 2 κ |u| 1,Ω ⩽ c 1 < ν.Then, there exists a constant c 2 = ν − c 1 > 0, independent on u, γ, β, h and ε such that By the same way as in lemma 11, there exists c 3 > 0, independent on u, γ, β, h and ε such that Applying Hölder inequality for q, r ∈ [2, +∞] such that 1 q + 2 r = 1, we obtain First, choosing q = +∞ and r = 2, and applying theorem 2, there exists a constant c 4 > 0, only depending of Ω and Γ D , such that where |u| 1,Ω + β u D 0,ΓD is uniformly bounded (see remark 9).Then, there exists c 5 > 0 such that Taking Second, choosing q = 2 and r = 4, applying Sobolev Embedding theorem and theorem 2, there exists c 4 > 0 such that Repeating the previous argument, there exists C 2 > 0 independent on γ, γ 1 and h such that proving this lemma.
Proof.Testing the two equations of ( 6) with φ and ξ, respectively, and integrating by parts some terms, we obtain Proceeding in the same way as in the proof of lemma 15, there exists c 1 ∈ (0, ν) and Applying Hölder inequality and theorem 2, there exists c 3 > 0 such that Then, The uniformly boundedness of |φ| 1,Ω is consequence of the uniformly boundedness of |u| 1,Ω .
Taking C = c 3 c 2 > 0, the lemma is proved.
Theorem 18.The map (γ, β) where [u ′ ′ , p ′ ′ ] is the unique weak solution of the problem where f and g are defined by An expression for the Fréchet second derivative of J (γ) on directions (γ 1 , ε 1 ) and (γ 2 , ε 2 ) is given by where, reasoning as in the proof of theorem 14, In consequence, In what follows, a second order optimality condition is proved.For this, a series of technical results are required.Let (γ, β) , . Using this, it is possible to obtain the following estimations.
Lemma 19.There exists C > 0 such that Proof.First, applying Hölder inequality, Sobolev Embedding theorem and theorem 2, there exists c 1 > 0 such that Applying lemmas 15 and 20, there exists c 2 > 0 and c 3 > 0 such that where C = c 1 c 2 c 3 .Thus, the lemma is proved.
Lemma 22.Let k, j ∈ {1, 2}, with j = k.There exists C > 0 such that Proof.Using theorems 4 and 2, we obtain Applying lemmas 17 and 20, there exists c 1 > 0 and c 2 > 0 such that where C = κc 2 1 c 2 > 0, proving the first estimate.The proof of the following two estimates are similar.Finally, the last three estimates can be obtained using theorem 5 instead of theorem 4 Proof.Applying triangular inequality, we obtain Corollary 24.Let (γ, β) ; (γ * , β * ) ∈ A. There exists L > 0 such that, for every θ ∈ [0, 1] Finally, a second order sufficient optimality condition is presented and proved.
Remark 26.The result obtained in theorem 25 depends of s, but is valid for every s ⩾ 0.

Numerical results
In this section, we present some numerical test for our minimization problem simulating a cardiac valve with 2D and 3D synthetic data in order to complement the theory presented in the previous sections.According to [4], it is possible to approximate the effect of the valves using an appropriate L 2 function in a virtual domain.For the 2D experiments, we propose a realistic domain with a piecewise smooth bicuspid valve which represent a longitudinal section of the structure of a cardiac valve in an symmetric arbitrary position.We obtained our reference velocity from the numerical solutions of the Navier-Stokes equation with our real domain, extending by zero in the virtual domain used.For the 3D experiments, we used a cylinder as a virtual domain simulating a tricuspid cardiac valve with a realistic shape to obtain the reference velocity.In both cases, 2D and 3D, we used complete or partial information to numerically solve the following minimization problem.
subject to where α 0 , α 1 ∈ R are positive constants.The theory from sections 3-5 is still valid for this new problem.In the following subsections, the configurations of the reference case is explained, as well as the numerical solutions of the inverse problems associated with synthetic MRI.

2D reference test
To define a reference geometry, we use a domain Ω 0 that represents the area around the aortic valve with the valves (see figure 1).The inflow Γ I follows a Poiseuille's Law with parabolic profile given by where x = (x, y) are the Cartesian coordinates of the domain, n is the outer normal vector and d is the diameter of the inflow.Later, we define the virtual domain Ω given for the realistic domain without the valves and the same boundary conditions.The valves were generated with a rotated parabola with a thickness of 0.1 mm.The parameters for the human blood flow are given by the kinematic viscosity equal to ν = 0.035 cm 2 s −1 , the blood density given by ρ = 1 g cm −3 , and the dimensions d = 2 cm and U = 30 cm s −1 , resulting in a peak Reynolds number on the inlet of The Navier-Stokes equations are discretized using the finite element method (FEM) with Taylor-Hood elements (P 2 for the velocity u and P 1 for the pressure) on an unstructured triangular mesh.To obtain the reference solution, we used a mesh for the realistic domain Ω 0 generated by domain triangulation with h = 0.02 cm, which corresponds to 37666 elements and 19209 nodes.We generated a second mesh for the virtual domain used in the minimization problem.That mesh were generated by domain triangulation of Ω with h = 0.05 cm, which corresponds to 9716 elements and 4986 nodes.
The solvers were implemented using the finite element library FEniCS [11] with the default configuration.To solve the nonlinear problems, the Newton method was used.We define the set O, that represents the valve inside of Ω, that is, O = Ω \ Ω 0 .The reference velocity u R is computed by the Lagrange interpolation on the virtual mesh of the velocity computed as the numerical solution of the Navier-Stokes in the realistic mesh, extended by 0 in O.

Numerical solution of the inverse problem in 2D
The valve geometry can be recovered using the γ function.Indeed, according to [4], the valves are modeled on the resistance term γu using the function γ.This function assumes a constant value M 1 in the regions where the valve is and assumes the value 0 where the valve is not.Using u OBS = u R as reference solution, the minimization problem ( 7) is numerically solved using FEniCS and dolfin-adjoint.
For this first example, we considered a measurement area ω = Ω, γ ∈ H 1 (Ω), and the values M 1 = 50, M 2 = 10 4 , α 0 = 10 −5 and α 1 = 10 −8 .The use of two different weights for the norm and seminorm is consistent with the theoretical analysis of the previous sections, so this problem has a solution.The dolfin-adjoint library [12] allows to implement automatic derivation of the discrete adjoint equations for PDE models and implement minimization algorithms from the Python 3 libraries.In particular, the L-BFGS-B algorithm (see section 4.3 in [10]) was used with the following stopping criteria on the step k To start the algorithm, γ 0 = 0 and β 0 = 0 were used as the initial solution.The parameters γ and β were rescaled to the interval [0, 1] for a correct implementation in dolfin-adjoint.As a way to define a valve reconstruction algorithm sketch, we follow the same steps we established in [1].
1. We defined an axis that crosses the domain from the inflow to the outflow.2. For a uniform discretization of the axis, we defined perpendicular lines.
3. The solution γ * obtained by the algorithm is interpolated on each of the lines.Three points are selected on each side of the axis.The first and second point are the limits of an interval where ∇γ * • n has the maximum positive values with 1% of tolerance.The third point is the local maximum closest to the interval.4.An average is obtained between the three points.5.A polyline is drawn on each side of the axis.Each polyline passes through all the average points.
Numerical results are presented in figure 2. The polyline is drawn in white, which presents a great approximation to the interface between the different values of the reference given by γ.The optimal γ * has values close to 0 between the valves, above and below the valves.Likewise, the magnitude and direction of u * is similar to u R , where u * corresponds to the optimal state.The optimal β * = 31.854 is very near to the reference value β = 30, showing empirically that the previously defined problem allows to obtain a good approximation of the maximum velocity magnitude in the inflow.

Measurements of MRI type in 2D
In several medical applications, the 4D MRI allows to register the blood flow in a volume unit, also called voxel, which can be reinterpreted as the average blood velocity in the voxel.Inside each voxel, we can assume the blood velocity as a constant.Since our problem is stationary, an approximation to a synthetic MRI is to project u R , extended by zero outside the virtual domain Ω, to a Q 0 FEM space (given by piecewise constant discontinuous functions on a quadrilateral mesh) using a new quadrilateral mesh that contains the virtual domain mesh obtaining our synthetic 2D MRI u MRI .That mesh were generated by uniform quadrilaterals of 1 mm ×1 mm.Unlike [1], where the authors used this synthetic MRI as an observation, we project it again to the original P 2 space in order to obtain the new observation u OBS that we can easily compare with the numerical states.
Figure 3 shows the synthetic MRI generated from the reference solution and the new reference solution u P , where we can verify that the reference is affected by the loss of resolution given by the MRI.
Then, the new minimization problem is given by ( 7), but replacing u OBS with u P , where the values M 1 = 10 4 , M 2 = 50α 0 = 10 −5 and α 1 = 10 −8 were used.It is possible to prove the existence of solution of this problem in the same way as in the proof of the theorem.Figure 4 shows the numerical results and the references.In comparison with figure 2, the reconstructed valve in figure 4 is located further from the center than in figure 2, because of synthetic MRI sampling, but the optimal values β * are similar.
The white noise intensity in the velocity measurements from MRI is proportional to the velocity encoding parameter (also called VENC [13]) of the scan.This parameter is configured with a value greater than the maximum expected velocity, in order to eliminate signal aliasing.Then, the noise in all voxels is proportional to the maximum velocity in the measurement area.In the clinical practice it can be expected that high-quality MRI contains a velocity noise of 10% of the maximum velocity in each voxel [13] in each direction.Gaussian noises were added to this MRI in every direction with a standard deviation of 5%, 10% and 20% of the maximum absolute value on each direction of u R .Figure 5 shows the results of this experiment with a 5% of Gaussian noise.The results are similar to the experiment without noise in terms to the tendency of the polyline to approximate the valve shape and draw lines parallel to the voxels.
This approximation is less accurate as noise increases, in the sense that the polyline has a lower quality in its approximation and γ * tends to overfit the data.Table 1 shows the mean square error (MSE) between the reconstructed valve given by the polylines obtained using MRI with different noise levels, and the polyline obtained in the reference test (see figure 2).To quantify this error, we consider only the points of the polyline in figure 2 that are at a distance less than or equal to 0.5 mm from O.There are minor differences between the valve  reconstructions for the cases with a noise level of 0% and 5%.However, the quality of the reconstruction decreases when the level noise increases up to 20%.

3D reference test
Here, the domain Ω is given by where R = 1.305 and L = 4, and a unstructured tetrahedral mesh with h = 0.05, with 59568 nodes and 329126 elements.The valves are modeled on the permeability term using the γ function.This function assumes the constant value M 1 in the regions where the valve is and assumes the value 0 where the valve is not.We used a parametric model of the tricuspid valve built using the methods reported in [14] to define the γ function, where the support of . This function is modeled as a P 1 function, where the nodal values of the function are given by γ = 10 10 or γ = 0, depending if the node lies or not in the valve.We still using the same kinematic viscosity ν = 0.035 cm 2 s −1 , density ρ = 1 g cm −3 and U = 30 cm s −1 .The inflow Γ I = (x, y, 0) ∈ R 3 | x 2 + y 2 ⩽ R 2 follows a Poiseuille's Law with parabolic profile given by where x = (x, y) are the Cartesian coordinates of the domain, n is the outer normal vector and d is the diameter of the inflow, while the DDN conditions are imposed on the outflow The walls of the structure are represented by Unlike the 2D case, due to computing efficiency, the Navier-Stokes equations were discretized using the FEM with the MINI element (P 1,bub = P 1 ⊕ V bub for the velocity u and P 1 for the pressure p, where V bub is the space of the bubble functions, see section 3.6.1 in [15]).We still using FEniCS [11] with the default configuration with the Newton method.The numerical experiments in 3D were computed on 48 Intel Xeon 2.5 GHz cores.We define the set O, that represents the valve inside of Ω, by Figures 6 and 7 show the reference valve and reference velocity field.The peak Reynolds number on the inlet is Re = Ud ν = 2237.

Measurements of MRI type in 3D
In this experiment, we present the covering of the γ function form a projection of u R , extended by zero outside the virtual domain Ω, on a Q 0 FEM space using a new quadrilateral mesh that contains the virtual domain mesh obtaining our synthetic 3D MRI u MRI .That mesh were generated by uniform hexahedrons of 1 mm ×1 mm ×1 mm.Unlike [1], where the authors used this synthetic MRI as a reference, we project again to the P 1 space in order to obtain the new velocity reference u OBS = u P that we can easily compare with the projection of numerical states on P 1 space.Then, we solved problem (7) considering this new functional where (u) P1 denotes the projection of u on the P 1 space.We decided to compare the velocity projections on P 1 instead of the velocity in P 1,bub since P 1 are better approximations to the velocity than the one obtained in P 1,bub (see section 2 in [16]).This first example in 3D follows the same configuration of the first example in 2D, choosing the measurement area ω = Ω and the values the values M 1 = 10 4 , M 2 = 50α 0 = 10 −5 and α 1 = 10 −8 .We still using the dolfinadjoint library [12] with the L-BFGS-B algorithm (see section 4.3 in [10]) and stopping criteria on the step k given by To start the algorithm, γ 0 = 0 and β = 0 were used as the initial solution.The parameters γ and β were also rescaled to the interval [0, 1] for a correct implementation in dolfin-adjoint.Numerical results are presented in figure 8.The optimal γ * has values close to 0 in zones before the valves and in the interior zone where there are no valves.If we choose the zones where γ * has values greater or equal to the threshold 0.4 max{γ * (x | x ∈ Ω)}, we can see that region is able to recover the space between the valves.The magnitude and direction of u * is similar to u R , where u * corresponds to the optimal state.The optimal β * = 35.6507 is near to  the reference value β = 30, but the difference with respect the reference is greater than the 2D case.
We also simulated a MRI with a velocity noise proportional to the maximum absolute velocity for each direction, following the same accepted medical parameters for a 4D flow MRI.Gaussian noises were added to this MRI in every direction with a standard deviation of 10% and 20% of the maximum absolute value on each direction of u MRI .Figures 9 and 10 show the results of this experiment with a 10% and 20% of Gaussian noise, respectively.This approximation seems weaker as noise increases, following a similar tendency given in the 2D case.The valve reconstruction obtained in figure 9 is similar to the one obtained in the previous 3D examples, but the γ * has some numerical noise and is not so similar to γ R .The results are worse when noise is increased up to 20%.

Conclusions
We have presented a distributed parameter identification problem for the Navier-Stokes equations, with the goal of detecting obstacles and deformations in fluid dynamics studies.Our approach establishes the existence of solutions and optimality conditions, providing validation for the use of optimization algorithms for differentiable functionals.We have extended our results from our previous article [1] with the following improvements: (i) from the Oseen equations to the Navier-Stokes equations, (ii) incorporating the recovery of the maximum velocity of the inflow, (iii) going from the do-nothing condition to the DDN condition to model backflow, and (iv) extending the identification of the case of simple 2D valve geometries to the case of more realistic 3D tricuspid valve geometries.Future work will focus on improving the uniqueness of solutions through the use of results from chapter 4 and other techniques.
Our numerical experiments have demonstrated improvements over previous research (see [1]), with better precision and stability in 2D experiments.Results for simulations without noise showed satisfactory accuracy in reconstructing a simulated valve from real data.Additionally, our post-processing of simulated MRI data improved the results of experiments with Gaussian noise.As expected, solution quality deteriorates with increasing noise levels.The 3D experiments showed similar results to the 2D experiments.However, the threshold criterion was not effective in reconstructing the space between valves for higher noise levels, and the solver execution time was high.

Remark 9 .
The uniformly boundedness of u and p are obtained since 0 ⩽ β ⩽ M 1 and γ ∞,Ω ⩽ M 2 for all (γ, β) ∈ A. Now we can state the result of existence of solution.

Figure 3 .
Figure 3. Original velocity u (left), synthetic MRI type velocity measurement u MRI (center) and u P (right).

Figure 6 .
Figure 6.Plots of the valve region O. Slice for x = 0 (left), frontal view (center) and outflow view (right).

Figure 7 .
Figure 7. Plots of the valve (left), the reference velocity field (center) and isovalues of reference velocity for a x = 0 cut (right).
The sequences {v n } n∈N and {p n } n∈N are also uniformly bounded on H1

Table 1 .
MSE of reconstructed valves using MRI with different noise levels.