Effective Filtering and Interpolation of 2D Discrete Velocity Fields with Navier-Stokes Equations

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1 Effective Filtering and Interpolation of 2D Discrete Velocity Fields with Navier-Stokes Equations Louis-Philippe Saumier, Boualem Khouider and Martial Agueh August 2, 2016 Abstract. We introduce a new variational technique to interpolate and filter a two-dimensional velocity vector field which is discretely-sampled in a region of R 2 and sampled only once in time, on a small time-interval [0, t]. The main idea is to find a solution of the Navier-Stokes equations that is closest to a prescribed field in the sense that it minimizes the l 2 norm of the difference between this solution and the target field. The minimization is performed on the initial vorticity by expanding it into radial basis functions of Gaussian type, with a fixed size expressed by a parameter ɛ. In addition, a penalty term with parameter k e is added to the minimizing functional in order to select a solution with a small kinetic energy. This additional term makes the minimizing functional strongly convex, and therefore ensures that the minimization problem is well-posed. The interplay between the parameters k e and ɛ effectively contributes to smoothing the discrete velocity field, as demonstrated by the numerical experiments on synthetic and real data. 1 Introduction There are many instances where one needs to filter and interpolate a discretely sampled velocity vector field. For example, in the context of Particle Image Velocimetry (PIV), tracer particles are seeded in a fluid and are Department of Mathematics and Statistics, University of Victoria (lsaumier@uvic.ca). ibid. (khouider@uvic.ca). ibid. (agueh@uvic.ca). 1

2 illuminated with a pulsing laser [13, 21]. A velocity vector field is then created from a sequence of images capturing the light scattered back by the particles to approximate the fluid flow. The field obtained is typically defined on a non-uniform grid given either by the interrogation windows for cross-correlation algorithms or by the particle locations in particle tracking algorithms [15, 16, 22]. In addition, the approximation given is usually stationary on the time interval [0, t] between two successive images. It may also contain corrupted vectors which need to be corrected. Many methods have been developed to interpolate discretely sampled velocity fields in the context of fluid flows. While direct approaches such as splines [18] or triangulations [17] may be used, velocity estimates can also be obtained from the solution of various inverse problems for parameters of the flow [6]. Given that in many cases the discrete field under study is likely to contain noisy data, one may formulate the inverse problem more specifically as a data assimilation problem. Data assimilation - the process by which observations of a system are incorporated into a model of the system - is typically employed in numerical weather forecasting, hydrology or geology. It can be viewed as a specific type of inverse problem [2, 5] and it has been used for example to reconstruct the circulation of the ocean from the positions of drifting floats [1, 11]. Out of the various data assimilation methods developed for fluid flows [2, 8], variational approaches have been proven successful to perform velocity fields extensions and filtering by using the Navier-Stokes equations [7, 12, 14, 20]. While some of these methods have been developed to handle fields sampled at multiple points in time (for example this allows to deal with entire sequences of PIV images [12, 14]), we focus here on fields which are given at only one point in time. The method we present could thus be used on its own, or for example to initialize more involved algorithms taking into account fields given at multiple points in time. In addition, as opposed to [12, 14], we do not assume that any other information is available (for example, additional information from the PIV images). We rely solely on the discrete vector field in order to make the technique applicable to a potentially wider range of problems. In [20], the authors have considered a similar situation where a corrupted velocity field is available only at one point in time. However, they assumed the flow was quasi-stationary and thus neglected the time derivatives. Our goal in this paper is to design a method which allows non-stationary flows and which also performs well when only a few vectors are available, i.e. 2

3 when the non-uniform grid on which the discrete field is defined is sparse. The new technique we propose finds the closest (in a certain way) Navier-Stokes solution to a two-dimensional velocity field which is discretely sampled in a region R 2 and sampled only once in time, on a small time interval [0, t]. More specifically, we use the vorticity form of the Navier-Stokes equations combined with a variational formulation of the error between the discretely sampled field and the target field. The functional corresponding to this error is minimized with respect to the initial vorticity ω 0 in the Navier-Stokes equations. However, on its own, this functional is convex but not strictly convex with respect to ω 0 and thus the minimization problem may not be well-posed. We therefore add a penalty term consisting of the kinetic energy of the solution, with associated penalty parameter k e. This penalty term has two important features. First, it makes the functional strongly convex on a time-interval [0, t] for t small enough. Second, it gives a bias towards a minimizer with a smaller kinetic energy, thus preventing unwanted small scale vortices to form when sparse grids are used. We use an approach similar to the vortex blob method [9] which consists in expanding the initial vorticity ω 0 into radial basis functions. Given the discrete sampling of velocity at some time t [0, t], the inverse problem considered is thus to recover a good approximation of the initial vorticity in terms of this expansion. For these basis functions, we use Gaussian blobs of fixed width given by another parameter ɛ, and solve the corresponding optimization problem with Fourier transforms and a quasi-newton method. The interplay between the parameters k e and ɛ contributes to filtering the vector field from noisy measurements. Numerical experiments demonstrate that with the appropriate combination of parameters k e and ɛ, one can obtain minimizers which are good approximations of the underlying target velocity field. This paper is organized as follows. Section 2 introduces the optimization problem. In section 3, we show why the associated functional is strongly convex, and present a linearization procedure. Then, the numerical algorithm we employ to solve the problem is given in section 4 and the results of numerical experiments are displayed in section 5. Finally, we give concluding remarks in section 6. 3

4 2 The Optimization Problem Let v D be a discrete velocity field defined on the set of discrete points {q 1, q 2,..., q Nq }, where for simplicity we select = [0, 1] 2. Ideally, the resulting field v E (x, t) obtained by extending v D to the entire physical domain and on the time-interval [0, t] would be a solution of the incompressible Navier-Stokes equation on : Dv = p + ν v, Dt div(v) = 0, (1) v(x, 0) = v 0 (x), x, t [0, t], where D/Dt := / t + (v ) is the material derivative, p is the fluid s pressure, ν the viscosity and v 0 (x) is the initial velocity field which is unknown. For the sake of simplicity, we will assume this system is equipped with periodic boundary conditions in our numerical experiments in sections 4 and 5. However, the theory presented in this section and the next one remains valid for any other boundary condition that make the problem in (1) well-posed. There might be many solutions of (1) that coincide (at some given time t in [0, t]) with the prescribed vectors v D (q i ), i = 1,..., N q. This is especially likely to be the case when the non-uniform grid given by these locations is sparse; the diameter of the associated set of Navier-Stokes solutions can be very large. One might therefore want to add a constraint to select a solution v E which does not add too many artificial features to the field. A natural choice is to pick a solution with a small kinetic energy t v(x, t) 2 dx dt. (2) 0 Also, given that v D may possibly contain erroneous vectors, it might be too stringent of a condition to perfectly match v E and v D at some time(s) t on [0, t]. We thus choose to minimize the total error N q i=1 v (q i, t ) v D (q i ) 2. (3) To summarize, we are looking for a spatial and temporal extension v E (x, t) of v D on [0, t] which solves (1) and minimizes the sum of (2) and (3). 4

5 Let us therefore build an optimization problem to find such a vector field v E. First, we point out that it is sufficient to minimize the initial kinetic energy of v E since the kinetic energy (2) of the solution to the Navier-Stokes equations (1) is time-decreasing [10]. Also, one common and easier way to solve the Navier-Stokes equations in 2D is to consider the associated vorticity equation Dω Dt = ν ω, (4) ω(x, 0) = ω 0 (x), x, t [0, t], where D/Dt := / t + (v ) is still the material derivative and ω = v is the vorticity of the field [10]. Again, we do not yet specify boundary conditions for, but we assume that the conditions selected make (4) wellposed. The velocity v is then determined from the vorticity, using the Biot- Savart law [10] v(x, t) = K 2 (x y)ω(y, t) dy, where K 2 is the 2D Biot-Savart Kernel on, and the pressure is determined by the Poisson equation p = tr( v) 2 = 1 i,j 2 v i x j v j x i. If we let ψ be the stream function for the fluid, then the Biot-Savart law is derived from the combination of the relationship between the velocity and stream function ( v = ψ = ψ, ψ ) x 2 x 1 and the relationship between the vorticity and the stream function ω = ψ. For simplicity, we will therefore sometimes denote v = 1 ω in the next sections. Let us now introduce the following optimization problem: Minimization Problem: N q } inf {F t (w 0 ) := k e v 0 (x) 2 dx + v(q i, t ) v D (q i ) 2 : w 0 C 2 (), i=1 (5) 5

6 where v 0 and v depend on ω 0 through v = 1 ω and (4), and t > 0, k e > 0 are constants. Essentially, the goal here is to find, among all suitable initial vorticities in, the one that will give the velocity field v E which minimizes F t. The space C 2 () is selected here so that (4) is defined in the classical sense. To obtain a numerical solution of (5), we use an expansion of the initial vorticity in radial basis functions N b e x b i 2 /ɛ 2 ω 0 (x) = α i (6) ɛ 2 i=1 akin to the one used in the context of vortex blob methods [9]. Here, b i is the center of blob i (b i b j for i j), N b is the number of blobs and ɛ > 0 is a parameter controlling the size of the blobs. With this expansion, (5) becomes a minimization problem in R N b for the weights αi, i = 1,..., R N b. We will present a proof of the strong convexity of F t (α) := F t (ω 0 ) with respect to the weights α = (α 1,..., α Nb ) for small t and for k e > 0 in the next section, but for now we assume that F t (α) has a unique minimizer in R N b. Besides making the functional F t strongly convex for small t, the parameter k e allows flexibility in this solution by controlling the size of the kinetic energy with respect to the error associated with the prescribed field. Indeed, a solution corresponding to a small k e should be very faithful to v D but may have too much kinetic energy and thus may present too many turbulent eddies. On the other hand, a solution associated with a larger k e will have less kinetic energy, at the expense of allowing v E to be further from v D at the particle locations. If the time t [0, t] at which we want the vector fields to be close to each other is unknown, or if the discrete vector field v D is a stationary approximation of the real physical velocity on [0, t], we can select t = t/2 for numerical experiments. One can thus think of (3) as a midpoint approximation of the time-integral of this error term over the interval [0, t]. Also, by considering the minimization of the total error (3), one allows the optimal solution to not go through every single v D (q i ) and thus potentially smooth out some of the noise. 6

7 3 Convexity and Linearization Let us now comment on the convexity of F t with respect to α. To ease notations, we introduce the two functionals: F 1 (ω 0 ) := v 0 (x) 2 dx = 1 ω 0 (x) 2 dx, F t 2 (ω 0 ) := N q N q v (q i, t ) v D (q i ) 2 = 1 ω (q i, t ) v D (q i ) 2, i=1 and we define F 1 (α) := F 1 (ω 0 ) and F t 2 (α) := F t 2 (ω 0 ) for ω 0 given by (6). Recall the definition of a strongly convex function: Definition 1. A function F (z) : R n R is said to be strongly convex if there is a constant λ > 0 such that the function g(z) = F (z) λ z 2 is convex on R n. In that case we call λ the modulus of strong convexity of F. We have the following theorem for the strong convexity of F t : Theorem 2. For k e > 0, and for a small enough t > 0, F t (α) is a strongly convex function on R N b. The modulus of strong convexity of F t, denoted λ F t, is bounded above by k e λ F1 where λ F1 is the modulus of strong convexity of F 1. Before proving this theorem, we state and prove the following lemma. Lemma 3. F 1 (α) is a strongly convex function on R N b. In addition, the modulus of strong convexity λ F1 of F 1 depends only on the blob s initial positions b i, i = 1,..., N b. Proof. For clarity and to avoid tedious computations, we will present the proof only for N b = 2, that is for α = (α 1, α 2 ) R 2. We will discuss how to extend the results to the general case in a remark after the proof. Let ω 0 (x) = α 1 e x b 1 2 /ɛ 2 for b 1 b 2. We have v 0 (x) = K 2 (x y)w 0 (y) dy ɛ 2 i=1 = α 1 K 2 (x y) e y b 1 2 /ɛ 2 dy + α ɛ 2 2 := α 1 v 01 (x) + α 2 v 02 (x) + α 2 e x b2 2/ɛ2 ɛ 2 (7) K 2 (x y) e y b 2 2 /ɛ 2 dy ɛ 2 7

8 where in the last line we defined v 01 (x) and v 02 (x) for convenience. Using this in F 1 gives F 1 (α) = v 0 (x) 2 dx = α 1 v 01 (x) + α 2 v 02 (x) 2 dx = α1 2 v 01 (x) 2 dx + 2α 1 α 2 v 01 (x), v 02 (x) dx + α2 2 v 02 (x) 2 dx, where, denotes the usual l 2 inner product in R 2. Let us now take g(α) := F 1 (α) λ α 2. We get ( ) g = [2α 1 v 01 2 dx λ + 2α 2 v 01, v 02 dx, ( ) ] 2α 2 v 02 2 dx λ + 2α 1 v 01, v 02 dx and then D 2 g = ( ) 2 v 01 2 dx λ 2 v 01, v 02 dx 2 The first leading principal minor of the Hessian is ( ) 2 v 01 2 dx λ 2 v 01, v 02 dx ( v 02 2 dx λ and it is positive for λ < v 01 2 dx. The second leading principal minor is ( ) ( ) 4 v 01 2 dx λ v 02 2 dx λ [ ( = 4 v 01 2 dx v 02 2 dx ( 4 v 01, v 02 dx [ 4λ ( v v 02 2 ) dx ). v 01, v 02 dx ) ] 2 ) 2 ] + 4λ 2. (8) 8

9 Using the Cauchy-Schwartz inequality, we have ( 2 ( v 01, v 02 dx) v 01, v 02 dx ( < v 01 2 dx ) 2 ) 2 v 01 v 02 dx v 02 2 dx where we get a strict inequality since v 01 is not proportional to v 02 for b 1 b 2. We thus set ( 2 v 01 2 dx v 02 2 dx v 01, v 02 dx) = m where m > 0 is a constant independent of λ. We can write (8) as ( ) 4λ 2 4λ ( v v 02 2 ) dx + 4m, (9) which is a quadratic equation in λ. The discriminant is ( 2 ( = 16 v 01 2 dx v 02 dx) v 01, v 02 dx and it is positive for b 1 b 2. This implies the equation has the two (positive) roots λ 1 = 1 [ ( ) 4 ( v v 02 2 ) dx ], 8 λ 2 = 1 [ ( ) 4 ( v v 02 2 ) dx + ]. 8 Therefore, taking λ < λ 1 gives that (9) is positive and we get that the second leading minor is positive. This means that the Hessian D 2 g is positive definite for λ < min{ v 01 2 dx, λ 1 }. In addition, as the ordering of v 01 and v 02 is arbitrary, we have the same inequality for v 02 as for v 01. We thus get that g is a convex function for { λ < min v 01 2 dx, ) 2 v 02 2 dx, λ 1 }. (10) We can conclude that F 1 (α) is a strongly convex function with modulus depending only on b 1, b 2 and satisfying (10). 9

10 Remark 4. To extend this proof to higher dimensions, one can recognize the Hessian D 2 F 1 as the integral of a Gram matrix. Gram matrices are known to be positive definite when the corresponding set of vectors is linearly independent (which is the case here for distinct b i due to the specific basis functions used). The Hessian D 2 F 1 can thus be shown to be positive definite, which in turn implies that all of its principal leading minors are strictly positive. A minor of the Hessian D 2 g can be written as the sum of a minor of D 2 F 1 plus an O(λ) term, therefore positive for small λ. However, the explicit calculations of the minors become cumbersome in higher dimensions and that is why we only presented the proof in R 2. We now prove Theorem 2. Proof. First, we point out that F2 0 is a convex function of α. This can directly be seen by computing the Hessian matrix of F2 0 in a similar way as in the proof of the previous lemma. Next, from the regularity theory of the 2D Navier-Stokes equations (4) [10, 19], we know that for ω 0 given by (6), the spatial Hessian D 2 F2 t (α) is continuous with respect to t at t = 0. Therefore, we have that for γ > 0 (to be chosen later), there exists δ > 0 such that if t < δ, for any ξ R 2, ξ 0. This gives which in turn yields ξ T D 2 F t 2 (α)ξ ξ T D 2 F 0 2 (α)ξ < γ ξ 2 ξ T D 2 F t 2 (α)ξ > ξ T D 2 F 0 2 (α)ξ γ ξ 2 ξ T D 2 F t (α)ξ = k e ξ T D 2 F 1 (α)ξ + ξ T D 2 F t 2 (α)ξ > k e ξ T D 2 F 1 (α)ξ + ξ T D 2 F 0 2 (α)ξ γ ξ 2 k e λ F1 ξ 2 + ξ T D 2 F 0 2 (α)ξ γ ξ 2 (k e λ F1 γ) ξ 2, where in the last line we used the fact that F2 0 is a convex function. We see that taking γ < k e λ F1 where λ F1 is the modulus of strong convexity of F 1 is enough to guarantee the strong convexity of F t. We also conclude that the modulus of strong convexity of F t, that is, λ F t = k e λ F1 γ < k e λ F1, is bounded above by k e λ F1 as desired. 10

11 Remark 5. At the end of the proof of Theorem 2, we saw that γ < k e λ was needed in order for F t to be strongly convex. If the value of k e selected is small, then a small value of δ (and thus t ) may be required to keep γ smaller than k e λ. Therefore, increasing the value of k e may allow for larger values of t to be selected without losing the strong convexity of F t (more weight is being put on the strongly convex function F 1 in the sum F t = k e F 1 + F2 t ). The addition of k e F 1 to F2 t can also be seen as adding a penalty term to the error functional F2 t in the context of penalty methods, ensuring solvability of the optimization problem at the expense of allowing less kinetic energy in the final solution. Next, in order to obtain the numerical solution of this optimization problem within reasonable time, we will employ a steepest descent algorithm. We therefore need to linearize the functional F t. We will compute the linearization with respect to a general ω 0 and then we can use it for the problem in R N b by specializing it to (6). Consider a small variation ω0 + ηh of ω 0. For F 1, we have F 1 [ω 0 + ηh] = 1 (ω 0 + ηh) 2 dx = 1 ω 0 2 dx + 2η 1 ω 0, 1 h dx + O(η 2 ) where, denotes the l 2 inner product in R 2. From this, we obtain the linearization of F 1 at ω 0 in direction h: DF 1 (ω 0 )(h) = 2 1 ω 0, 1 h dx. (11) Let us denote the solution of (4) by ω(t) = S t ω 0 where S t is the flow operator associated with the solution of (4). For F t 2, we have F t 2 [ω 0 + ηh] = N q i=1 1 S t (ω 0 + ηh)(q i ) v D (q i ) 2 N q (ω 1 0 (q i ) + ηh(q i ) + t ) t S t(ω 0 + ηh)(q i ) v D (q i ) t=0 i=1 where in the second line we used a first order in time Taylor expansion to approximate S t. Note that this approximation is reasonable since the O(t 2 ) 11 2,

12 term missing is small in our case due to the small t. Using (4) to replace the time derivative, we reach N q i=1 1 ω 0 (q i ) + η 1 h(q i ) ) + t (ν (ω ηh) ( 1 (ω 0 + ηh) )(ω 0 + ηh) (q i ) v D (q i ) 2 N q = 1 ω 0 (q i ) + t 1 ν ω 0 (q i ) i=1 t 1 ( 1 ω 0 (q i ) ω 0 (q i )) v D (q i ) ( + η 1 h(q i ) + t ν 1 h(q i ) t 1 ( 1 h(q i ) ω 0 (q i )) ) t 1 ( 1 ω 0 (q i ) h(q i )) + O(η 2 ) 2. Expanding the norm as an inner product and gathering the linear terms in η yields the following formula: DF2 t (ω 0 )(h) = N q ( ) 2 1 ω 0 (q i ) + t ν ω 0 (q i ) t 1 ω 0 (q i ) ω 0 (q i ) v D (q i ), i=1 Thus, we obtain (h(q 1 i ) + t ν h(q i ) t 1 h(q i ) ω 0 (q i ) ) t 1 ω 0 (q i ) h(q i ). (12) DF t (ω 0 )(h) = k e DF 1 (ω 0 )(h) + DF t 2 (ω 0 )(h), (13) where DF 1 (ω 0 )(h) and DF t 2 (ω 0 )(h) are defined by (11) and (12), respectively. 4 Numerical Algorithm As opposed to using a Lagrangian discretization of (4) which is typical of vortex methods, we impose instead periodic boundary conditions on and 12

13 use Fourier transforms with a finite differences scheme to solve the resulting first-order in time ODE in Fourier space. When the domain is the twodimensional torus T 2, the vorticity formulation of (1) becomes ω t + [(v 0 + ṽ) ]ω = ν ω, ω(x, 0) = ω 0 (x), x, t [0, t] (14) where ω = curl ṽ, v 0 = v 0 dx and ṽ is recovered from the periodic version of the Biot-Savart law ṽ(x, t) = k 0 ( k 2, k 1 ) t 2πi k 2 e 2πix k ω(k, t) (15) and ω is the Fourier transform of the vorticity ω [10]. Let us now give the specific details of the discretization employed. First, we embed in a larger numerical domain n to avoid boundary effects, and we impose the periodic boundary conditions on n instead of. We lay a uniform grid of the centers b i for the Gaussian blobs in. We then consider (5) as an unconstrained minimization problem in R Nb and use MATLAB s fminunc quasi-newton algorithm to find the minimum of F t with respect to the weights α i in (6). This algorithm uses the gradient information supplied by (13) as well as an approximation of the Hessian of F t with the BFGS method to iterate in the direction of steepest descent [3]. At each step, given a set of weights {α i }, ω 0 is reconstructed on a uniform numerical grid on n of size N = N 1 N 2 through (6). Then, v 0 is recovered with the periodic Biot- Savart law (15) using the FFT algorithm (all the computations of 1 required in the derivative of F t are done this way). Once v 0 is obtained, the periodic Navier-Stokes equations in vorticity form (14) are solved in Fourier space with the FFT and the trapezoidal method for the resulting ODE to get ω and v on the time-domain [0, t], which is itself discretized with N t points. In the last step, F2 t is computed at t = t/2, all the integrals are approximated by a 2D Simpon s method, and F t as well as its derivative are computed with the same techniques. Finally, we use the stopping criterion provided by the fminunc package, which stops the iterations when either the size of the objective function F t or the size of a step becomes smaller than the specified tolerance TOL fminunc = Also, as we are looking for an optimal solution with a minimum kinetic energy, it is natural to take ω 0 = 0 as an initial guess in the quasi-newton algorithm. 13

14 (a) Target Vector Field. (b) Random selection of 100 vectors in a). Figure 1: Fields for Test 1. 5 Numerical Experiments We now conduct several numerical experiments to analyze the behavior of the minimizer in (5). In all the following tests, we run the algorithm using a spatial grid size of N = N 1 N 2 = and a temporal grid size of N t = 100, using a fixed time-step t = 0.1. The viscosity is set to ν = m 2 /s, which is roughly the (kinematic) viscosity of distilled water at 20 degrees Celsius. We will vary the parameters (k e, ɛ, and N b ) to study their effect on the resulting fields. The computations were all performed on a personal laptop computer of type Intel(R) Core(TM) i7-3537u 2.00GHz with a parallel implementation using all 4 cores to compute the components of the gradient of F t at every step. The computing times for a typical experiment presented in this section (when N b = 64) are around 15 minutes. As most of the work goes into computing the gradient at every step and that each component of the gradient is independent of the others, using more than 4 cores would greatly improve the performances of the current implementation. Also, we point out that our code has not been professionally optimized to minimize the computing time. Finally, in what follows, the images displayed all correspond to the optimal vector fields v E at time t = t/2 and the various norms are computed with the same fields. Test 1 For the first numerical experiment, we consider the vector field, on the 14

15 F F 1 F Number of Iterations Figure 2: Iterations details for Test 1 with k e = 0.1, ɛ = 0.05 and N b = 64. square [0, 1] 2, given by ) ( v(x 1, x 2 ) = (x 2 0.5, 0.5 x 1 exp ) 2((x 1 0.5) 2 + (x 2 0.5) 2 ). (16) This vector field cannot be recovered exactly with a single term in the expansion (6) since its curl is given by ( ) ( ) v = 4(x 1 0.5) 2 +4(x 2 0.5) 2 2 exp 2((x 1 0.5) 2 + (x 2 0.5) 2 ). In addition, even though it is divergence free, this field is not a solution of (14) due to the small viscosity term. We randomly select 100 vectors in (16) in order to simulate a vector field on a non-uniform grid. The resulting field is displayed in Figure 1. Figure 2 then displays the details of the iterations when the model parameters are set to k e = 0.1, ɛ = 0.05 and N b = 64. We observe that for the first few iterations, both kinetic energy F 1 and error with prescribed field F2 t are decreasing. After that, the algorithm slightly increases F 1 while decreasing F2 t. This behavior is fairly typical for the algorithm, especially for small values of k e. 15

16 k e F F2 t lnorm 2 error e rel error l error Table 1: Results for Test 1 when ɛ = 0.05 and N b = 64. The errors presented are computed for every grid point in. Let us now analyze the effect of the parameter k e. Figure 3 and Table 1 present the results of the algorithm when ɛ and N b are fixed to ɛ = 0.05 and N b = 64 but k e is varied. We see from Figure 3 that for small values of k e, the velocity field obtained displays artificial features (at the bottom right of the target vortex) which disappear for k e = 10. Indeed, the algorithm allows more kinetic energy for small k e in order to decrease F2 t as much as possible. We also observe that it is not desirable to select k e too large since the optimal solution becomes very close to 0, which corresponds to a field with almost no kinetic energy. These claims are quantified with the norms given in Table 1. The first two rows of this table give the values of F 1 and F2 t obtained for different values of k e. The last three rows give the error between the target field (16) and the field obtained by the algorithm at time t = t/2 with the lnorm 2 and the l norms. Note that lnorm 2 is the usual normalized l 2 norm and e rel is the relative error with respect to that norm. As k e varies from 1000 to 0, we see that the kinetic energy F 1 steadily increases while the error with the prescribed field F2 t steadily decreases. The lnorm 2 error is best when k e = 10, and significantly increases when k e is increased or reduced. A similar behavior is displayed for the l error, which is best at k e = 1. These results confirm the usefulness of k e, not only to make the functional strongly convex, but also to prevent the formulation of artificial features (i.e. small scale vortices) for sparse grids. Note that some residual error is expected since v in (16) is not a solution of the Navier-Stokes equations (14). In addition, the solutions presented for k e = 0 may not be unique, but initiating ω 0 = 0 possibly helps creating a bias towards a field with a small kinetic energy. Now that we analyzed the effect of k e, let us fix k e = 1 and N b = 64 and vary ɛ. Figure 4 presents the different vector fields obtained for different values of ɛ. The behavior observed is similar to the behavior of k e : taking 16

17 (a) k e = 1000 (b) k e = 100 (c) k e = 10 (d) k e = 1 (e) k e = 0.1 (f) k e = 0 Figure 3: Results for Test 1 when ɛ = 0.05 and N b =

18 (a) ɛ = 1.0 (b) ɛ = 0.5 (c) ɛ = 0.2 (d) ɛ = 0.1 (e) ɛ = 0.05 (f) ɛ = 0.02 Figure 4: Results for Test 1 when k e = 1 and N b =

19 (a) N b = 4 (b) N b = 16 (c) N b = 64 (d) N b = 256 Figure 5: Results for Test 1 when k e = 0.1 and ɛ = 0.1. ɛ F F2 t lnorm 2 error e rel error l error Table 2: Results for Test 1 when k e = 1 and N b = 64. The errors presented are computed for every grid point in. ɛ too small or too large decreases the quality of the approximation. Indeed, by taking ɛ to be too large, the algorithm cannot fully recover the size of 19

20 N b F F2 t lnorm 2 error e rel error l error Table 3: Results for Test 1 when k e = 0.1 and ɛ = 0.1. The errors presented are computed for every grid point in. Errors for Figure 6 (b) Figure 6 (d) Figure 6 (f) lnorm 2 error e rel error l error Table 4: Results for Test 1 with noise for k e = 1, ɛ = 0.2 and N b = 64. the target vortex, and by taking ɛ too small, the algorithm allows smaller variations in the field which are not physical in this case. This observation from the images is confirmed with the numbers in Table 2: the best field obtained is when ɛ = 0.2 for the lnorm 2 error and ɛ = 0.1 for the l error. Both the l 2 norm and l norm increase when ɛ is increased or decreased. In addition, F 1 and F2 t also become larger as ɛ increases or decreases from its best values. This behavior is inherent to the vortex blob method; an optimal ɛ is expected to exist for any given grid resolution. Let us now fix k e = 0.1, ɛ = 0.1 and vary N b. The results for grids N b = 2 2, N b = 4 4, N b = 8 8 and N b = are presented in Figure 5 and Table 3. We see on the images that the quality of the approximation increases as N b gets larger, which is to be expected. The size of F2 t roughly decreases by a factor 10 for each increase in N b. The results are best when N b = 256, but this comes at the expense of a much longer computational time required to obtain the approximate field. Indeed, this time jumps from about 15 minutes for N b = 64 to about 6.5 hours for N b = 256 (still on the personal laptop of specifications described earlier). We also investigate the effect of noise added to the field. More specifically, we add one erroneous velocity vector to the field of Figure 1 (b) and investigate the resulting error on the recovered field. This situation is common when using particle tracking methods for PIV: a mismatch between particles 20

21 (a) Same vector field as in Figure 1 (b). (b) Results of algorithm applied to (a). (c) One erroneous vector added to (a) at position x 1 = 0.68, x 2 = (d) Results of algorithm applied to (c). (e) Same erroneous vector added to (a), but at position x 1 = 0.77, x 2 = (f) Results of algorithm applied to (e). Figure 6: Results of Test 1 with noise for k e = 1, ɛ = 0.2 and N b =

22 may give a vector which is clearly going against the flow. First, we consider the case where the erroneous vector is placed close to a group of (correct) vectors. Figure 6 (c) gives the input field with the erroneous vector added while (d) gives the result of the algorithm applied to (c). In addition, Table 4 shows the different errors between the result in Figure 6 (d) and the target in Figure 1 (a). We see that the l 2 norm is almost unchanged (1% increase) and that the l norm decreases significantly. If we change instead the location of the erroneous vector so that it is further from other vectors as in Figure 6 (e), we obtain the field in Figure 6 (f) with the algorithm. Table 4 also shows the errors associated with this field. We see that the l 2 norm error now increases by about 7%, but the l error still remains smaller than the one for the field without noise. The algorithm can therefore smooth out some of the noise added, unless the erroneous vectors are in a region more sparsely filled with trustworthy vectors, in which case it is not as effective. We now briefly discuss the importance of the time derivative of the recovered solution. As previously mentioned in the introduction, our method is designed to produce non-stationary velocity fields on the interval [0, t]. Even though all the Figures presented thus far in Section 5 display the recovered fields at time t = t/2, the velocity fields we obtain visibly vary from t = 0 to t = t. In fact, for the examples of Test 1, we typically observe that the l norm of v/ t is about 0.25 m/s (assuming units in meters per second to match the selected viscosity) and that this is achieved close to the center of the vortex. For a velocity vector of length 0.15 at the position where the maximum time derivative is achieved, this corresponds to a change of about 2.5% in each of the components of that vector on the time interval [0, 0.01] selected for the current experiment. This small variation might not seem like much, but is is enough to see the vortex move on [0, 0.01] just by looking at a plot of v. In addition, if the goal of the experiment is to obtain a very precise approximation of the physical velocity field (as it is the case in PIV), then a 2.5% error may not be deemed negligible. As a final comment with the selection of 100 vectors for Test 1, we mention that if number of vectors is increased from 100 to 500, the results significantly improve. For example, in the case where the parameters are ɛ = 0.1, k e = 0.1 and N b = 64, l 2 norm drops from to , e rel from to and l from to The solution recovered is thus closer to the target when more information is available, which is of course to be expected. 22

23 (a) Field obtained with Optimal Transport method for PIV given in [16]. ɛ = (b) Result of test 2 for N b = 256, k e = 0.1 and 3. (c) Result of test 2 for N b = 256, k e = 0.1 and ɛ = 0.8. (d) Result of test 2 for N b = 256, k e = 0.1 and ɛ = 0.3. Figure 7: Results of Test 2. Test 2 In this second test, we apply the procedure to a real example. We select a discrete field obtained in [16] by using the Optimal Transport algorithm for Particle Image Velocimetry on the first two images in a dataset given by [23]. This dataset consists of PIV images showing a slightly turbulent air flow seeded with small water droplets. The results are presented in Figure 7. We display the fields associated with three different values of ɛ to show an interesting feature of the algorithm. For ɛ = 3, we see that the general direction of the flow is recovered, but none of the finer structures are present. 23

24 For ɛ = 0.8, the flow recovered is fairly close to the one given in b) and c) and the finer variations are displayed. However, the flow given by ɛ = 0.3 gives too small features which do not seem to be accurate in the target flow. The field obtained for ɛ = 0.8 also effectively smooths the prescribed field and corrects some apparently erroneous vectors. Keep in mind that we do not have access to the exact solution as this dataset is taken from a real experiment, but we can nonetheless observe that varying ɛ may help to visualize different regimes of the target velocity field and provide different levels of smoothing. Also, even though a large value of N b was selected for Test 2, good results can also be obtained for N b = Conclusions We introduced a variational method to find the vector field defined on the whole spatial domain and time interval that is the closest to a discretely sampled vector field given at one point in time. The closest field is selected to be a solution of the Navier-Stokes equations with minimal kinetic energy. To obtain it, we assumed that the initial vorticity was given by a sum of Gaussian blobs of fixed width determined by a parameter ɛ. The minimal kinetic energy constraint was enforced as a penalty term on the error with the prescribed field, with corresponding penalty parameter k e. The interplay between the parameters k e and ɛ contributed to filtering the flow from erroneous approximated vectors. We found that larger values of ɛ effectively smoothed the prescribed field by limiting the presence of small-scale features. On the other hand, larger values of k e made the functional strongly convex with respect to the weights in the expansion of the initial vorticity and helped reducing the presence of artificial features for sparse grids when smaller values of ɛ are employed. The method presented only uses information from the discrete vector field and thus could be applied to a variety of situations. It would be interesting in the future to extend it in order to interpolate and filter three-dimensional vector fields as they are becoming more common in some applications, for example in 3D-PIV [4]. Acknowledgement The authors are supported by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) and by a fellowship from the University of Victoria. 24

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