19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

Adaptive sampling in two dimensions for point-wise experimental measurement
techniques
R. Theunissen1,*, P. Gjelstrup2

1: Dept. of Aerospace Engineering, University of Bristol, BS81TR, UK
2: Dantec Dynamics, DK-2740, Denmark* Correspondent author: r.theunissen@bristol.ac.uk
Keywords: Adaptive Sampling, Radial Basis Function, scale-dependent sampling, LDA, Design of Experiments, Thin Plate Spline

ABSTRACT
When performing point-wise measurements within a pre-defined domain, the experimentalist is faced with the
problem of defining the spatial locations where to collect data based on an a-priori unknown underlying signal. While
structured sampling grids are most common, these are rarely optimal from a time-efficiency perspective. In this work
an adaptive process is presented for point-wise measurement techniques to guide the spatial distribution of sampling
locations in two-dimensions. Thin Plate Splines of varying degree of smoothing are utilised to iteratively obtainsurrogate models of the spatial distribution of the quantities of interest. Adaptive sampling criteria reflecting typicalhuman decisions are combined into a unique objective function, allowing the derivation of suitable measurement
positions. The overall implementation is elaborated and the methodology is assessed on the basis of numericalsimulations and an experimental test case. Results support the new method's advocated superiority compared to
traditional full-factorial sampling in terms of accuracy and reliability.

1. Introduction
While full-field image-based measurement techniques such as Particle Image Velocimetry (PIV)
are capable of providing velocity data within planar or even volumetric sections of the scrutinised
flow (Kähler et al., 2016), highest spatial resolutions are attained with more traditional point-wisemethodologies such as e.g. hot-wire anemometry (HWA) and laser Doppler anemometry (LDA)(Lavoie et al., 2007). On the other hand, because measurement volumes are typically hundreds oforders of magnitude smaller compared to the flow region of interest, such techniques require someform of traversing. The experimentalist is thus forced to compromise between data content andtime efficiency: the more samples of the flow, the longer the measurement time, the higher runningcosts of the experimental facility yet the more flow detail can be obtained. When faced with thetask of characterising an a-prioiri unknown flow phenomenon, every experimentalist is therefore
ultimately faced with the same problem: Where to measure?. Of the various sampling strategies,
space-filling is the most common and includes, among others, the distribution of equidistantsamples across the measurement domain (full-factorial sampling: FFS). As the user refines the19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

sampling grid, the number of measurement points becomes rapidly impractical. It is for this reason
that more intelligent sampling approaches are needed, capable of focussing, and thereby
minimising, the data accrual to flow regions in which a higher number of measurements are
beneficial in the flow field reconstruction and subsequently understanding, of the underlying
physics.
The aim of the current work is to devise an autonomous routine capable of guiding the spatialsampling process in a typical experiment in which point-wise measurement techniques are
utilized to survey a prescribed domain in terms of scalar quantities such as e.g. pressure, velocity,temperature, etc. At each sampling location the statistical average of multiple readings serves as
the quantity of interest in the characterisation of the underlying phenomenon. Given the
exponential increase of the number of sampling positions with dimensionality of the problem andconsequently the experiment run time, considered applications are limited to two-dimensions. The
positioning of the data extraction locations within the 2D domain is performed through an
adaptive process similar to Theunissen et al. (2015} but extended into higher dimensions while
rendered more efficient. It is important to note that the importance of efficient collection of
meaningful data spreads beyond the boundaries of experimental fluid mechanics and has abroader application referred to as the design of experiments (DoE). As such, the proposed
methodology and associated implementation is expected to benefit a wider range of research.
2. Surrogate modelling
In its simplest form the surrogate model in two dimensions consists of an interpolation of available
data f(x ), x=(x,y) onto the evaluation grid. Because of their simplicity and ease of implementationkpolynomial regression and RBF interpolation are the most common methods (Fasshauer, 2007).
Given N data points (k=1,…,N), the value of the surrogate model s(x ) at a new random location xssin two dimensional space is formulated through an augmented RBF model with polynomialsummands asNs (x s ) = å a ky ( x k - x s ) + q(x s ) x Î ! 2

(1)

k =1

where y (r ) accounts for the contribution of the k basis function in function of the radial distancethr = xk - x s

between the new location and the kthdata point. The polynomial partq(xs ) = å j b j p j (xs ) is added to guarantee polynomial accuracy up to degree 2 and is given by

q = b1 + b 2 x + b3 y + b 4 x 2 + b5 y 2 + b 6 xy . Scaling coefficients a k , b k Î ! are obtained straightforward
by solving Eq.1 for the known data points, i.e. the interpolation conditions s(x )=f(x ) with the 6kNkadditional constraints å a k p j (x k ) = 0, j = 1,..., 6 . This is equivalent to solving the matrix systemk =1

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

é Y + l I P ù éa ù é f ù(2)
ê PTú ê b ú = ê0úOëûë û ë û
where Yi , j = y ( xi - x j ), i, j = 1,..., N , P contains the polynomial terms of the q, l=1,…,6, O is a 6´6i.lzero matrix, I is the N´N identity matrix, a = [a1 ,..., a N ]T , b = [ b1 ,..., b N ]T , f = [ f (x1 ),..., f (x N )]T and
0 is a 1´N vector. The additional term λI is referred to as ridge-regression and introduces a degreeof smoothing, which can be controlled through λ. Having calculated the coefficients for each data
point, these are then substituted into Eq.1 to solve for the interpolation at any random location.
Thin Plate Splines (TPS) have been chosen as radial basis kernels in the current work since they
do not require shape parameters (Fasshauer, 2007). However, they are inhibitive as they are
intrinsically not compact (Li et al., 2007) and thus involve solving large matrix systems (Wood,2003). However, in the experimental applications considered the number of sampling locations
will ordinarily lie in the order of 10 rendering the matrix system (depending on the choice of3

evaluation grid density) of only low to moderate complexity. More importantly though,considerable work has been done in optimizing the ridge regression parameter λ to accommodate
splines for noisy data (Wahba, 1990). Following Duchon (1977) the appropriate degree of thepolynomial term when using TPS equals m=1 and m=2 for TPS kernels involving 2 (TPS2) and 3ndrd(TPS4) order basis functions. On uniform grids, the TPS4 has a condition number which increases
as O(N ) compared to O(N ) for TPS2 (Boyd and Gildersleeve, 2011). This translates in ill7

4

conditioned, and subsequently, unstable solutions for the coefficient vectors a and b in Eq.2. To
overcome this problematic, the interpolation matrix is preconditioned. For two-dimensional
problems the preconditioning strategy proposed by Dyn et al. (1986) is specifically tailored to TPS2
and has been extended here for TPS4. It was shown in Dyn et al. (1986) that this type of
preconditioner reduces the condition number by several orders of magnitude. Solutions for the
TPS coefficients a are retrieved through an iterative conjugate gradient scheme.
Measurement data will unavoidably contain some level of uncertainty due to noise originatingfrom temporal fluctuations in the data, limited spatial resolution of the measurement system,
electronic noise, etc. In view of adjusting the sampling process to signal curvature, such regions ofjitter will unnecessarily attract samples and continue to do so. A degree of filtering is therefore
required. The choice of λ when performing ridge regression must be in relation to the noise level,
yet over-smoothing must be avoided as oscillations may genuinely originate from smaller scale
phenomena such as e.g. turbulence. To this extent the adaptive sampling routine is nested in a
primary process in which the degree of smoothing is progressively reduced Fig.1a. In the first loop,
values of λ are obtained on the basis of average data scatter (Forrester et al., 2008) and are updated(1)with each new adaptively placed data sample. Once the adaptive sampling process has converged,

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018the second loop is initiated in which a smoothed TPS2 spline is estimated recursively using ageneralised cross-validation (GCV) scheme λGCVas suggested and implemented by Bates et al.

(1987).

(a)(b)

Fig. 1 (a) Illustration of progressive smoothing using ridge regression. The underlying signalf ( x) = 10sin ( 2p x L ) + 2cos ( 20p x L ) is sampled with 50 randomly positioned points superimposed with an

(

)

(

)

uncertainty s i =| 5g cos 100p x L sin 160p x L | where g is a random number between 0 and 1 with uniform
probability. Measurement noise n is defined as ni = 3t with t a random number drawn from a normal probability
idistribution N(0,1). The first iterative value of λ is based on the ensemble average! of data scatter. As the value of

λ is reduced, the underlying surrogate model reflects smaller scale oscillations. The inset shows a close-up of thegraph portion bounded by the rectangle. (b) Adaptive evaluation gridding when sampling Franke's bivariant test
function: f(5(x-0.6),5(y-0.3)). Hollow circles indicate adaptively placed measurement positions. As points are locatedmore densely in regions where the underlying signal varies, the evaluation grid becomes denser.

3. Adaptive sampling objective function
3.1 Curvature dependent sampling
To properly capture the spatial variation in regions of higher signal curvature, the first criterion JC

ensures an increased number of samples in such regions, similar to mesh-refinement heuristics in
Computational Fluid Dynamics (CFD). Curvature has been estimated on the basis of the Laplacian
D of the RBF-based surrogate models, yielding the criterion J L = Ds(x) = å k =1a k Dy ( x - xk ) + Dq(x)N. To compensate for the detrimental effect of stochastic noise, J was combined with a measure forL

local signal change JD =|s-s | where sMAMAconstituted the moving average filtered response

evaluated with a filter of approximately 30% domain width.

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

3.2 Uncertainty-based sampling
The second criterion ensures regions of higher uncertainty are adequately sampled. While eachvariable is assigned one unique value at each sampling location, these values typically correspondto the average of N independent measurements taken at each location. The number of readings N
is typically set to 10,000 to retrieve reliable mean statistics or can be locally adjusted according the
local data scatter to ensure a constant relative error across the domain. Subsequently, the objective
component pertaining error can related to the local data scatter s. The s values at the data points
can also be interpolated using TPS as per section 2 yielding a continuous function J .
E

3.3 Improvement-driven sampling
Sampling locations whose addition does not alter the surrogate model are evidently of low
importance. Conversely, regions where the addition of new data yields a change in the previous
underlying approximant merit further exploration. This thought is captured in the improvement
objective J , quantifying the relative additive information content, W(i ) = Q(i ) (1+ | s(i ) |)-1 whereI

Q(i ) =| s (i ) - s (i -1) | quantifies the model's change at the i iteration . To retain historic model change,th

the improvement factor combines the four most recent()

W's in a weighted average;

J I = å m=0 wi -mW(i -m) . Currently all weights are equal to 0.25. Changes in surrogate model are not3nnecessarily linked to the addition of novel data, but can be due to alterations in the ridgeregression parameter (λ is updated every iteration), uncertainty in the TPS due to uncertainty inthe acquired data or limitations in interpolation accuracy. Only values of Q (x) exceeding the(i)estimators related to the above three effects are considered sufficiently high to be changes in thesurrogate model genuinely caused by the addition of newly accrued data and are taken into
account in the calculation of W .
(i)

3.4 Iteration dependent exploratory sampling
New measurement locations will be prescribed by the objective function. However, each region
within the domain should ideally be sampled at least once to ensure a representative objectivefunction. To this extent oversampling of local extrema in objective function should be avoided. In
this work this is achieved through a separation function J (x) attaining zero values at existinghmeasurement locations and tending to unity away from the sample location. The influence of eachpoint should thus be radially symmetric and finite, making the compactly supported Wendland
functions ideal kernels (Wendland, 2004). Since increasingly smaller scales are envisaged within
the iterative process (Section 2), samples must be allowed to be positioned in closer proximity. This

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

can be achieved by adjusting the Wendland functions' kernel support in function of the changing
fill distance and iteration in ridge regression.
In line with the concept behind domain exploration, it would not be efficient to sample the domain
in sequential regions. Instead, areas of interest must receive a higher number of samples, while
avoiding global under-sampling. This is accounted for by introducing a secondary weighting tothe objective function, particular to the m sample point and dependent on the iteration number i;th3i - jm ö
where j designates the iteration number in which the m sample point was
wi ,m = 1 - æç1 Ni ÷ø+èthm

introduced. The weighting starts from 0 and gradually reaches a value of 1 after N iterations.
iParameter N thus essentially dictates how fast a region can be re-sampled and has been set toi

N =15 in the current study. The cut-off function is defined as (X) =X for X³0 and 0 else. Thei+interpolated form of w , J (x) thus serves as a continuous weighting for the objective function andi,mwwill avoid recently sampled regions from continuously accruing the majority of the samples insubsequent iterations, yet gradually allows further investigation of already explored domain
sections.
3.5 Combined Objective
At each sampling station multiple variables can be measured. In the case of two-dimensional LDA
for example, the velocity magnitudes in two directions is obtained. Each of the N variables willv

therefore have an associated J , J and J . For each variable these elementary functions are combined
c

E

I

as
J o,v (x) = (kC + JC ,v (x) )(k I + J I ,v (x) )(k E + J E ,v (x) )(3)

The empirical offset constants k were set as k =0.5, k =0.1 and k =0.5. The definitive objectiveCfunction is finally given byæ Nv
ö
J O (x) = ç Õ J O ,v (x) ÷ 1 - ( J h (x) )n J w (x)è v =1
ø

(

)

I

E

(4)

4. Sample allocation
The evaluation grid defines the underlying nodes at which the surrogate models, andconsequently the objective function defined in Eq.4, is computed. Also here adaptivity isimplemented by partitioning the global domain into regular blocks and adapting local grid
spacing in each block to the current sample spacing. Local grid spacing is based on the smallest
nearest neighbour distance between measurement samples such that local spatial fluctuations canbe adequately described following the Nyquist sampling criterion. Grid refinement is then applied19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

only if the the new sample spacing will yield a considerable change in number of evaluation
points.
While peaks in the objective function indicate the most suitable sampling locations, such a process
would rapidly lead to clustering of measurement stations. To avoid over-sampling samples areallocated based on local objective function peak geometry. Having defined the objective function,a threshold is defined based on the average of all local maxima to select potential regions forfurther investigation. Regions are sorted on the basis of parameter G serving as a heuristic of thelocal volume and height; Gi = ò J O (x)dx ´ max ( J O (x) ) . Having identified regions in which J

O

xÎWi

Wi

exceeds the threshold value, the N regions associated with the highest G values are selected, whereS

N equals the number of new samples to be distributed. Starting with the region of largest G, eachS()

region is attributed N S ,i = max 1, êGi G ú with G = N1 åi Gi .
ëûS
5. Numerical assessment
In the numerical assessment two numerical fields were considered; a modified version of Franke's
function (Fig.1b) and a lid driven cavity flow at Re=100. In both cases, the initial samples werepositioned on a 7-by-7 structured grid.
Fig. 2 gives an overview of the algorithm performance with respect to the traditional full factorial
sampling (FFS) approach in the analysis of the modified Franke function. As the number ofsamples imposed per iteration N is increased, fewer iterations are needed to reach convergenceS(Fig.2a). The total number of samples increases however in a near linear fashion (Fig.2b). As aresult, the maximum error and standard deviation in the error will initially decrease, reaching
asymptotic values equal for both FFS and adaptive sampling. To juxtapose the "goodness" of theadaptive sampling distribution with equispaced full factorial sampling (spacing is given as éê N -0.5 ùú, a quality heuristic Q was introduced defined as Q = N1 å i =1| Df (xi ) | . High values of Q indicate theNsample distribution to be denser in regions of higher signal curvature, in line with curvaturedependency criterion. The sampling quality is consistently higher for adaptive sampling as the
samples are located in appropriate regions. This can be clearly seen in Fig.3. With N =71 the regions
Sof higher curvature are clearly visible through the heightened sample concentrations. However,these regions become oversampled as the interpolation accuracy improves due to the reduced
sample spacing and additional samples are placed to further enhance the surrogate model (section
3.3). In view of the time required to perform the experiment, a total number of samples O(N)~10

3

is desirable, in which case the typical number of samples to be added each iteration should be lessthan 20. Comparing the error distributions (Fig.4) it can be seen that little is gained going from

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

N =5 to N =11. Both AS and FFS have the largest errors in the regions of largest spatial variance,SS

but already for N =5 AS reaches error levels comparable to those achievable with N =11. It is suchS

S

proposed that N =5 is most suitable.
S

(a)

(b)

(d)

(e)

(c)

Fig. 2 Evolution in (a) number of iterations before convergence, (b) total number of samples N, (c) rms error se
normalised with the maximum value of the underlying signal A , (d) normalised maximum error e /A and (e) Q
o

max

o

heuristic, with imposed samples per iteration N for the case of the modified Franke function (Fig.1b).
S

(a)

(b)

(c)

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

Fig. 3 Sample distributions for the modified Franke function (Fig.1b) (a) N =1, (b) N =5, (c) N =71. Colors indicate theS

SS

reconstructed signal from the data samples.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4 Error distribution across the modified Franke function's domain obtained with (top) adaptive sampling (bottom)
equispaced sampling. (a),(d) N =1; (b),(e) N =5; (c),(f) N =11.
S

S

S

When applied to the lid-driven cavity flow (Fig.5a), the total number of samples rapidly increases
with N (Fig.5b). Once again N =5 is the most appropriate choice. Samples are distributed inS

Sregions of strong gradients and allow proper sampling of the center of the larger central
recirculation region as well as the smaller corner ones (Fig.5d). It should be noted that in this test
case the sampling is based on surrogate models of both the horizontal (U) and vertical (V) velocity
component, although only the U component is depicted herein. Compared to the full factorial
approach, the local error in the reconstructed signal thus improves (Fig.5f vs. c) leading to a
reduction in global error (Fig.5c).
6. Experimental assessment
The final assessment consisted of the application of the adaptive methodology to an experimental
test. The case of an open cavity flow with a length-to-depth (L/H) ratio of 2 was considered at a
Reynolds number Re ~37´10 . Experiments were performed in the open-return low-speed windH3

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018tunnel of the University of Bristol at a flow speed of approximately 10.33m/s and a turbulence
intensity of approximately 2%. 2D2C Particle Image Velocimetry measurements were performed

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 5 (a) Illustration of the lid-driven cavity flow field at Re=100. The colors relate to the horizontal velocity
component. Evolution in (b) total number of samples N and (c) standard deviation in reconstruction error withnumber of imposed samples per iteration N . (d) Distribution of samples imposing N =5. Colors refer to theS

Sreconstructed horizontal velocity component. Spatial distribution in the error in the horizontal displacement
component as per (e) adaptive sampling (f) equispaced sampling, imposing N =5, yielding N~900 - see (b).
Son a cavity 110mm long and 55mm deep using a Dantec Dynamics system consisting of a 200mJ
Litron laser and 4MP FlowSense EO camera. A sketch of the setup and exemplary PIV image
illustrating the location of the cavity walls is provided in Fig.6a. The 2072\times 2072 pixels sensor
2

corresponded to a field of view of 115´115mm and images were analysed using initial
2

interrogation windows of 65´65 pixels , recursively reduced to 17´17 pixels . A mutual window
2

2

overlap of 75% yielded a final vector spacing of approximately 0.2mm. In total 1000 instantaneous
velocity fields were considered in the calculation of the mean and fluctuating velocity components,103

u and v respectively, where u = 1013 å uq ( x) . The extracted PIV data subsequently served as input
q =1

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

into the adaptive sampling process adopting linear data interpolation from the structured PIV data
to the intermediate locations.
The adaptive process was initiated by taking measurements at 49 (7´7) equispaced samples. Each
iteration five new samples (N =5) were added. Typical LDA traverse increments are in the order ofS

0.1mm. In terms of the unit domain, this implies that sample locations must be located on
multiples of 0.1/115. This was taken into account when attributing new data positions. Useful data
can only be extracted within the flow region. To this extent cavity walls are to be excluded when
attributing new sampling locations. This was accounted for by setting the objective function (Eq.4)
to zero within the corresponding regions as evidenced when overlaying the FOV with thepredicted sampling locations (Fig.6b).
The number of consecutive iterations to establish convergence in each of the smoothing phases
through ridge regression was automatically set to 16, 15 and 15 respectively. A higher
concentration of samples is allocated within the cavity compared to the outer flow region. A zoneof recirculation can be noted in the upstream corner of the cavity as visualised by the streamlines
and this region, as well as the upstream and downstream cavity stagnation points are adaptively
allocated more samples. Contrary to the numerical cases, which are void of any fluctuation insignal, the distribution of new samples is now influenced by the local data uncertainty, i.e. J . ThisE

is manifested when samples are superimposed onto the product of local Reynolds stresses u '2 ´ v '2
(Fig.6c). Whereas the adaptive approach yields a consistent reduction in maximum error withincreasing number of samples (Fig.6d), the standard approach shows stronger fluctuations. Not
only does AS attain error levels inferior to those of FF (Fig.6d,e), but, as per the numerical cavity
flow, adding more data does not ensure either the maximum, or the rms error of FF to reduce. On
the contrary, adding samples can actually increase the maximum error as samples are not
optimally placed in flow regions of spatially rapid changes such as the cavity corners. This leads
to poor representations of the flow as manifested in the error distribution plots (Fig.6f-i). In thebulk of the flow where spatial gradients in time-averaged velocity are low, differences between
full factorial and adaptive sampling are marginal. However, in the close vicinity of the cavity walls,
adaptive sampling clearly yields a representation in better agreement with the PIV results.
Both this test case and foregoing numerical assessments corroborate AS to be more reliable andefficient in the measurement of an a-priori unknown flow field19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

(a)

(b)

(d)

(e)

(f)

(c)

(g)

(h)

(i)

Fig. 6 (a) Sketch of the open cavity experimental setup (L =200mm, H=55mm, L=110mm, W=200mm) at Re =37´10 .
oH3

The green square indicates the PIV field-of-view. (b) Contours of the longitudinal velocity component withsuperimposed streamlines (white lines) and adaptively placed sampling positions (grey dots). The dashed black line
represents the dividing streamline issued from (x/L,y/H)=(0,0). (c) Contours of the product between the timeaveraged fluctuations in the horizontal and vertical velocity component. Evolution in (d) maximum and (e) rms error
with total number of samples. Error distribution obtained with adaptive sampling in (f) horizontal and (g) verticalvelocity component compared with PIV. Error distribution as per full factorial sampling in (h) U and (i) V component.

7. Conclusions
With the aim of improving experimental efficiency when surveying e.g. flow regions by means ofpoint-wise metrologies, an adaptive strategy has been proposed to properly sample 2D areas ofinterest in an automated manner. The methodology incorporates sampling criteria based on local

19th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics・LISBON | PORTUGAL ・JULY 16 – 19, 2018

signal curvature, improvement, uncertainty and domain exploration. In addition, initialsmoothing is applied and progressively reduced as to mitigate the effect of local data jitter due tomeasurement noise. The different processes are elaborated within this paper. Thin Plate Splines
finally allow the reconstruction of the surrogate model onto a self-adapting evaluation grid.
Automated stopping criteria have been implemented to imply convergence is reached and thenumerical and computational implications are discussed within the document.
The performance of the adaptive sampling strategy is assessed and juxtaposed with results
obtained by traditional, equispaced sampling (full factorial sampling) based on Franke's function
and the lid driven cavity flow. In addition, both approaches are applied to an open cavity
experiment. All test cases attest regions of interest to be consistently attributed a higher numberof samples through adaptive sampling. The optimum number of samples to add with each
iteration was found to be 5. This yields a drastic reduction (typically factor 2-3) in error compared
to traditional full factorial sampling, independent of the initial number of samples. To reach thesame level of error as the adaptive approach, traditional factorial sampling thus requires at least
twice the number of samples. More importantly, whereas adaptive sampling yields lower error
levels with increasing number of samples, full factorial sampling does not guarantee such atendency. However, without any a-priori knowledge of the flow field, the experimentalist is unable
to determine an appropriate number of samples when adopting equispaced sampling. Traditional
sampling thus demands the experimentalist to frequently re-evaluate the reconstructed signal andaugment the number of samples until presumed convergence is reached. Such a processintrinsically replicates the concept behind adaptive sampling and, combined with the fact that the
automatic stopping criteria implemented have shown to be effective in the assessments provided,
corroborates the adaptive approach to be more reliable and efficient in the measurement of an
unknown signal/flow field.
Acknowledgement
This work was supported by the Engineering and Physical Sciences Research Council Impact
Acceleration Account (Grant number EP/K503824/1).
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