Concienciación del Uso de los Plaguicidas

In this section, we will assume that a fixed experimental design {yl}L_l=1and the

corresponding set of model evaluations {g (yl)}L_l=1are available. Accordingly, the

computational costs attributed to the number of model evaluations are now fixed and equal to L. Given an oversampling coefficient C such that L ≥ C M, where M denotes the number of approximation terms, equivalently, the size of the gPC polynomial basis, a TP or TD basis of suitable size can be constructed. However, the TP and TD bases defined by the multi-index sets ΛTP

pmaxin (4.11) and Λ TD

pmaxin (4.12),

respectively, are isotropic, i.e. the same maximum polynomial degree is considered for every parameter. As in the dimension-adaptive stochastic collocation case, we

can exploit parameter anisotropies and construct adaptively a multi-index set Λ and the corresponding anisotropic gPC basis and approximation. Assuming that the QoI is not equally sensitive to all parameters, the anisotropic gPC expansion will be more accurate than its isotropic TP or TD counterparts for the same number of approximation terms.

Similarly to Section 3.3, we will employ a dimension-adaptive approach based on a sequence of nested, downward-closed (see Definition 2.4) multi-index sets (Λk)_k≥0, such that Λ0 ⊂ Λ1⊂ · · · ⊂ Λk ⊂ · · · . At every iteration, the multi-index

set Λkis expanded by exactly one multi-indexpk+1∈ Λadmk , such that Λk+1= Λk∪

{pk+1}. Expanding the multi-index set with multiple coefficients at each step has

also been considered in [102]. The multi-indexp_k+1 is chosen to be the one with the maximum contribution to the already available gPC approximation among all admissible multi-indicesp ∈ Λadm

k . The contribution indicators ηp are now given

by the absolute values of the admissible gPC coefficients, such that ηp :=

 sp

 ,

p ∈ Λadm

k . The use of the gPC coefficients as contribution indicators is motivated

by the fact that they are expected to decay quickly for smooth QoIs [115]. We consider two different approaches for computing the admissible gPC coefficients, detailed below.

We will refer to the first approach as the “all-in” approach, due to the fact that the gPC coefficients are computed by solving the LS problem (4.18) using a polynomial basisΨp

p∈ΛLS_k , where ΛLSk = Λk∪ Λadmk , i.e. the LS problem is solved including all

admissible multi-indicesp ∈ Λadm

k . Then, the multi-indexpk+1is chosen as

p_{k+1= arg max} p∈Λadm_k ηp= arg max p∈Λadm_k  sp   , (4.24)

wheresp,p ∈ Λadmk , are the coefficients resulting from the solution of the regres-

sion problem (4.18). This procedure is continued iteratively until a number of LS unknowns_{M, equivalently, a gPC basis cardinality #Λ = M has been reached, such} that_{C M ≥ L, or until the maximum or total contribution of the admissible set Λ}adm

k

is below a specified tolerance ε. The adaptive basis construction with the all-in ap- proach is shown in Algorithm 4.1. After the termination of the algorithm, the gPC approximation is constructed using the multi-index set Λ = Λk∪ Λadmk in order to

include the already computed contributions of the admissible multi-indices. The second approach will be referred to as the “one-to-one” approach, because now the LS problem (4.18) is solved once per admissible multi-index p ∈ Λadm

k ,

each time using the multi-index set ΛLS

k = Λk∪{p}, p ∈ Λadmk , and the corresponding

polynomial basisΨ_{p p∈Λ}LS

k . Each one of the LS solutions yields a gPC coefficient

s_p,p ∈ Λadm

Algorithm 4.1:Adaptive gPC polynomial basis construction with the “all-in”

approach.

Data:mapg (y), initial multi-index set Λ₀, tolerance ε, experimental design

{yl, g (yl)}L_l=1, oversampling coefficientC

Result:multi-index set Λ, gPC polynomial basisΨ_{p p∈Λ}, gPC coefficients 

s_{p p∈Λ}

1 k = 0

2 whileTRUEdo

3 Compute admissible set Λadm_k :=p ∈ Λ+_k : p /∈ Λ_k and {p}−⊂ Λ_k . 4 Construct LS multi-index set ΛLS_k = Λ_k∪ Λadm_k and polynomial basis



Ψ_{p p∈Λ}LS

k .

5 UsingΨ_{p p∈Λ}LS

k , compute the gPC coefficients by solving

s = arg min_ˆs∈RMkDˆs − gk₂.

6 Compute contribution indicators η_p=

 sp

, ∀p ∈ Λadm

k .

7 Compute current number of LS unknownsM_k= #ΛLS_k = # Λ_k∪ Λadm_k

.

8 Compute current total (or maximum) contribution εk=

P p∈Λadm k ηp  or εk= max_p∈Λadm k ηp  . 9 ifC M_k≥ LORεk≤ ε then 10 break loop 11 end

12 Find new multi-indexp_k+1= arg max_p∈Λadm

k ηp.

13 Update activated set Λ_k+1= Λ_k∪ {p_k+1}. 14 k = k + 1

15 end

16 Construct the gPC polynomial basisΨ_{p p∈Λ}and compute the gPC coefficients

 sp

p∈Λ for Λ = Λk∪ Λadmk .

that Λ_{k+1= Λ}k∪ {pk+1}, where pk+1is given by (4.24), until the same termination

criteria are met. The one-to-one procedure is depicted in Algorithm 4.2. Contrary to the all-in case, we cannot consider the multi-index set Λ = Λk∪ Λadmk after

the termination of the algorithm, due to the fact that the size of the experimental design is not sufficient for computing the corresponding LS problem. Compared to the all-in approach, the one-to-one algorithm requires the solution of a significantly larger number of LS problems, however, of smaller size.

Algorithm 4.2:Adaptive gPC polynomial basis construction with the “one-to-

one” approach.

Data:mapg (y), initial multi-index set Λ₀, tolerance ε, experimental design

{yl, g (yl)}L_l=1, oversampling coefficientC

Result:multi-index set Λ, gPC polynomial basisΨ_{p p∈Λ}, gPC coefficients 

s_{p p∈Λ}

1 k = 0

2 whileTRUEdo

3 Compute admissible set Λadm_k :=p ∈ Λ+_k : p /∈ Λ_k and {p}−⊂ Λ_k . 4 forEVERY MULTI-INDEXp ∈ Λadm_k do

5 Construct LS multi-index set ΛLS_k = Λ_k∪ {p} and polynomial basis



Ψ_{p p∈Λ}LS

k .

6 UsingΨ_{p p∈Λ}LS

k , compute the gPC coefficients by solving

s = arg min_ˆs∈RMkDˆs − gk₂.

7 end

8 Compute contribution indicator η_p=

 sp

, ∀p ∈ Λadm

k .

9 Compute current number of LS unknownsM_k= #ΛLS_k = #Λ_k+ 1. 10 Compute current total (or maximum) contribution ε_k=P_p∈Λadm

k ηp  or ε_{k= maxp∈Λ}adm k ηp  . 11 ifC M_k≥ LORεk≤ ε then 12 break loop 13 end

14 Find new multi-indexp_k+1= arg max_p∈Λadm

k ηp.

15 Update activated set Λ_k+1= Λ_k∪ {p_k+1}. 16 k = k + 1

17 end

18 Construct the gPC polynomial basisΨ_{p p∈Λ}and compute the gPC coefficients



s_{p p∈Λ} for Λ = Λk.