I made public for discussion a first version (subject to changes) of a talk that will be given at the Particles 2019 conference. The bold points of this presentation are the following : 1) We present new to my knowledge, sharp, estimations for Monte-Carlo type methods. 2) These estimations can be used in a wide variety of context to perform a sharp error analysis. 3) We present a class of numerical methods that we refer to as Transported Meshfree Methods. This class of methods can be used for a wide variety of problems based on Partial Differential Equations, among which Artificial Intelligence problems belongs. 4) We can guarantee, thanks to the error analysis, a worst-error while computing with transported meshfree methods. We can also check that this error matches optimal convergence rate.

1. Integration with Kernel Methods, Transported
Meshfree methods.
P.G. LeFloch 1, J.M. Mercier 2
1CNRS, 2MPG-Partners
19 10 2019
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 1 / 8

2. Local integration with kernel methods
Monte Carlo estimations - consider the following error estimation (µ probability
measure, Y = (y1
, . . . , yN
) ∈ RN×D
)
RD
ϕ(x)dµ −
1
N
N
n=1
ϕ(yn
) ≤ E Y , HKµ ϕ HKµ
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 2 / 8

3. Local integration with kernel methods
Monte Carlo estimations - consider the following error estimation (µ probability
measure, Y = (y1
, . . . , yN
) ∈ RN×D
)
RD
ϕ(x)dµ −
1
N
N
n=1
ϕ(yn
) ≤ E Y , HKµ ϕ HKµ
1 example 1 : Y i.i.d. → E(Y , HKµ ) ∼ 1√
N
and HKµ ∼ L2
(RD
, |x|2
dµ) (stat.).
2 ex 1 : Y SOBOL, µ = dxΩ, Ω = [0, 1]D
. Then HK = BV (Ω) (bounded variations),
E(Y , HK ) ≥ ln(N)D−1
N
( Koksma-Hlawka sharp estimate conjecture).
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 2 / 8

4. Local integration with kernel methods
Monte Carlo estimations - consider the following error estimation (µ probability
measure, Y = (y1
, . . . , yN
) ∈ RN×D
)
RD
ϕ(x)dµ −
1
N
N
n=1
ϕ(yn
) ≤ E Y , HKµ ϕ HKµ
1 example 1 : Y i.i.d. → E(Y , HKµ ) ∼ 1√
N
and HKµ ∼ L2
(RD
, |x|2
dµ) (stat.).
2 ex 1 : Y SOBOL, µ = dxΩ, Ω = [0, 1]D
. Then HK = BV (Ω) (bounded variations),
E(Y , HK ) ≥ ln(N)D−1
N
( Koksma-Hlawka sharp estimate conjecture).
3 Others examples ...quantifiers, wavelet, deep feed-forward neural networks ...
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 2 / 8

5. Local integration with kernel methods
Monte Carlo estimations - consider the following error estimation (µ probability
measure, Y = (y1
, . . . , yN
) ∈ RN×D
)
RD
ϕ(x)dµ −
1
N
N
n=1
ϕ(yn
) ≤ E Y , HKµ ϕ HKµ
1 example 1 : Y i.i.d. → E(Y , HKµ ) ∼ 1√
N
and HKµ ∼ L2
(RD
, |x|2
dµ) (stat.).
2 ex 1 : Y SOBOL, µ = dxΩ, Ω = [0, 1]D
. Then HK = BV (Ω) (bounded variations),
E(Y , HK ) ≥ ln(N)D−1
N
( Koksma-Hlawka sharp estimate conjecture).
3 Others examples ...quantifiers, wavelet, deep feed-forward neural networks ...
4 A general method Let K(x, y), an admissible kernel, then
E2
(Y , HKµ ) =
R2D
K(x, y)dµx dµy +
1
N2
N
n,m=1
K(yn
, ym
)−
2
N
N
n=1 RD
K(x, yn
)dµx
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 2 / 8

6. Local integration with kernel methods
Monte Carlo estimations - consider the following error estimation (µ probability
measure, Y = (y1
, . . . , yN
) ∈ RN×D
)
RD
ϕ(x)dµ −
1
N
N
n=1
ϕ(yn
) ≤ E Y , HKµ ϕ HKµ
1 example 1 : Y i.i.d. → E(Y , HKµ ) ∼ 1√
N
and HKµ ∼ L2
(RD
, |x|2
dµ) (stat.).
2 ex 1 : Y SOBOL, µ = dxΩ, Ω = [0, 1]D
. Then HK = BV (Ω) (bounded variations),
E(Y , HK ) ≥ ln(N)D−1
N
( Koksma-Hlawka sharp estimate conjecture).
3 Others examples ...quantifiers, wavelet, deep feed-forward neural networks ...
4 A general method Let K(x, y), an admissible kernel, then
E2
(Y , HKµ ) =
R2D
K(x, y)dµx dµy +
1
N2
N
n,m=1
K(yn
, ym
)−
2
N
N
n=1 RD
K(x, yn
)dµx
5 We can compute sharp discrepancy sequences and optimal discrepancy error as
Y = arg inf
Y ∈RD×N
E(Y , HKµ ), EHKµ
(N, D) = E(Y , HKµ )
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 2 / 8

7. Our local kernels : lattice-based and transported kernels
Without loss of generality, consider µ = dx[0,1]D . We use two kind of kernels :
1 Lattice-based kernel : Let L a Lattice, L∗
its dual Lattice. Consider any discrete
function satisfying φ(α∗
) ∈ 1
(L∗
), φ(α∗
) ≥ 0, φ(0) = 1.
Kper (x, y) =
1
|L|
α∗∈L∗
φ(α∗
) exp2iπ<x−y,α∗
>
x
y
z
Matern
x
y
k
Multiquadric
x
y
k
Gaussian
x
y
k
Truncated
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 3 / 8

8. Our local kernels : lattice-based and transported kernels
Without loss of generality, consider µ = dx[0,1]D . We use two kind of kernels :
1 Lattice-based kernel : Let L a Lattice, L∗
its dual Lattice. Consider any discrete
function satisfying φ(α∗
) ∈ 1
(L∗
), φ(α∗
) ≥ 0, φ(0) = 1.
Kper (x, y) =
1
|L|
α∗∈L∗
φ(α∗
) exp2iπ<x−y,α∗
>
x
y
z
Matern
x
y
k
Multiquadric
x
y
k
Gaussian
x
y
k
Truncated
2 Transported kernel : S : Ω → RD
a transport map. Ktra(x, y) = K(S(x), S(y)).
x
y
k
Matern
x
y
k
Gaussian
x
y
k
Multiquadric
x
y
k
Truncated
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 3 / 8

9. An example : Monte-Carlo integration with Matern kernel
1 Kernel, random and computed sequences Y . N=256,D=2.
x
y
z
Matern
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
random points
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
computed points for lattice Matern
x
y
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 4 / 8

10. An example : Monte-Carlo integration with Matern kernel
1 Kernel, random and computed sequences Y . N=256,D=2.
x
y
z
Matern
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
random points
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
computed points for lattice Matern
x
y
2 Optimal discrepancy error → Koksma-Hlavka type estimate
EHK (N, D) ∼ n>N
φ(α∗n)
N
∼
ln(N)D−1
N
, φ(α) = ΠD
d=1
2
1 + 4π2α2
d /τ2
D
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 4 / 8

11. An example : Monte-Carlo integration with Matern kernel
1 Kernel, random and computed sequences Y . N=256,D=2.
x
y
z
Matern
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
random points
x
y
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
computed points for lattice Matern
x
y
2 Optimal discrepancy error → Koksma-Hlavka type estimate
EHK (N, D) ∼ n>N
φ(α∗n)
N
∼
ln(N)D−1
N
, φ(α) = ΠD
d=1
2
1 + 4π2α2
d /τ2
D
3 E(Y , HK ) random – vs E(Y , HK ) computed - vs theoretical EHK (N, D)
D=1 D=16 D=128
N=16 0.228 0.304 0.319
N=128 0.117 0.111 0.115
N=512 0.035 0.054 0.059
D=1 D=16 D=128
N=16 0.062 0.211 0.223
N=128 0.008 0.069 0.077
N=512 0.002 0.034 0.049
D=1 D=16 D=128
N=16 0.062 0.288 0.323
N=128 0.008 0.077 0.105
N=512 0.002 0.034 0.043
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 4 / 8

12. Application : Transported Meshfree Methods (TMM)
If µ(t, x) solution to non-linear hyperbolic-parabolic Fokker-Planck equations
∂t µ − Lµ = 0, L = · (bµ) + 2
· (Aµ), A :=
1
2
σσT
1 FORWARD : Compute µ(t) ∼ 1
N
δy1(t) + . . . + δyN (t) as best discrepancy
sequences
RD
ϕ(x)dµ(t, x) −
1
N
N
n=1
ϕ(yn
(t)) ≤ E Y (t), HKµ(t,·)
ϕ HKµ(t,·)
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 5 / 8

13. Application : Transported Meshfree Methods (TMM)
If µ(t, x) solution to non-linear hyperbolic-parabolic Fokker-Planck equations
∂t µ − Lµ = 0, L = · (bµ) + 2
· (Aµ), A :=
1
2
σσT
1 FORWARD : Compute µ(t) ∼ 1
N
δy1(t) + . . . + δyN (t) as best discrepancy
sequences
RD
ϕ(x)dµ(t, x) −
1
N
N
n=1
ϕ(yn
(t)) ≤ E Y (t), HKµ(t,·)
ϕ HKµ(t,·)
2 CHECK optimal rate : E Y (t), HKµ(t,·)
∼ EHKµ(t)
(N, D)
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 5 / 8

14. Application : Transported Meshfree Methods (TMM)
If µ(t, x) solution to non-linear hyperbolic-parabolic Fokker-Planck equations
∂t µ − Lµ = 0, L = · (bµ) + 2
· (Aµ), A :=
1
2
σσT
1 FORWARD : Compute µ(t) ∼ 1
N
δy1(t) + . . . + δyN (t) as best discrepancy
sequences
RD
ϕ(x)dµ(t, x) −
1
N
N
n=1
ϕ(yn
(t)) ≤ E Y (t), HKµ(t,·)
ϕ HKµ(t,·)
2 CHECK optimal rate : E Y (t), HKµ(t,·)
∼ EHKµ(t)
(N, D)
3 BACKWARD : interpret t → yn
(t), n = 1 . . . N as a moving, transported, PDE
grid. Solve with it the Kolmogorov equation. ERROR ESTIMATION :
RD
P(t, ·)dµ(t, ·) −
1
N
N
n=1
P(t, yn
(t)) ≤ EHKµ(t)
(N, D) P(t, ·) HKµ(t,·)
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 5 / 8

15. Application : Transported Meshfree Methods (TMM)
If µ(t, x) solution to non-linear hyperbolic-parabolic Fokker-Planck equations
∂t µ − Lµ = 0, L = · (bµ) + 2
· (Aµ), A :=
1
2
σσT
1 FORWARD : Compute µ(t) ∼ 1
N
δy1(t) + . . . + δyN (t) as best discrepancy
sequences
RD
ϕ(x)dµ(t, x) −
1
N
N
n=1
ϕ(yn
(t)) ≤ E Y (t), HKµ(t,·)
ϕ HKµ(t,·)
2 CHECK optimal rate : E Y (t), HKµ(t,·)
∼ EHKµ(t)
(N, D)
3 BACKWARD : interpret t → yn
(t), n = 1 . . . N as a moving, transported, PDE
grid. Solve with it the Kolmogorov equation. ERROR ESTIMATION :
RD
P(t, ·)dµ(t, ·) −
1
N
N
n=1
P(t, yn
(t)) ≤ EHKµ(t)
(N, D) P(t, ·) HKµ(t,·)
4 HYPERBOLIC CASE (σ ≡ 0) : LAGRANGIAN MESHFREE METHODS.
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 5 / 8

16. Illustration : the 2D SABR process, widely used in Finance
SABR process d
Ft
αt
= ρ
αt Fβ
t 0
0 ναt
dW 1
t
dW 2
t
, with 0 ≤ β ≤ 1,
ν ≥ 0,ρ ∈ R2×2
. The Fokker-Planck equation associated to SABR is
∂t µ + L∗
µ = 0, L∗
µ = ρ
x2
2
2
xβ
1 0
0 ν2
2
x2
ρT
· 2
µ.
(Loading SABR200)
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 6 / 8

17. academic tests, business cases, curse of dimensionality
1 Academic tests : finance, non-linear hyperbolic systems
1 Revisiting the method of characteristics via a convex hull algorithm
2 Numerical results using CoDeFi
3 A new method for solving Kolmogorov equations in mathematical finance
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 7 / 8

18. academic tests, business cases, curse of dimensionality
1 Academic tests : finance, non-linear hyperbolic systems
1 Revisiting the method of characteristics via a convex hull algorithm
2 Numerical results using CoDeFi
3 A new method for solving Kolmogorov equations in mathematical finance
2 Business cases : finance
1 Hedging Strategies for Net Interest Income and Economic Values of
Equity (http://dx.doi.org/10.2139/ssrn.3454813, with S.Miryusupov).
2 Compute metrics for big portfolio of Autocalls depending on several
underlyings (unpublished).
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 7 / 8

19. academic tests, business cases, curse of dimensionality
1 Academic tests : finance, non-linear hyperbolic systems
1 Revisiting the method of characteristics via a convex hull algorithm
2 Numerical results using CoDeFi
3 A new method for solving Kolmogorov equations in mathematical finance
2 Business cases : finance
1 Hedging Strategies for Net Interest Income and Economic Values of
Equity (http://dx.doi.org/10.2139/ssrn.3454813, with S.Miryusupov).
2 Compute metrics for big portfolio of Autocalls depending on several
underlyings (unpublished).
3 CURSE of dimensionality in finance : Price and manage a complex option
written on several underlyings. Result ? We can solve at any order of accuracy :
RD
P(t, ·)dµ(t, ·) −
1
N
N
n=1
P(t, yn
(t)) ≤
P(t, ·) HKµ(t,·)
Nα
...
...where α ≥ 1/2 is ANY NUMBER ! Choose it according to your desired
electricity bill ! BUT ...beware to low-regularity problems if the dimension D is too
big (e.g. american-type options or Autocall)
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 7 / 8

20. Summary and Conclusions
We presented in this talk:
1 News, sharps, estimations for Monte-Carlo methods...
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 8 / 8

21. Summary and Conclusions
We presented in this talk:
1 News, sharps, estimations for Monte-Carlo methods...
2 ...that can be used in a wide variety of context to perform a sharp
error analysis.
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 8 / 8

22. Summary and Conclusions
We presented in this talk:
1 News, sharps, estimations for Monte-Carlo methods...
2 ...that can be used in a wide variety of context to perform a sharp
error analysis.
3 A new method for numerical simulations of PDE : Transported
Meshfree Methods....
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 8 / 8

23. Summary and Conclusions
We presented in this talk:
1 News, sharps, estimations for Monte-Carlo methods...
2 ...that can be used in a wide variety of context to perform a sharp
error analysis.
3 A new method for numerical simulations of PDE : Transported
Meshfree Methods....
4 ... that can be used in a wide variety of applications (hyperbolic /
parabolic equations, artificial intelligence, etc...)...
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 8 / 8

24. Summary and Conclusions
We presented in this talk:
1 News, sharps, estimations for Monte-Carlo methods...
2 ...that can be used in a wide variety of context to perform a sharp
error analysis.
3 A new method for numerical simulations of PDE : Transported
Meshfree Methods....
4 ... that can be used in a wide variety of applications (hyperbolic /
parabolic equations, artificial intelligence, etc...)...
5 ..for which the error analysis applies : we can guarantee a worst
error estimation, and we can check that this error matches an
optimal convergence rate.
P.G. LeFloch 1
, J.M. Mercier 2
(1
CNRS, 2
MPG-Partners)Integration with Kernel Methods, Transported Meshfree methods. 19 10 2019 8 / 8