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Adapting quasi-Monte Carlo methods to simulation problems in weighted Korobov spaces

Christian Irrgeher

joint work with G. Leobacher

RICAM Special Semester – Workshop 1 “Uniform distribution and quasi-Monte Carlo methods” October 2013, Linz

Christian Irrgeher (JKU Linz) 1

Problem formulation

◮ Efficient computation of E(g(B))

◮ B . . . standard Brownian motion with index set [0, T] ◮ g . . . suitable function Christian Irrgeher (JKU Linz) 2

Problem formulation

◮ Efficient computation of E(g(B))

◮ B . . . standard Brownian motion with index set [0, T] ◮ g . . . suitable function

◮ Examples in finance, biology, physics,. . . ◮ e.g.: Financial derivative pricing

◮ Gaussian financial market models ◮ European-style options Christian Irrgeher (JKU Linz) 2

Numerical simulation – quasi-Monte Carlo (QMC)

• 1. Discretization

◮ E(g(B)) ≈ E(gd(B T

d , . . . , Bd T d )) = E(fd(X1, . . . , Xd)) =: I(fd)

◮ (X1, . . . , Xd) are independent N(0, 1) Christian Irrgeher (JKU Linz) 3

Numerical simulation – quasi-Monte Carlo (QMC)

• 1. Discretization

◮ E(g(B)) ≈ E(gd(B T

d , . . . , Bd T d )) = E(fd(X1, . . . , Xd)) =: I(fd)

◮ (X1, . . . , Xd) are independent N(0, 1)

• 2. QMC integration

◮ I(fd) ≈ 1

N

N

j=1 fd(xj) =: Qd,N(fd)

◮ {x1, . . . , xN} ⊂ Rd deterministic point set Christian Irrgeher (JKU Linz) 3

Numerical simulation – quasi-Monte Carlo (QMC)

• 1. Discretization

◮ E(g(B)) ≈ E(gd(B T

d , . . . , Bd T d )) = E(fd(X1, . . . , Xd)) =: I(fd)

◮ (X1, . . . , Xd) are independent N(0, 1)

• 2. QMC integration

◮ I(fd) ≈ 1

N

N

j=1 fd(xj) =: Qd,N(fd)

◮ {x1, . . . , xN} ⊂ Rd deterministic point set

◮ Error of QMC algorithm Qd,N

err := |E(g(B)) − Qd,N(fd)|

Christian Irrgeher (JKU Linz) 3

Error estimate

◮ First estimate:

err ≤

• E(g(B)) − I

fd

• +
• I

fd − Qd,N(fd)

• discretization error

integration error

Christian Irrgeher (JKU Linz) 4

Error estimate

◮ First estimate:

err ≤

• E(g(B)) − I

fd

• +
• I

fd − Qd,N(fd)

• discretization error

integration error

◮ Analysis of both errors

◮ emphasis on integration error ◮ but discretization error not negligible Christian Irrgeher (JKU Linz) 4

Discretization error

◮ Discretization (with step size 1/d)

◮ Euler-Maruyama method ◮ Milstein method ◮ . . . Christian Irrgeher (JKU Linz) 5

Discretization error

◮ Discretization (with step size 1/d)

◮ Euler-Maruyama method ◮ Milstein method ◮ . . .

◮ Discretization error

errdisc ≤ c1d−p with convergence rate p > 0 and constant c1 > 0

Christian Irrgeher (JKU Linz) 5

Discretization error

◮ Discretization (with step size 1/d)

◮ Euler-Maruyama method ◮ Milstein method ◮ . . .

◮ Discretization error

errdisc ≤ c1d−p with convergence rate p > 0 and constant c1 > 0

◮ Convergence rate depends on

◮ discretization method ◮ function g Christian Irrgeher (JKU Linz) 5

Gaussian measure and Hermite polynomials

◮ Density of the (standard) Gaussian measure ϕ(x) = 1 √ 2πe− x

⊤x 2 Christian Irrgeher (JKU Linz) 6

Gaussian measure and Hermite polynomials

◮ Density of the (standard) Gaussian measure ϕ(x) = 1 √ 2πe− x

⊤x 2

◮ L2(Rd, ϕ) = {f : Rd −

→ R : f measurable,

• Rd f(x)2ϕ(x)dx < ∞}

Christian Irrgeher (JKU Linz) 6

Gaussian measure and Hermite polynomials

◮ Density of the (standard) Gaussian measure ϕ(x) = 1 √ 2πe− x

⊤x 2

◮ L2(Rd, ϕ) = {f : Rd −

→ R : f measurable,

• Rd f(x)2ϕ(x)dx < ∞}

◮ Univariate Hermite polynomials

Hk(x) = (−1)k √ k! e

x2 2 dk

dxk e− x2

2 Christian Irrgeher (JKU Linz) 6

Gaussian measure and Hermite polynomials

◮ Density of the (standard) Gaussian measure ϕ(x) = 1 √ 2πe− x

⊤x 2

◮ L2(Rd, ϕ) = {f : Rd −

→ R : f measurable,

• Rd f(x)2ϕ(x)dx < ∞}

◮ Univariate Hermite polynomials

Hk(x) = (−1)k √ k! e

x2 2 dk

dxk e− x2

2

◮ Multivariate Hermite polynomials

Hk(x) =

d

• j=1

Hkj(xj)

Christian Irrgeher (JKU Linz) 6

Gaussian measure and Hermite polynomials

◮ Density of the (standard) Gaussian measure ϕ(x) = 1 √ 2πe− x

⊤x 2

◮ L2(Rd, ϕ) = {f : Rd −

→ R : f measurable,

• Rd f(x)2ϕ(x)dx < ∞}

◮ Univariate Hermite polynomials

Hk(x) = (−1)k √ k! e

x2 2 dk

dxk e− x2

2

◮ Multivariate Hermite polynomials

Hk(x) =

d

• j=1

Hkj(xj)

◮ {Hk}k is an ONB of L2(Rd, ϕ)

Christian Irrgeher (JKU Linz) 6

Hermite expansion

◮ Hermite expansion of f ∈ L2(Rd, ϕ)

f(x) =

• k∈Nd

ˆ f(k)Hk(x) in L2

◮ k-th Hermite coefficient ˆ

f(k) =

• Rd f(x)Hk(x)ϕ(x)dx

Christian Irrgeher (JKU Linz) 7

Hermite expansion

◮ Hermite expansion of f ∈ L2(Rd, ϕ)

f(x) =

• k∈Nd

ˆ f(k)Hk(x) in L2

◮ k-th Hermite coefficient ˆ

f(k) =

• Rd f(x)Hk(x)ϕ(x)dx

Theorem

Let f ∈ L2(Rd, ϕ) ∩ C(Rd) and

k∈Nd

0 ˆ

f(k) < ∞. Then f(x) =

• k∈Nd

ˆ f(k)Hk(x) for all x ∈ Rd.

Christian Irrgeher (JKU Linz) 7

Korobov space of functions on R

◮ Let α > 1, γ > 0. Define for k ∈ N0:

r(α, γ, k) :=

• 1

if k = 0 γk−α if k = 0

Christian Irrgeher (JKU Linz) 8

Korobov space of functions on R

◮ Let α > 1, γ > 0. Define for k ∈ N0:

r(α, γ, k) :=

• 1

if k = 0 γk−α if k = 0

◮ Introduce inner product:

f, gα,γ :=

∞

• k=0

r(α, γ, k)−1 ˆ f(k)ˆ g(k)

Christian Irrgeher (JKU Linz) 8

Korobov space of functions on R

◮ Let α > 1, γ > 0. Define for k ∈ N0:

r(α, γ, k) :=

• 1

if k = 0 γk−α if k = 0

◮ Introduce inner product:

f, gα,γ :=

∞

• k=0

r(α, γ, k)−1 ˆ f(k)ˆ g(k)

◮ Corresponding norm: fα,γ =

• f, fα,γ

Christian Irrgeher (JKU Linz) 8

Korobov space of functions on R

◮ Let α > 1, γ > 0. Define for k ∈ N0:

r(α, γ, k) :=

• 1

if k = 0 γk−α if k = 0

◮ Introduce inner product:

f, gα,γ :=

∞

• k=0

r(α, γ, k)−1 ˆ f(k)ˆ g(k)

◮ Corresponding norm: fα,γ =

• f, fα,γ

◮ Function space:

Hα,γ(R, ϕ) := {f ∈ L2(R, ϕ) ∩ C(R) : fα,γ < ∞}

Christian Irrgeher (JKU Linz) 8

Korobov space of functions on R

◮ Hα,γ(R, ϕ) is a reproducing kernel Hilbert space

Christian Irrgeher (JKU Linz) 9

Korobov space of functions on R

◮ Hα,γ(R, ϕ) is a reproducing kernel Hilbert space ◮ Reproducing kernel function Kα,γ : R × R −

→ R

◮ Kα,γ(·, y) ∈ Hα,γ(R, ϕ)

∀y ∈ R

◮ f, Kα,γ(·, y)α,γ = f(y)

∀y ∈ R ∀f ∈ Hα,γ(R, ϕ)

Christian Irrgeher (JKU Linz) 9

Korobov space of functions on R

◮ Hα,γ(R, ϕ) is a reproducing kernel Hilbert space ◮ Reproducing kernel function Kα,γ : R × R −

→ R

◮ Kα,γ(·, y) ∈ Hα,γ(R, ϕ)

∀y ∈ R

◮ f, Kα,γ(·, y)α,γ = f(y)

∀y ∈ R ∀f ∈ Hα,γ(R, ϕ)

◮ Series representation of the reproducing kernel

Kα,γ(x, y) = 1 + γ

∞

• k=1

k−αHk(x)Hk(y)

Christian Irrgeher (JKU Linz) 9

Korobov space of functions on R

◮ There are interesting functions in this space. ◮ Define differential operator D x := d dx − x

Christian Irrgeher (JKU Linz) 10

Korobov space of functions on R

◮ There are interesting functions in this space. ◮ Define differential operator D x := d dx − x

Theorem (I. & Leobacher)

Let β > 2 be an integer and f : R − → R be a β times differentiable function such that (i) Dj

xf(x)ϕ(x)

1 2 ∈ L1(R) for each j ∈ {1, . . . , β} and

(ii) Dj

xf(x) = O

ex2/(2c) as |x| → ∞ for each j ∈ {0, . . . , β − 1} and

some c > 1. Then f ∈ Hα,γ(R, ϕ) with 1 < α < β − 1.

Christian Irrgeher (JKU Linz) 10

Korobov space of functions on Rd

◮ For non-increasing weights γ = (γ1, . . . , γd)

Hα,γ(Rd, ϕ) := Hα,γ1(R, ϕ) ⊗ . . . ⊗ Hα,γd(R, ϕ)

Christian Irrgeher (JKU Linz) 11

Korobov space of functions on Rd

◮ For non-increasing weights γ = (γ1, . . . , γd)

Hα,γ(Rd, ϕ) := Hα,γ1(R, ϕ) ⊗ . . . ⊗ Hα,γd(R, ϕ)

◮ Inner product: f, gα,γ = k∈Nd

0 r(α, γ, k)−1 ˆ

f(k)ˆ g(k) with r(α, γ, k) = d

j=1 r(α, γj, kj)

Christian Irrgeher (JKU Linz) 11

Korobov space of functions on Rd

◮ For non-increasing weights γ = (γ1, . . . , γd)

Hα,γ(Rd, ϕ) := Hα,γ1(R, ϕ) ⊗ . . . ⊗ Hα,γd(R, ϕ)

◮ Inner product: f, gα,γ = k∈Nd

0 r(α, γ, k)−1 ˆ

f(k)ˆ g(k) with r(α, γ, k) = d

j=1 r(α, γj, kj) ◮ Hα,γ(Rd, ϕ) is a weighted reproducing kernel Hilbert space

◮ Reproducing kernel Kα,γ(x, y) = d

j=1 Kα,γj(xj, yj)

◮ Influence of variables is given by the choice of weights γ Christian Irrgeher (JKU Linz) 11

Integration Error

◮ Let fd ∈ Hα,γ(Rd, ϕ).

◮ I(fd) =

• Rd fd(x)ϕ(x)dx =
• Rdfd, Kα,γ(·, x)α,γϕ(x)dx

= fd,

• Rd Kα,γ(·, x)ϕ(x)dxα,γ

◮ Qd,N(fd) = fd, 1

N

N

n=1 Kα,γ(·, xn)α,γ

Christian Irrgeher (JKU Linz) 12

Integration Error

◮ Let fd ∈ Hα,γ(Rd, ϕ).

◮ I(fd) =

• Rd fd(x)ϕ(x)dx =
• Rdfd, Kα,γ(·, x)α,γϕ(x)dx

= fd,

• Rd Kα,γ(·, x)ϕ(x)dxα,γ

◮ Qd,N(fd) = fd, 1

N

N

n=1 Kα,γ(·, xn)α,γ

◮ Integration error

errQMC = |I(fd) − Qd,N(fd)| ≤

• fd
• α,γ
• Rd Kα,γ(·, x)ϕ(x)dx − 1

N

N

• n=1

Kα,γ(·, xn)

• α,γ
• =:ed,N(x1,...,xN)

Christian Irrgeher (JKU Linz) 12

Worst case error of integration

◮ Considering the Gaussian-weighted root-mean-square error for QMC

integration: ¯ ed,N :=

• RdN e2

d,N(x1, . . . , xN)ϕ(x1) . . . ϕ(xN)d(x1, . . . , xN)

1

2 Christian Irrgeher (JKU Linz) 13

Worst case error of integration

◮ Considering the Gaussian-weighted root-mean-square error for QMC

integration: ¯ ed,N :=

• RdN e2

d,N(x1, . . . , xN)ϕ(x1) . . . ϕ(xN)d(x1, . . . , xN)

1

2

Theorem (I. & Leobacher)

The Gaussian-weighted root-mean-square error for QMC integration in the Korobov space Hα,γ(Rd, ϕ) is ¯ ed,N = 1 √ N

 

d

• j=1

(1 + γjζ(α)) − 1

 

1 2

.

Christian Irrgeher (JKU Linz) 13

Worst case error of integration and tractability

◮ There exists a point set {x1, . . . , xN} such that

ed,N ≤ 1 √ N exp

ζ(α)

2

d

• j=1

γj

• Christian Irrgeher (JKU Linz)

14

Worst case error of integration and tractability

◮ There exists a point set {x1, . . . , xN} such that

ed,N ≤ 1 √ N exp

ζ(α)

2

d

• j=1

γj

• ◮ Strong tractability, if lim supd

d

j=1 γj < ∞.

Christian Irrgeher (JKU Linz) 14

Worst case error of integration and tractability

◮ There exists a point set {x1, . . . , xN} such that

ed,N ≤ 1 √ N exp

ζ(α)

2

d

• j=1

γj

• ◮ Strong tractability, if lim supd

d

j=1 γj < ∞. ◮ Polynomial tractability, if lim supd

d

j=1 γj

ln(d)

< ∞.

Christian Irrgeher (JKU Linz) 14

Worst case error of integration and tractability

◮ There exists a point set {x1, . . . , xN} such that

ed,N ≤ 1 √ N exp

ζ(α)

2

d

• j=1

γj

• ◮ Strong tractability, if lim supd

d

j=1 γj < ∞. ◮ Polynomial tractability, if lim supd

d

j=1 γj

ln(d)

< ∞.

◮ err ≤ c1d−p +

• fd
• α,γc2N−1/2dq

◮

fd

• α,γ can grow in d

Christian Irrgeher (JKU Linz) 14

Orthogonal transforms

◮ For any orthogonal transform U on Rd:

• Rd fd(x)ϕ(x)dx =
• Rd fd(Ux)ϕ(x)dx

Christian Irrgeher (JKU Linz) 15

Orthogonal transforms

◮ For any orthogonal transform U on Rd:

• Rd fd(x)ϕ(x)dx =
• Rd fd(Ux)ϕ(x)dx

◮ Every orthogonal transform corresponds to an Brownian path

construction (Papageorgiou 2002)

Christian Irrgeher (JKU Linz) 15

Orthogonal transforms

◮ For any orthogonal transform U on Rd:

• Rd fd(x)ϕ(x)dx =
• Rd fd(Ux)ϕ(x)dx

◮ Every orthogonal transform corresponds to an Brownian path

construction (Papageorgiou 2002)

◮ Classical construction methods

◮ Forward construction ◮ Brownian bridge construction ◮ PCA construction Christian Irrgeher (JKU Linz) 15

Orthogonal transforms

◮ For any orthogonal transform U on Rd:

• Rd fd(x)ϕ(x)dx =
• Rd fd(Ux)ϕ(x)dx

◮ Every orthogonal transform corresponds to an Brownian path

construction (Papageorgiou 2002)

◮ Classical construction methods

◮ Forward construction ◮ Brownian bridge construction ◮ PCA construction

◮ Equivalence principle (Wang & Sloan 2011)

◮ Roughly spoken: every construction that is good for one function is

bad for another

Christian Irrgeher (JKU Linz) 15

Orthogonal transforms

◮ Bound of the integration error:

errQMC ≤

• fd ◦ U
• α,γed,N

Christian Irrgeher (JKU Linz) 16

Orthogonal transforms

◮ Bound of the integration error:

errQMC ≤

• fd ◦ U
• α,γed,N

◮ In general,

• fd
• α,γ =
• fd ◦ U
• α,γ

Christian Irrgeher (JKU Linz) 16

Orthogonal transforms

◮ Bound of the integration error:

errQMC ≤

• fd ◦ U
• α,γed,N

◮ In general,

• fd
• α,γ =
• fd ◦ U
• α,γ

◮ Goal: Find U such that fd ◦ Uα,γ grows slower in d than fdα,γ or

is even bounded.

◮ LT-method (Imai & Tan 2007) ◮ Regression algorithm (I. & Leobacher 2012) Christian Irrgeher (JKU Linz) 16

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using forward method

Christian Irrgeher (JKU Linz) 17

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using forward method ◮ Discrete problem fd(x) = exp( 1 √ d

d

j=1 xj)

Christian Irrgeher (JKU Linz) 17

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using forward method ◮ Discrete problem fd(x) = exp( 1 √ d

d

j=1 xj) ◮ Hermite coefficients of fd:

• fd(k) = √e d−|k|/2

√ k!

Christian Irrgeher (JKU Linz) 17

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using forward method ◮ Discrete problem fd(x) = exp( 1 √ d

d

j=1 xj) ◮ Hermite coefficients of fd:

• fd(k) = √e d−|k|/2

√ k!

◮ fd2 α,γ = k∈Nd

0 r(α, γ, k)−1

fd(k)2 ≈ exp(1 + 1

d

d

j=1 γ−1 j )

Christian Irrgeher (JKU Linz) 17

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using forward method ◮ Discrete problem fd(x) = exp( 1 √ d

d

j=1 xj) ◮ Hermite coefficients of fd:

• fd(k) = √e d−|k|/2

√ k!

◮ fd2 α,γ = k∈Nd

0 r(α, γ, k)−1

fd(k)2 ≈ exp(1 + 1

d

d

j=1 γ−1 j ) ◮ If γj = j−2,

◮ ed,N is bounded by cN −1/2 with constant c > 0, ◮ But fdα,γ = O(ed) Christian Irrgeher (JKU Linz) 17

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using Brownian bridge

construction

Christian Irrgeher (JKU Linz) 18

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using Brownian bridge

construction

◮ Discrete problem (fd ◦ U)(x) = exp(x1)

Christian Irrgeher (JKU Linz) 18

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using Brownian bridge

construction

◮ Discrete problem (fd ◦ U)(x) = exp(x1) ◮ Hermite coefficients of fd ◦ U:

• fd ◦ U(k) =

  

√e

1 √k1!

if k2 = . . . = kd = 0 else

Christian Irrgeher (JKU Linz) 18

Example

Compute E(exp(B1))

◮ Simulation of B on time grid 1 d, 2 d, . . . , d d using Brownian bridge

construction

◮ Discrete problem (fd ◦ U)(x) = exp(x1) ◮ Hermite coefficients of fd ◦ U:

• fd ◦ U(k) =

  

√e

1 √k1!

if k2 = . . . = kd = 0 else

◮ Norm is independent of d

fd ◦ U2

α,γ =

• k∈Nd

r(α, γ, k) fd ◦ U(k)2 = e +

∞

• k=1

γ−1

1 kα 1

e k1! = e + γ−1

1 e2c(α)

Christian Irrgeher (JKU Linz) 18

Hermite coefficients and orthogonal transforms

◮ Define:

◮ AU : L2(Rd, ϕ) −

→ L2(Rd, ϕ) by AUf = f ◦ U

◮ Hm := span{Hk(x) : |k| = m} ◮ AU,m := AU

• Hm

Christian Irrgeher (JKU Linz) 19

Hermite coefficients and orthogonal transforms

◮ Define:

◮ AU : L2(Rd, ϕ) −

→ L2(Rd, ϕ) by AUf = f ◦ U

◮ Hm := span{Hk(x) : |k| = m} ◮ AU,m := AU

• Hm

◮ Then:

◮ AU is a Hilbert space automorphism of L2(Rd, ϕ) ◮ AU,m is a Hilbert space automorphism of Hm ◮ L2(Rd, ϕ) =

m≥0 Hm

◮ AU =

m≥0 AU,m

Christian Irrgeher (JKU Linz) 19

Hermite coefficients and orthogonal transforms in Hm

◮ Hk(x) = ∂|k| ∂tk G(x, t)

• t=0 with exponential generating function G

Christian Irrgeher (JKU Linz) 20

Hermite coefficients and orthogonal transforms in Hm

◮ Hk(x) = ∂|k| ∂tk G(x, t)

• t=0 with exponential generating function G

◮ For any (β1, . . . , βm) ∈ {1, . . . , d}m:

∂ ∂tβm · · · ∂ ∂tβ1 G(Ux, t)

• t=0 =

=

d

• ξ1,...,ξm=1

Uβmξm · · · Uβ1ξ1

• ∂

∂tξm · · · ∂ ∂tξ1 G(x, t)

• t=0
• Christian Irrgeher (JKU Linz)

20

Hermite coefficients and orthogonal transforms in Hm

◮ Hk(x) = ∂|k| ∂tk G(x, t)

• t=0 with exponential generating function G

◮ For any (β1, . . . , βm) ∈ {1, . . . , d}m:

∂ ∂tβm · · · ∂ ∂tβ1 G(Ux, t)

• t=0 =

=

d

• ξ1,...,ξm=1

Uβmξm · · · Uβ1ξ1

• ∂

∂tξm · · · ∂ ∂tξ1 G(x, t)

• t=0
• ◮ Many (β1, . . . , βm) correspond to the same multi-index k

Christian Irrgeher (JKU Linz) 20

Hermite coefficients and orthogonal transforms in Hm

◮ Space Km := ℓ2({1, . . . , d}m)

◮ takes account of the order of differentiation Christian Irrgeher (JKU Linz) 21

Hermite coefficients and orthogonal transforms in Hm

◮ Space Km := ℓ2({1, . . . , d}m)

◮ takes account of the order of differentiation

◮ Isometries:

◮ Jm : Hm −

→ Km

◮ J⊤

m : Km −

→ Hm

Christian Irrgeher (JKU Linz) 21

Hermite coefficients and orthogonal transforms in Hm

◮ Space Km := ℓ2({1, . . . , d}m)

◮ takes account of the order of differentiation

◮ Isometries:

◮ Jm : Hm −

→ Km

◮ J⊤

m : Km −

→ Hm

Hm

AU,m

− − − − − → Hm

Jm

 

• 

J⊤

m

Km − − − − →

U⊗m

Km

Christian Irrgeher (JKU Linz) 21

Hermite coefficients and orthogonal transforms in L2

L2(Rd, ϕ)

AU

− − − − − − − − → L2(Rd, ϕ)

J

 

• 

J⊤

K − − − − − − − − − →

• m≥0 U⊗m

K with K =

• m≥0

Km, J =

• m≥0

Jm, J⊤ =

• m≥0

J⊤

m.

Christian Irrgeher (JKU Linz) 22

Hermite coefficients and orthogonal transforms in L2

L2(Rd, ϕ)

AU

− − − − − − − − → L2(Rd, ϕ)

J

 

• 

J⊤

K − − − − − − − − − →

• m≥0 U⊗m

K with K =

• m≥0

Km, J =

• m≥0

Jm, J⊤ =

• m≥0

J⊤

m. ◮ Thus A = J⊤ m≥0 U⊗m

J

Christian Irrgeher (JKU Linz) 22

Remarks and open problems

◮ Is AU well defined on the Korobov space? ◮ Other weight structures (choice of weights) ◮ Sobolev space setting (RKHS, tractability) ◮ Concrete point sets (worst case error analysis, tractability)

Christian Irrgeher (JKU Linz) 23

Thank you for your attention

Christian Irrgeher (JKU Linz) 24