quasi-monte carlo algorithms with applications in ... · quasi-monte carlo algorithms with...
Embed Size (px)
TRANSCRIPT
Quasi-Monte Carlo Algorithms withApplications in Numerical Analysis and
FinanceReinhold F. Kainhofer
Rigorosumsvortrag, Institut fur Mathematik
Graz, 16. Mai 2003
Inhalt
1. Quasi-Monte Carlo Methoden
2. Sublineare Dividendenschranken in der Risikotheorie
3. Runge-Kutte QMC Methoden fur retardierte Differentialgleichungen
(a) Losungsskizze des Algorithmus
(b) RKQMC Losungsmethoden (Hermite Interpolation, QMC Methoden)
(c) Konvergenzbeweis
(d) Numerische Beispiele
4. QMC Methoden fur singulare gewichtete Integration
(a) Konvergenztheorem und -beweis
(b) Hlawka-Muck Konstruktion
(c) Numerisches Beispiel (Bewertung von asiatischen Optionen)
Quasi-Monte Carlo methods
Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s
f (x)dx =1
N
N∑i=1
f (xi)
Quasi-Monte Carlo methods
Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s
f (x)dx =1
N
N∑i=1
f (xi)
MC methods: xi random points
QMC methods: xi low-discrepancy sequences
Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)
s, good uniform distribution
Quasi-Monte Carlo methods
Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s
f (x)dx =1
N
N∑i=1
f (xi)
MC methods: xi random points
QMC methods: xi low-discrepancy sequences
Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)
s, good uniform distribution
discrepancy of the point set S
DN(S) = supJ⊆[0,1]s
∣∣∣∣A(J , S)
N− λs(J )
∣∣∣∣
Quasi-Monte Carlo methods
Integrals approximated by a discrete sum over N (quasi-)random points:∫[0,1]s
f (x)dx =1
N
N∑i=1
f (xi)
MC methods: xi random points
QMC methods: xi low-discrepancy sequences
Low discrepancy sequences: deterministic point sequences {xi}1≤n≤N ∈[0, 1)
s, good uniform distribution
discrepancy of the point set S
DN(S) = supJ⊆[0,1]s
∣∣∣∣A(J , S)
N− λs(J )
∣∣∣∣Koksma-Hlawka inequality (f of bounded Variation):∣∣∣∣∣ 1
N
N∑n=1
f(xn)−∫
[0,1)s
f(u)du
∣∣∣∣∣ ≤ V ([0, 1)s, f)D∗N (x1, . . . , xN ) .
Low-Discrepancy sequences
D∗N (S) ≤ O
((log N)s
N
)
Low-Discrepancy sequences
D∗N (S) ≤ O
((log N)s
N
)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the
comma
• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals
Low-Discrepancy sequences
D∗N (S) ≤ O
((log N)s
N
)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the
comma
• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Monte Carlo, 3000 points
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Halton sequence, 3000 points
Low-Discrepancy sequences
D∗N (S) ≤ O
((log N)s
N
)• Halton-sequence in bases (b1, . . . bs): inversion of digit expansion of n in base bi at the
comma
• (t, s) nets in base b (Niederreiter, Sobol, Faure): net-like structure → best possible uniformdistribution on elementary intervals
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Monte Carlo, 3000 points
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Halton sequence, 3000 points
Problem: correlations between elements
Risk Model with constant interest forceand non-linear dividend barrier
������������������
������������������
tt
������������������������������������������������������������������������������������������������������������������������������������������������������������
������������������������������������������������������������������������������������������������������������������������������������������������������������
timet
claims ~ F(y)premiums
xb
dividend barrier breserve R
dividends
ruin
φ(u, b) . . . probability of survival
W (u, b) . . . expected value of discounted dividend payments
Integro-differential equation for W (u, b):
(c+i u) ∂W∂u + 1
α m bm−1∂W∂b − (i+λ) W +λ
u∫0
W (u−z,b)dF (z)=0
with boundary condition ∂W∂u
∣∣u=b
= 1.
Solution is fixed point of integral operator ⇒ apply it iteratively orrecursively
Ag(u, b) =
∫ t∗
0λe−(λ+i)t
∫ (c′+u)eit−c′
0g
((c′ + u)eit − c′ − z,
(bm +
t
α
)1/m)
dF (z)dt
+
∫ ∞
t∗λe−(λ+i)t
∫ (bm+ tα)
1/m
0g
((bm +
t
α
)1/m
− z,
(bm +
t
α
)1/m)
dF (z)dt
+
∫ ∞
t∗λe−λt
∫ t
t∗e−is
((c + i u)eis − 1
mα(bm + s
α
)1−1/m
)ds dt,
=⇒ Solution is just a high-dimensional integration problem.
Numerical solution of delayed differen-tial equations using QMC methods
1. Sketch of the numerical solution
2. The RKQMC solution methods (Hermite Interpolation, QMC meth-ods)
3. Convergence proofs
4. Numerical examples
The problem
Heavily varying delay differential equations (DDE) or DDE with heavilyvarying solutions.
y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) , for t ≥ t0, k ≥ 1,
y(t) = φ(t), for t ≤ t0 ,
with
The problem
Heavily varying delay differential equations (DDE) or DDE with heavilyvarying solutions.
y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) , for t ≥ t0, k ≥ 1,
y(t) = φ(t), for t ≤ t0 ,
with
f (t, y(t), yret(t)) . . .piecewise smooth in y and yret,
bounded and Borel measurable in t
y(t) . . .solution, d-dimensional real-valued function
τ1(t), ..., τk(t) . . .cont. delay functions, bounded from below by τ0 > 0,
satisfy t1 − τj(t1) ≤ t2 − τj(t2) for t1 ≤ t2
φ(t) . . .initial function, cont. on
[inf
t0≤t,1≤j≤k(t− τj(t)) , t0
].
Sketch of the numerical solution
For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.
• Hermite interpolation for retarded argument ⇒ ODE
Sketch of the numerical solution
For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.
• Hermite interpolation for retarded argument ⇒ ODE
• use RKQMC methods for ODE:Large Runge-Kutta error for heavily oscillating DEIdea (Stengle, Lecot): integrate over whole step size
Sketch of the numerical solution
For heavily oscillating DDE: conventional Runge-Kutta methods unsta-ble.
• Hermite interpolation for retarded argument ⇒ ODE
• use RKQMC methods for ODE:Large Runge-Kutta error for heavily oscillating DEIdea (Stengle, Lecot): integrate over whole step size
– Runge Kutta: Integration over y and t discretized
– RK(Q)MC: Integration over y discretized, numerical Integrationin t (using MC or QMC integration to minimize the error)
Hermite interpolation
Use hermite interpolation for the retarded arguments:
Hermite interpolation
Use hermite interpolation for the retarded arguments:
zj(t) := y(t− τj(t)) =
{φ(t− τj(t)), if t− τj(t) ≤ t0
Pq(t− τj(t); (yi); (y′i)) otherwise
Hermite interpolation
Use hermite interpolation for the retarded arguments:
zj(t) := y(t− τj(t)) =
{φ(t− τj(t)), if t− τj(t) ≤ t0
Pq(t− τj(t); (yi); (y′i)) otherwise
DDE transforms to a ODE:
y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .
Hermite interpolation
Use hermite interpolation for the retarded arguments:
zj(t) := y(t− τj(t)) =
{φ(t− τj(t)), if t− τj(t) ≤ t0
Pq(t− τj(t); (yi); (y′i)) otherwise
DDE transforms to a ODE:
y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .
Solution has to be piecewise r/2-times continuously differentiable in t.
Hermite interpolation
Use hermite interpolation for the retarded arguments:
zj(t) := y(t− τj(t)) =
{φ(t− τj(t)), if t− τj(t) ≤ t0
Pq(t− τj(t); (yi); (y′i)) otherwise
DDE transforms to a ODE:
y′(t) = f (t, y(t), y(t− τ1(t)), . . . , y(t− τk(t))) ≈≈ f (t, y(t), z1(t), . . . , zk(t)) =: g(t, y(t)) .
Solution has to be piecewise r/2-times continuously differentiable in t.Resulting ODE has to fulfill requirements for RKQMC methods (Borel-measurable in t, continously differentiable in y(t)).
Runge Kutta QMC methods for ODE
G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty
y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0
f smooth in y, bounded and Borel measurable in t.
Runge Kutta QMC methods for ODE
G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty
y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0
f smooth in y, bounded and Borel measurable in t.
f is Taylor-expanded only in y ⇒ integral equation in t.
Runge Kutta QMC methods for ODE
G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty
y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0
f smooth in y, bounded and Borel measurable in t.
f is Taylor-expanded only in y ⇒ integral equation in t.
yn+1 = yn +hn
s!N
∑0≤j<N
Gs (tj,n; y)
Gs (tj,n; y) . . . differential increment function of scheme
Runge Kutta QMC methods for ODE
G. Stengle, Ch. Lecot, I. Coulibaly, A. Koudiraty
y′(t) = f (t, y(t)), 0 < t < T, y(0) = y0
f smooth in y, bounded and Borel measurable in t.
f is Taylor-expanded only in y ⇒ integral equation in t.
yn+1 = yn +hn
s!N
∑0≤j<N
Gs (tj,n; y)
Gs (tj,n; y) . . . differential increment function of scheme
G1 (u; y) = f (u, y)
G2 (u; y) = f (u1, y) +1
βf (u2, y)) +
1
αf (u2, y + αhnf (u1, y)))
G3 (u; y) = a1f(u1, y) +L2∑l=1
a2,lf(u2, y + b2,lhnf(u1, y)
)+
+L3∑l=1
a3,l
(u3, y + b
(1)3,l hnf (u1, y) + b
(2)3,l hnf
(u2, yn + c3,lhn(u1, yn)
))
RKQMC for Volterra functional equations
y′(t) = f (t, y(t), z(t)) t ≥ t0
z(t) = (Fy)(t)
y(s) = φ(s) s ≤ t0
RKQMC for Volterra functional equations
y′(t) = f (t, y(t), z(t)) t ≥ t0
z(t) = (Fy)(t)
y(s) = φ(s) s ≤ t0
• (Fy)(t) contains dependence on retarded values.
• F and f are Lipschitz continuous in all arguments except t.
RKQMC for Volterra functional equations
y′(t) = f (t, y(t), z(t)) t ≥ t0
z(t) = (Fy)(t)
y(s) = φ(s) s ≤ t0
• (Fy)(t) contains dependence on retarded values.
• F and f are Lipschitz continuous in all arguments except t.
RKQMC method (n ≥ 0):
yn+1 = yn + hn
N∑i=1
Gs (tn,i; yn, z(t))
z(t) = (F yj)(t)
Convergence: RKQMC for DDE
Theorem[K., 2002] If
(i) RKQMC method converges for ordinary differential equations withorder p
(ii) the increment function Gs of the method and F are Lipschitz
(iii) the interpolation fulfills a Lipschitz condition
(iv) Hermite interpolation is used with order r
(v) the initial error ‖e0‖ vanishes
then the method converges.
Convergence: RKQMC for DDE
Theorem[K., 2002] If
(i) RKQMC method converges for ordinary differential equations withorder p
(ii) the increment function Gs of the method and F are Lipschitz
(iii) the interpolation fulfills a Lipschitz condition
(iv) Hermite interpolation is used with order r
(v) the initial error ‖e0‖ vanishes
then the method converges. If at least ii and iii hold, the error is boundedby
‖ej+1‖ ≤ ‖e0‖ eLtj +(etjL − 1)
L(∥∥EODE
G
∥∥ + L1L2
∥∥E interpolr
∥∥) .
Special case: one retarded argument
Choose the RKQMC method:
• Gs Lipschitz (L2) in 2nd and 3rd argument, bounded variation (in the sense of Hardy andKrause).
• ∃ c1, c2, c3 such that for a p > 0
loc. trunc. error ‖εn‖ ≤ c1(hn)hpn
RK error ‖δn‖ ≤ c2(hn) ‖en‖QMC error ‖dn‖ ≤ c3(hn)D∗
N (S)
• interpolation order p, fulfils a certain Lipschitz condition
Special case: one retarded argument
Choose the RKQMC method:
• Gs Lipschitz (L2) in 2nd and 3rd argument, bounded variation (in the sense of Hardy andKrause).
• ∃ c1, c2, c3 such that for a p > 0
loc. trunc. error ‖εn‖ ≤ c1(hn)hpn
RK error ‖δn‖ ≤ c2(hn) ‖en‖QMC error ‖dn‖ ≤ c3(hn)D∗
N (S)
• interpolation order p, fulfils a certain Lipschitz condition
Then the error ‖en‖ = ‖yn − y(tn)‖ of the method is bounded fromabove by:
‖en‖ ≤ ‖e0‖ etn(c2+L2s! ) +
etn(c2+L2s! ) − 1
c2 + L2
s!
·
·{
c3D∗N(X) +
L2
s!MHq + c1H
p
}
Numerical examples
y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0
y(t) = 1, t ≤ 0 ,
Numerical examples
y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0
y(t) = 1, t ≤ 0 ,
0 5 10 15 20Log2HΛL
-12
-10
-8
-6
-4
Log2HSHmethL,ΛL
RKQMC3, N=10RKQMC3, N=100RKQMC3, N=1000RKQMC2, N=1000RKQMC1, N=1000RungeHeunButcher
Fig: Error of the RK and RKQMC methods, hn = 0.001 for RK, hn = 0.01 for RKQMC
Numerical examples
y′(t) = 3y(t− 1) sin(λt) + 2y(t− 1.5) cos(λt), t ≥ 0
y(t) = 1, t ≤ 0 ,
0 5 10 15 20Log2HΛL
-12
-10
-8
-6
-4
Log2HSHmethL,ΛL
RKQMC3, N=10RKQMC3, N=100RKQMC3, N=1000RKQMC2, N=1000RKQMC1, N=1000RungeHeunButcher
Fig: Error of the RK and RKQMC methods, hn = 0.001 for RK, hn = 0.01 for RKQMC
slowly varying (small λ): conventional RK betterrapidly varying (high λ): RKQMC outperform higher order RK
Advantage of RKQMC for unstable DDE
y′(t) = πλ
2
(y
(t− 2− 3
2λ
)− y
(t− 2− 1
2λ
)), t ≥ 0
y(t) = sin(λtπ), t < 0 ,
Advantage of RKQMC for unstable DDE
y′(t) = πλ
2
(y
(t− 2− 3
2λ
)− y
(t− 2− 1
2λ
)), t ≥ 0
y(t) = sin(λtπ), t < 0 ,
Exact Solution: y(t) = sin(λtπ)
• No discontinuities in any derivative
• k-th derivative only bounded by λk → ”exploding” error
Advantage of RKQMC for unstable DDE
y′(t) = πλ
2
(y
(t− 2− 3
2λ
)− y
(t− 2− 1
2λ
)), t ≥ 0
y(t) = sin(λtπ), t < 0 ,
Exact Solution: y(t) = sin(λtπ)
• No discontinuities in any derivative
• k-th derivative only bounded by λk → ”exploding” error
5 10 15 20Log2HΛL5
10
15
20
Log2HSΛ,HmethLL
RKQMC3, N=1000
RKQMC2, N=1000
RKQMC1, N=1000
Runge
Heun
Butcher
RKQMC schemes can delay the instability of the solution for heavilyoscillating delay differential equations.
Time-corrected error
QMC integration is more expensive than 1 evaluation (Runge-Kutta)
Time-corrected error
QMC integration is more expensive than 1 evaluation (Runge-Kutta)→ Compare a time-corrected error
Time-corrected error
QMC integration is more expensive than 1 evaluation (Runge-Kutta)→ Compare a time-corrected error
5 10 15 20Log2HΛL5
10
15
Log2HSTΛ,HmethLL
RKQMC3, N=1000
RKQMC2, N=1000
RKQMC1, N=1000
Runge
Heun
Butcher
Fig: Time-corrected error of the RK and RKQMC methods
Result: RKQMC methods loose some advantage, but still better thanRunge-Kutta for heavily oscillating DDE.
Non-uniform QMC integration of singu-lar integrands
1. Convergence theorem
2. Sketch of proof
3. Hlawka-Muck construction for h-distributed low-discrepancy se-quences
4. Numerical example (pricing of Asian options)
H-Diskrepancy
H-discrepancy of a sequence ω = (y1, y2, . . . ):
DN,H(ω) = supJ⊆L
∣∣∣∣ 1
NAN(J, ω)−H(J)
∣∣∣∣ ,with L . . . support of H ,
J . . . intervals[~a,~b], AN . . . number of elements of ω lying in J .
H-Diskrepancy
H-discrepancy of a sequence ω = (y1, y2, . . . ):
DN,H(ω) = supJ⊆L
∣∣∣∣ 1
NAN(J, ω)−H(J)
∣∣∣∣ ,with L . . . support of H ,
J . . . intervals[~a,~b], AN . . . number of elements of ω lying in J .
Koksma-Hlawka inequality (arbitrary distribution)
Let f a function of bounded variation (in the sense of Hardy andKrause) on L and ω = (y1, y2, . . . ) a sequence on L. Then∣∣∣∣∣
∫L
f (x)dH(x)− 1
N
N∑n=1
f (yn)
∣∣∣∣∣ ≤ V (f )DN,H(ω).
Convergence of the multidimensional QMC estimator
Convergence of the multidimensional QMC estimator
Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),
Convergence of the multidimensional QMC estimator
Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),
and let furthermore
cN,j = min1≤n≤N
y(j)n ∧ aj < cj ≤ cN,j.
Convergence of the multidimensional QMC estimator
Theorem[Hartinger, K., Tichy, 2003] Let f (x) be a function on L =[a, b] with singularities only at the left boundary of the definition interval(i.e. f → ±∞ only if x(j) → aj for at least one j),
and let furthermore
cN,j = min1≤n≤N
y(j)n ∧ aj < cj ≤ cN,j.
If the improper integral exists, and if
DN,H (ω) · V[c,b](f ) = o(1),
then the QMC estimator converges to the value of the improper integral:
limN→∞
1
N
N∑n=1
f (yn) =
∫[a,b]
f (x) dH(x).
Hlawka-Muck Transformation (1 dimension)
yk =1
N
N∑r=1
b1 + xk −H(xr)c =1
N
N∑r=1
χ[0,xk](H(xr))
Hlawka-Muck Transformation (1 dimension)
yk =1
N
N∑r=1
b1 + xk −H(xr)c =1
N
N∑r=1
χ[0,xk](H(xr))
DN,H(ω) ≤ (1 + 2sM)DN(ω)
0 might appear among the yk ⇒ adapt the construction as follows(needed for singular integrands): We define the sequence ω for i =1, . . . , N as:
yk =
{yk wenn yk ≥ 1
N,
1N
wenn yk = 0.
0 might appear among the yk ⇒ adapt the construction as follows(needed for singular integrands): We define the sequence ω for i =1, . . . , N as:
yk =
{yk wenn yk ≥ 1
N,
1N
wenn yk = 0.
The discrepancy of this sequence can be bounded as
DN,H(ω) ≤ (1 + 4M)sDN(ω).
Sample problem: Valuing Asian options
arithmetic mean until expiration time
Pay-Off (discrete Asian option, call)
P (ST ) =
(1
n
n∑i=1
Sti −K
)+
(St)t≥0 ... price process, K ... strike price
St = eXt with Levy process (Xt)t≥0
Increments hi = Xi −Xi−1 with distribution H(e.g. NIG, Variance-Gamma, Hyperbolic, ...)
NIG distribution
Use the NIG distribution for the increments hi ∼ HQ.Advantage: closed under convolution ⇒ dimension reduction, sampleonly weekly instead of daily
Valuation
Using fundamental theorem (Schachermayer):
Ct0 := e−r(tn−t0)EQ
[(1
n
n∑i=1
Sti −K
)+]r ... constant interest rateQ ... equivalent martingale measure (Esscher measure)
NIG distribution
Use the NIG distribution for the increments hi ∼ HQ.Advantage: closed under convolution ⇒ dimension reduction, sampleonly weekly instead of daily
Valuation
Using fundamental theorem (Schachermayer):
Ct0 := e−r(tn−t0)EQ
[(1
n
n∑i=1
Sti −K
)+]r ... constant interest rateQ ... equivalent martingale measure (Esscher measure)
Straigforward simulation (crude Monte Carlo): sample N price pathesand take the mean
Quasi-Monte Carlo schemes
Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)
Quasi-Monte Carlo schemes
Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)
2 Solutions:
1. Hlawka-Muck method for direct creation of (xn)1≤n≤N
i.i.d.∼ NIG⇒ direct QMC calculation of the expectation value
Quasi-Monte Carlo schemes
Problem: QMC numbersi.i.d.∼ NIG(α, β, δ, µ)
2 Solutions:
1. Hlawka-Muck method for direct creation of (xn)1≤n≤N
i.i.d.∼ NIG⇒ direct QMC calculation of the expectation value
2. Transformation of the integral using a suitable density (Ratio ofuniforms, ”Hat”) ⇒ variance reduction (if done right)
Transformation
Using a distribution K(~x) = u:∫Rn
P (~x)fQH (~x)d~x =
∫[0,1]n
P (K−1(u))fQ
H (K−1(u))
k(K−1(u))du
Problem: ”Usual” transformation F (x) = u leads to unbounded varia-tion
Discrepancy for Hlawka-Muck
The discrepancy of (yk)1≤k≤N can be bounded by
DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))
with M(ρ) = sup ρ.
Discrepancy for Hlawka-Muck
The discrepancy of (yk)1≤k≤N can be bounded by
DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))
with M(ρ) = sup ρ.
QMC estimator
1. Creation of low-discrepancy points with densityfQ
H (K−1(x))k(K−1(x))
(Transformation of the integral Rn to [0, 1]n using double-exponential distribution K(x))
Discrepancy for Hlawka-Muck
The discrepancy of (yk)1≤k≤N can be bounded by
DN ((yk), ρ) ≤ (2 + 6sM(ρ))DN ((yk))
with M(ρ) = sup ρ.
QMC estimator
1. Creation of low-discrepancy points with densityfQ
H (K−1(x))k(K−1(x))
(Transformation of the integral Rn to [0, 1]n using double-exponential distribution K(x))
2. Transformation [0, 1]n
to Rn of the sequence using double-exponential distribution K−1(x).
Estimator similar to crude Monte Carlo
Numerical results (4 dimensions)
210 212 214 216 218N
-14
-12
-10
-8
-6
-4
Log2HerrL 4 dimensions
double exp.
HM IS
Hlawka-Mück
Ratio of unif.
Import. samp.
MC
ROU and Hlawka-Muck are considerably better than Monte Carlo andcontrol variate
Dimension 12
210 212 214 216 218N
-9
-8
-7
-6
-5
-4
Log2HerrL 12 dimensions
double exp.
HM IS
Hlawka-Mück
Ratio of unif.
Import. samp.
MC
• ROU looses performance
• Hlawka-Muck gives best results