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On parameter identification in linear stochastic differential - - PowerPoint PPT Presentation

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work On parameter identification in linear stochastic differential equations by Gaussian statistics Shuai Lu

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

On parameter identification in linear stochastic differential equations by Gaussian statistics

Shuai Lu (Fudan University, Shanghai)

Jointed work with Pingping Niu and Jin Cheng (Fudan University)

New Trends in Parameter Identification for Mathematical Models 30/Oct - 03/Nov, Rio de Janeiro, Brazil

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 1 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Outline

1

Introduction Motivation and Introduction Existing results

2

Parameter identification by direct observation Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters

3

Parameter identification by indirect observation Coupled systems Numerical illustration

4

Conclusion and future work

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 2 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Motivation

Parameter identification of stochastic partial differential equations For x ∈ [−π,π] and t ∈ [t0,+∞), ∂v(x,t) ∂t = −c∂v(x,t) ∂x −dv(x,t)+ µ ∂ 2v(x,t) ∂x2 +f(x)h(t)+g(x) ˙ W(t) where parameters c, d, µ are constant. Direct Problem: calculate the random field v(x,t) given c,d,µ and functions f(x),h(t),g(x); Inverse Problem: recover the functions f(x) and g(x) by the measurement data v(x,t) with t ≥ t0.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 3 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Motivation

We expand the 2π−periodic solution in the spital direction in the Fourier series: v(x,t) =

∞

∑

k=−∞

ˆ vk(t)eikx, ˆ v−k = ˆ v∗

k,

where each ˆ vk(t),k > 0 solves the stochastic ODEs:

  • dˆ

vk(t) = −(d + µk2 +ick)ˆ vk(t)dt +fkh(t)dt +gkdWk(t), ˆ vk(0) = ˆ vk,0.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 4 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Motivation

Ornstein–Uhlenbeck process with constant parameters dv(t) dt = −γ(v(t)− ˆ v)+σ ˙ W(t) v(t0) = v0. where,γ, ˆ v and σ are parameters and {W(t),t ≥ t0} is a Brownian

  • motion. v0 ∼ N(m0,C0) and independent of W(t),t ≥ t0.

Direct Problem: calculate v(t) given parameters and initial state. Inverse Problem: recover the parameters given (continuous)

  • bservations of v(t).

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 5 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Few existing results

Pointwise observation (multiple (wave-number) & paths) Li, Peijun An inverse random source scattering problem in inhomogeneous media. Inverse Problems 27 (2011), no. 3, 035004, 22 pp. Bao, Gang; Xu, Xiang An inverse random source problem in quantifying the elastic modulus of nanomaterials. Inverse Problems 29 (2013), no. 1, 015006, 16 pp. Bao, Gang; Chow, Shui-Nee; Li, Peijun; Zhou, Haomin An inverse random source problem for the Helmholtz equation. Math. Comp. 83 (2014), no. 285, 215–233. Bao, Gang; Chen, Chuchu; Li, Peijun Inverse random source scattering problems in several dimensions. SIAM/ASA J. Uncertain. Quantif. 4 (2016), no. 1, 1263–1287. Lee, Wonjung; Stuart, Andrew Derivation and analysis of simplified

  • filters. Commun. Math. Sci. 15 (2017), no. 2, 413–450.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 6 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Few existing results

Continuous observation (single path) Papaspiliopoulos Omiros; Pokern Yvo; Roberts Gareth; Stuart Andrew Nonparametric estimation of diffusions: a differential equations

  • approach. Biometrika 99 (2012), 511–531.

Dunker, Fabian; Hohage, Thorsten On parameter identification in stochastic differential equations by penalized maximum likelihood. Inverse Problems 30 (2014), no. 9, 095001, 20 pp

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 7 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Main techniques

References by G. Bao/ P . Li/ X. Xu et al. Equation: u′′(x,ω)+ω2(1+q(x))u(x,ω) = g(x)+h(x)W′

x

Technique: 1. order reduction; 2. multiple wavenumber data with multiple boundary data (Multifrequency) References by A. Stuart et al. & T. Hohage et al. Equation: dXt = µ(Xt)dt +σdWt Non-Gaussian framework, σ known Technique: Density function & Fokker-Planck equation

∂ ∂tu = div

  • −µu+ 1

2σσTgradu

  • and regularization theory

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 8 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Main techniques

References by G. Bao/ P . Li/ X. Xu et al. Equation: u′′(x,ω)+ω2(1+q(x))u(x,ω) = g(x)+h(x)W′

x

Technique: 1. order reduction; 2. multiple wavenumber data with multiple boundary data (Multifrequency) References by A. Stuart et al. & T. Hohage et al. Equation: dXt = µ(Xt)dt +σdWt Non-Gaussian framework, σ known Technique: Density function & Fokker-Planck equation

∂ ∂tu = div

  • −µu+ 1

2σσTgradu

  • and regularization theory

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 8 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Main techniques

References by W. Lee and A. Stuart Equation: True non-Gaussian model:

  • du = −γudt +σudW

γ ∈ {γ+,γ−}, Poisson process Approximate Gaussian/non-Gaussian model:

  • d¯

u = − ¯ γ ¯ udt +σudW ¯ γ = constant

  • r

dˆ u = − ˆ γ ˆ udt +σudWu d ˆ γ = − ν

ε ( ˆ

γ − µ)dt + σ

√ε dWγ.

Technique: Identify the constant parameters by fitting the expectation and variance at certain time T.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 9 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Outline

1

Introduction Motivation and Introduction Existing results

2

Parameter identification by direct observation Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters

3

Parameter identification by indirect observation Coupled systems Numerical illustration

4

Conclusion and future work

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 10 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Parameter identification problems

Direct problems dv(t) dt = −γ(v(t)− ˆ v)+σ ˙ W(t) v(t0) = v0. where,γ, ˆ v and σ are parameters and {W(t),t ≥ t0} is a Brownian

  • motion. v0 ∼ N(m0,C0) and independent of W(t),t ≥ t0.

Inverse problems and techniques Identify the unknown parameter γ, ˆ v and σ by the asymptotic behavior

  • f v(t).

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 11 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Gaussian statistics

Exact solution v(t) = v0e−γ(t−t0) + ˆ v

  • 1−e−γ(t−t0)

+σ

t

t0

e−γ(t−s)dW(s).

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 12 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Gaussian statistics

Gaussian statistics Mean: E(v(t)) = E(v0)e−γ(t−t0) + ˆ v(1−e−γ(t−t0)) Variance: V(v(t)) = V(v0)e−2γ(t−t0) + σ2(1−e−2γ(t−t0)) 2γ Covariance: R(t,t +τ) = E[(v(t)−E(v(t)))(v(t +τ)−E(v(t +τ)))] = V(v(t))e−γτ

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 13 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Gaussian statistics

Definition of decorrelation time: Autocorrelation function and decorrelation time:

  • ρ(t,t +τ) = R(t,t+τ)

V(v(t)) = e−γτ

Tv(t) =

∞

0 ρ(t,t +τ)dτ = 1 γ

Gaussian statistics      limt→∞ E(v(t)) = ˆ v limt→∞ V(v(t)) = σ2

2γ

limt→∞ Tv(t) = 1

γ

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 14 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Minimization process

Parameter identification by fitting the Gaussian statistics

  • γ, ˆ

v,σ2 = min

γ>0, ˆ v, σ 2≥0

|E(v(γ, ˆ v,σ))−Eemp|2 +|V(v(γ, ˆ v,σ))−Vemp|2 +|Tv(γ)−Temp|2.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 15 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Minimization process

Empirical measurements Discrete observation vo(ti),i = 1,2,...,N at discrete time steps ti,i = 1,...,N after a burn-in period

  • Eemp = 1

N ∑N i=1 vo(ti)

Vemp =

1 N−1 ∑N i=1(vo(ti)−Eemp)2.

Temp is calculated by the autocorrelation function (autocorr.m) in Matlab.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 16 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Langevin equation

Langevin equation dv = −γ(t)(v(t)− ˆ v(t))dt +σ(t)dW(t) The path-wise solution is v(t) = v(0)e−

t

0 γ(τ)dτ +

t

0 e−

t

s γ(τ)dτγ(s)ˆ

v(s)ds+

t

0 e−

t

s γu(τ)dτσ(s)dWs

(Point-wise) decorrelation time Additional definition of (point-wise) decorrelation time Tv(t) =

∞

0 e−

t+τ

t

γ(s)dsdτ.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 17 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Asymptotic Gaussian statistics Assume that parameters {γ(t), ˆ v(t),σ(t)} are periodic functions of the same period T satisfying

T

0 γ(t)dt > 0. Let v(t) be the solution.

Denoting t = kT +ζ and letting k → ∞, the asymptotic statistics of v(t) are          lim

k→∞E(v(kT +ζ)) =

T

0 K(γ,ζ,s)γ(s)ˆ

v(s)ds := Fγ,ˆ

v(ζ)

lim

k→∞V(v(kT +ζ)) =

T

0 K(2γ,ζ,s)σ2(s)ds := Gγ,σ2(ζ)

Tv(t) = Tv(kT +ζ) = e

ζ

0 γ(s)ds T

0 K(γ,ζ,s)e−

s

0 γ(s′)ds′ds

  • .

Discontinous kernel function K(γ,ζ,s) =     

1 1−e−

T

0 γ e−

ζ

s γ

s < ζ

e−

T

0 γ

1−e−

T

0 γ e−

ζ

s γ

s ≥ ζ,

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 18 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Minimization approach

Minimization approach            γ(ζ) = min

γ(ζ)

  • ζ

0 γ(s)ds−Temp

  • 2

L2(0,T) +α0γ(ζ)2 L2(0,T),

  • ˆ

v(ζ),σ2(ζ)

  • =

min

ˆ v(ζ),σ2(ζ)≥0

  • Fγ,ˆ

v(ζ)−Femp2 L2(0,T) +α1ˆ

v(ζ)2

L2(0,T)

+Gγ,σ2(ζ)−Gemp2

L2(0,T) +α2σ2(ζ)2 L2(0,T)

  • Shuai Lu (Fudan University, Shanghai)

On parameter identification in linear SDEs 19 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Outline

1

Introduction Motivation and Introduction Existing results

2

Parameter identification by direct observation Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters

3

Parameter identification by indirect observation Coupled systems Numerical illustration

4

Conclusion and future work

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 20 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Parameter identification by indirect observation

Coupled systems dv = −γv(v− ˆ v)dt +σvdWv(t), v(0) = v0 ∼ N(mv,0,Cv,0), du = −γuudt +v(t)dt +σudWu(t), u(0) = u0 ∼ N(mu,0,Cu,0).

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 21 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Parameter identification by indirect observation

Path-wise solutions

  • v(t) = v0e−

t

0 γv(s)ds +

t

0 e−

t

s γv(s′)ds′γv(s)ˆ

v(s)ds+

t

0 e−

t

s γv(s)dsσv(s)dWv(s),

u(t) = u0e−

t

0 γu(s)ds +

t

0 e−

t

s γu(s)dsv(s)ds+

t

0 e−

t

s γu(s)dsσu(s)dWu(s).

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 22 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Coupled system with constant parameters

Asymptotic behavior of coupled system with constant parameters Assume that parameters {γv, ˆ v,σv,γu,σu} in the coupled system are all constants with γv > 0 and γu > 0. Let v(t), u(t) be the solutions of the coupled system. When t tends to ∞, the asymptotic statistics of u(t) are        limt→∞ Eu(t) = ˆ

v γu ,

limt→∞ Vu(t) = σ2

u

2γu + σ2

v

2γuγv(γu+γv) := ˆ

U, Tu = 1

γu + 1 γv − σ2

u

2γuγv ˆ U .

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 23 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Coupled system with periodic parameters

Asymptotic behavior of coupled system with periodic parameters: I

Assume that parameters {ˆ v(t),σv(t),σu(t),γu(t),σu(t)} in the coupled system are all periodic functions with the same period T and {

T

0 γv,

T

0 γu} are positive. Let v(t), u(t)

be the solutions of the coupled system. Denote t = kT +ζ and let k → ∞, the asymptotic mean of u(t) is, lim

k→∞Eu(kT +ζ) =

T

0 K(γu,ζ,s)

T

0 K(γv,s,s′)γv(s′)ˆ

v(s′)ds′ ds. The asymptotic variance of u(t) is lim

k→∞Vu(kT +ζ) =

T

0 K(2γu,ζ,s)

  • σ2

u (s)+2

T

0 K(γu +γv,s,s′)

T

0 K(2γv,s′,s′′)σ2 v (s′′)ds′′ds′

  • ds.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 24 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Coupled system with periodic parameters

Asymptotic behavior of coupled system with periodic parameters: II

The asymptotic decorrelation time of u(t) is lim

k→∞Tu(kT +ζ) =

ζ+T

ζ

e

τ 0 γudτ

1−e−

T 0 γu

+ h(ζ) Gu(ζ) J(ζ) :=

ζ+T

ζ

e−

τ 0 γudτ

1−e−

T 0 γu

+ h(ζ) Gu(ζ)p(ζ) ζ+T

ζ

T

0 K(γu,s,τ)p(s)dsdτ

1−p(T) −

ζ+T

ζ

e

τ 0 γudτ

1−e

T 0 γuds

ζ

0 e

s 0 γup(s)ds

  • by defining p(ζ) = e−

ζ 0 γv(s)ds where

h(ζ) =

T

0 K(γu +γv,ζ,s)

T

0 K(2γv,s,s′)σ2 v (s′)ds′ds.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 25 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Direct observation

dv = −γ(t)(v(t)− ˆ v(t))dt +σ(t)dW(t) with γ = cos(2πζ)+1;σ = 0.2cos(2πζ)+0.3 and ˆ v = sin(2πζ +π/4).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5

Direct Observation: γ(ζ) in [0,T]

ζ

Recovered γ by continuous observation True γ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Direct Observation: σ2(ζ) in [0, T] ζ Recovered σ2 by continuous observation True σ2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.5 0.5 1 1.5

Direct Observation: ˆ u(ζ) in [0, T]

ζ

Recovered ˆ u by continuous observation True ˆ u

Figure: From left to right: recovered γv, σ2

v and ˆ

v with observation N = 107 and burn-in time is [0, 3

4N].

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 26 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Direct observation: evolution of accuracy

dv = −γ(t)(v(t)− ˆ v(t))dt +σ(t)dW(t) with γ = cos(2πζ)+1.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 Direct Observation: γ in [0,T] True γ Recovered γ with N=105 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 Direct Observation: γ in [0,T] True γ Recovered γ with N=106 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 Direct Observation: γ(ζ) in [0,T] ζ Recovered γ by continuous observation True γ

Figure: From left to right: recovered γv with different observation N = 105, N = 106, N = 107 and burn-in time is [0, 3

4N].

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 27 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Indirect observation

dv = −γv(t)(v(t)− ˆ v(t))dt +σv(t)dW(t), du = −γuu(t)dt +v(t)dt +σudWu(t) with γ = cos(2πζ)+1;σ = 0.2cos(2πζ)+0.3 and ˆ v = sin(2πζ +π/4).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Indirect Observation: γv(ζ) in [0, T ] Recovered γv with indirect observation True γv 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.05 0.1 0.15 0.2 0.25 Indirect Observation: σ2

v(ζ) in [0, T ]

Recovered σ2

v with indirect observation

True σ2

v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Indirect Observation: ˆ v(ζ) in [0, T ] Recovered ˆ v with indirect observation True ˆ v

Figure: From left to right: recovered γv, σ2

v and ˆ

v with observation N = 4∗107 and burn-in time is [0, 3

4N].

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 28 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Indirect observation: evolution of accuracy

dv = −γv(t)(v(t)− ˆ v(t))dt +σv(t)dW(t), du = −γuu(t)dt +v(t)dt +σudWu(t) with ˆ v = sin(2πζ +π/4).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.5 0.5 1 1.5 Indirect Observation: ˆ v(ζ) with N = 4 ∗ 105 Recovered ˆ v with N = 4 ∗ 105 True ˆ v

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.5 0.5 1 1.5

Indirect Observation: ˆ v(ζ) with N = 4 ∗ 106 Recovered ˆ v with N = 4 ∗ 106 True ˆ v 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Indirect Observation: ˆ v(ζ) in [0, T ] Recovered ˆ v with indirect observation True ˆ v

Figure: From left to right: recovered ˆ v with different observation N = 4∗105, N = 4∗106, N = 4∗107 and burn-in time is [0, 3

4N].

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 29 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Outline

1

Introduction Motivation and Introduction Existing results

2

Parameter identification by direct observation Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters

3

Parameter identification by indirect observation Coupled systems Numerical illustration

4

Conclusion and future work

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 30 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Conclusions and Future Work

Conclusions

1

Parameter identification of (in)directly observed Ornstein–Uhlenbeck process and Langevin equation;

2

Minimization approach based on the empirical Gaussian statistics. Future work Conditional Gaussian framework; High-dimensional SDE; Application in data assimilation.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 31 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Conclusions and Future Work

Conclusions

1

Parameter identification of (in)directly observed Ornstein–Uhlenbeck process and Langevin equation;

2

Minimization approach based on the empirical Gaussian statistics. Future work Conditional Gaussian framework; High-dimensional SDE; Application in data assimilation.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 31 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Ongoing work

SPEKF model du(t) = (−γ(t)+iω)u(t)dt +b(t)dt +f(t)dt +σdW(t), db(t)= (−γb +iωb)(b(t)− ˆ b)dt +σbdWb(t), dγ(t)= −dγ(γ(t)− ˆ γ)dt +σγdWγ(t).

Majda, Andrew J.; Harlim, John Filtering complex turbulent systems. Cambridge University Press, Cambridge, 2012. x+357 pp.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 32 / 34

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Chemnitz Symposium on Inverse Problems (2006)

Thank you, Bernd!

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 33 / 34

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Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work

Thank you for your attention!

Shuai Lu School of Mathematical Sciences, Fudan University Email: slu@fudan.edu.cn Homepage: http://homepage.fudan.edu.cn/shuailu/

Pingping Niu, Shuai Lu and Jin Cheng On parameter identification in linear stochastic differential equations by Gaussian statistics. In preparation.

Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 34 / 34