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
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Outline Introduction 1 Motivation and Introduction Existing results Parameter identification by direct observation 2 Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters Parameter identification by indirect observation 3 Coupled systems Numerical illustration Conclusion and future work 4 Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 2 / 34
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 ∈ [ t 0 , + ∞ ) , − dv ( x , t )+ µ ∂ 2 v ( x , t ) ∂ v ( x , t ) = − c ∂ v ( x , t ) + f ( x ) h ( t )+ g ( x ) ˙ W ( t ) ∂ x 2 ∂ t ∂ x 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 ≥ t 0 . Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 3 / 34
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Motivation We expand the 2 π − periodic solution in the spital direction in the Fourier series: ∞ v k ( t ) e ikx , v ∗ ∑ v ( x , t ) = v − k = ˆ k , ˆ ˆ k = − ∞ where each ˆ v k ( t ) , k > 0 solves the stochastic ODEs: � = − ( d + µ k 2 + ick ) ˆ v k ( t ) v k ( t ) d t + f k h ( t ) d t + g k d W k ( t ) , dˆ v k ( 0 ) ˆ = ˆ v k , 0 . Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 4 / 34
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Motivation Ornstein–Uhlenbeck process with constant parameters d v ( t ) v )+ σ ˙ = − γ ( v ( t ) − ˆ W ( t ) d t v ( t 0 ) = v 0 . where, γ , ˆ v and σ are parameters and { W ( t ) , t ≥ t 0 } is a Brownian motion. v 0 ∼ N ( m 0 , C 0 ) and independent of W ( t ) , t ≥ t 0 . Direct Problem: calculate v ( t ) given parameters and initial state. Inverse Problem: recover the parameters given (continuous) observations of v ( t ) . Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 5 / 34
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
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
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 )+ ω 2 ( 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: d X t = µ ( X t ) d t + σ d W t Non-Gaussian framework, σ known Technique: Density function & Fokker-Planck equation ∂ � − µ u + 1 2 σσ T grad u � ∂ t u = div and regularization theory Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 8 / 34
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 )+ ω 2 ( 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: d X t = µ ( X t ) d t + σ d W t Non-Gaussian framework, σ known Technique: Density function & Fokker-Planck equation ∂ � − µ u + 1 2 σσ T grad u � ∂ t u = div and regularization theory Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 8 / 34
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: � d u = − γ u d t + σ u d W γ ∈ { γ + , γ − } , Poisson process Approximate Gaussian/non-Gaussian model: � u = − ¯ γ ¯ u d t + σ u d W d¯ γ = constant ¯ or � dˆ γ ˆ u d t + σ u d W u u = − ˆ γ = − ν γ − µ ) d t + σ ε ( ˆ √ ε d W γ . d ˆ 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
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Outline Introduction 1 Motivation and Introduction Existing results Parameter identification by direct observation 2 Ornstein–Uhlenbeck process with constant parameters Langevin equation with periodic parameters Parameter identification by indirect observation 3 Coupled systems Numerical illustration Conclusion and future work 4 Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 10 / 34
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Parameter identification problems Direct problems d v ( t ) v )+ σ ˙ = − γ ( v ( t ) − ˆ W ( t ) d t v ( t 0 ) = v 0 . where, γ , ˆ v and σ are parameters and { W ( t ) , t ≥ t 0 } is a Brownian motion. v 0 ∼ N ( m 0 , C 0 ) and independent of W ( t ) , t ≥ t 0 . Inverse problems and techniques Identify the unknown parameter γ , ˆ v and σ by the asymptotic behavior of v ( t ) . Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 11 / 34
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Gaussian statistics Exact solution � t v ( t ) = v 0 e − γ ( t − t 0 ) + ˆ � 1 − e − γ ( t − t 0 ) � e − γ ( t − s ) d W ( s ) . + σ v t 0 Shuai Lu (Fudan University, Shanghai) On parameter identification in linear SDEs 12 / 34
Introduction Parameter identification by direct observation Parameter identification by indirect observation Conclusion and future work Gaussian statistics Gaussian statistics Mean: E ( v ( t )) = E ( v 0 ) e − γ ( t − t 0 ) + ˆ v ( 1 − e − γ ( t − t 0 ) ) Variance: V ( v ( t )) = V ( v 0 ) e − 2 γ ( t − t 0 ) + σ 2 ( 1 − e − 2 γ ( t − t 0 ) ) 2 γ 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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