Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun Park Pete Sauer, and Richard Sowers Department of Aerospace Engineering University of Illinois at Urbana-Champaign February 5, 2008 Banff International Research Station NSF CMMI 04-01412, DMS 05-04581 and CNS-0540216 February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Specific Physically-Motivated Problems Specific Physically-Motivated Problems 1 Dynamic Data Driven Electric Power System Nonlinear Filtering in Multi-Scale Environment 2 Signal & Observation Processes Nonlinear Filtering & Zakai Equation The objectives Dimensional Reduction for Noisy Nonlinear Systems 3 Original signal process Reduced State Space Reduced Markov Process Reduced Order Nonlinear Filters: Dynamic Data Assimilation 4 Dimensional Reduction of Nonlinear Filters Interaction between scaling and filtering (Park & Sowers) Data Assimilation in the Detection of Vortices (Barreiro & Liu) February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Specific Physically-Motivated Problems Dynamic Data Driven Power Systems ( DDDPS ) The main objectives: Combine computational models with sensor data to predict the 1 dynamics of large-scale evolving systems. Figure: Electric Power Grid of United States Improve the ability to dynamically steer large-scale complex 2 systems and the measurement processes. February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Specific Physically-Motivated Problems 1 Dynamic Data Driven Electric Power System Nonlinear Filtering in Multi-Scale Environment 2 Signal & Observation Processes Nonlinear Filtering & Zakai Equation The objectives Dimensional Reduction for Noisy Nonlinear Systems 3 Original signal process Reduced State Space Reduced Markov Process Reduced Order Nonlinear Filters: Dynamic Data Assimilation 4 Dimensional Reduction of Nonlinear Filters Interaction between scaling and filtering (Park & Sowers) Data Assimilation in the Detection of Vortices (Barreiro & Liu) February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Signal & Observation Processes We consider a nonlinear R d -valued signal process X ε dX ε t = b ε ( X ε t ) dt + σ ε ( X ε X ε t ) dV t , 0 = ξ and an R n -valued observation process Y ε given by the SDE dY ε t = h ε ( X ε Y ε t ) dt + dB t , 0 = 0 where V and B are independent Wiener processes and ξ is a random initial condition which is independent of V and B . Signal process X ε is composed of slow and fast variables such that the generator L ε of the Markov process X ε is of the form L ε ϕ = 1 ε L F ϕ + L S ϕ, for all ε ∈ ( 0 , 1 ) (denotes the scale separation) and all ϕ ∈ C ∞ ( R d ) , where L F and L S represent generators of fast and slow variables. February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment What is Filtering Estimate the signal X ε t at time t based on the information in the observation Y ε up to time t ; def Y ε = σ { Y ε s : 0 ≤ s ≤ t } . t More precisely for each t ≥ 0, we want to compute the conditional law of X ε t given Y ε t t ( A ) def π ε = P { X ε t ∈ A | Y ε t } for all A ∈ B ( R d ) . We then have (with respect to Lebesgue measure on R d ) � � x ∈ A u ε ( t , x ) dx π ε p ε ( t , x ) dx = t ( A ) = x ∈ R d u ε ( t , x ) dx , � x ∈ A where we can directly define u ε ( t , x ) p ε ( t , x ) def = x ′ ∈ R d u ε ( t , x ′ ) dx ′ . � February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Filtering Equations One can construct a linear recursive filtering equation for the un-normalized density u ε ( t , x ) via a stochastic PDE (Zakai equation): du ε ( t , x ) = L ∗ ε u ε ( t , x ) dt + u ε ( t , x ) h ε ( x ) dY ε u ε ( 0 , · ) = p ξ t , where L ∗ ε is the adjoint operator of the L ε and the initial condition ξ has density p ξ . The main difficulty: The numerical solutions of such stochastic PDEs get prohibitively expensive as the state dimension increases. February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Goals: Find a data-driven low-order model to extract useful information for more accurate and long-term assesment of the system for real-time detection of extreme events Procedures: combine two ingredients, namely, stochastic dimensional reduction and nonlinear filtering. Dimensional Reduction of Nonlinear Filters 1 ◮ Reduction of Signal Processes: X ε t ⇒ X t ◮ Reduction of Nonlinear Filters: π ε ⇒ ¯ π Approximate Filters via Particle Methods: Construction of 2 lower-dimensional particle filters . ◮ lim N →∞ (¯ Π N ( t ) , φ ) = ¯ π t ( φ ) Application 3 ◮ Vortex Dynamics February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Goals: Find a data-driven low-order model to extract useful information for more accurate and long-term assesment of the system for real-time detection of extreme events Procedures: combine two ingredients, namely, stochastic dimensional reduction and nonlinear filtering. Dimensional Reduction of Nonlinear Filters 1 ◮ Reduction of Signal Processes: X ε t ⇒ X t ◮ Reduction of Nonlinear Filters: π ε ⇒ ¯ π Approximate Filters via Particle Methods: Construction of 2 lower-dimensional particle filters . ◮ lim N →∞ (¯ Π N ( t ) , φ ) = ¯ π t ( φ ) Application 3 ◮ Vortex Dynamics February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Goals: Find a data-driven low-order model to extract useful information for more accurate and long-term assesment of the system for real-time detection of extreme events Procedures: combine two ingredients, namely, stochastic dimensional reduction and nonlinear filtering. Dimensional Reduction of Nonlinear Filters 1 ◮ Reduction of Signal Processes: X ε t ⇒ X t ◮ Reduction of Nonlinear Filters: π ε ⇒ ¯ π Approximate Filters via Particle Methods: Construction of 2 lower-dimensional particle filters . ◮ lim N →∞ (¯ Π N ( t ) , φ ) = ¯ π t ( φ ) Application 3 ◮ Vortex Dynamics February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
Nonlinear Filtering in Multi-Scale Environment Goals: Find a data-driven low-order model to extract useful information for more accurate and long-term assesment of the system for real-time detection of extreme events Procedures: combine two ingredients, namely, stochastic dimensional reduction and nonlinear filtering. Dimensional Reduction of Nonlinear Filters 1 ◮ Reduction of Signal Processes: X ε t ⇒ X t ◮ Reduction of Nonlinear Filters: π ε ⇒ ¯ π Approximate Filters via Particle Methods: Construction of 2 lower-dimensional particle filters . ◮ lim N →∞ (¯ Π N ( t ) , φ ) = ¯ π t ( φ ) Application 3 ◮ Vortex Dynamics February 5, 2008 Banff International Research N. Sri Namchchivaya Andrea Barreiro, Shanshan Liu, Kristjan Onu, Jun ParkPete Sauer, and Richard Sowers (UIUC) Data Assimilation and Detection in Multi-Sensor & Multi-Scale Environments / 29
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