Inverse Problems and Propagation of Uncertainty in Dynamical Systems H. T. Banks Center for Research in Scientific Computation (CRSC) Center for Quantitative Sciences in Biomedicine (CQSB) North Carolina State University Raleigh, NC 27695 RICAM Workshop on Inverse and Partial information Problems Special Semester on Stochastics with Emphasis on Finance Linz, Austria October 27-31, 2008 C enter for Q uantitative S ciences in B iomedicine North Carolina State University 1
(A) Special Semester on Stochastics with Emphasis on Finance (B) Workshop on Inverse and Partial Information Problems: Methodology and Applications (1) Stochastics (2) Inverse Problems: Methodology and Applications (3) Finance Meatloaf:“Two out of three ain’t bad” (want, need, love) 2
Inverse Problems with Uncertainty and Aggregate Data Inverse Problems with Uncertainty and Aggregate Data i) Individual Dynamics- ii) Aggregate Dynamics-(measure dependent dynamics) Related to: a) Relaxed Controls (sliding regimes, chattering controls) b) Preisach Hysteresis in materials c) Mixing Distributions/Random Effects in Statistical Inverse Problems
GENERIC INVERSE PROBLEM:”Individual” Dynamics E ∼ Aggregate Data: d C [x( ; ) : t q P ] i i dx dt = ∈ f t x t q ( , ( ), ) q Q Dynamics: where f can represent ordinary, functional, or partial differential equation ∑ E 2 = − J P ( ) C [ ( ; ): ] x t q P d Minimize i i i ∈ = P P (Q) { probability measures over Q} over Includes as special cases usual problems with constant R.V.’s (i.e., usual vector or function space parameters)
Here ≡ ∫ = E x t P ( ; ) [ ( ; ) : x t q P ] x t q dP q ( ; ) ( ) Q In this case, one has individual dynamics for each realization q of a random variable with distribution P. This class of problems involves a mathematical model (the system dynamics for the parameter dependent state x(t;q)), a statistical model (in this case,assumptions about the data-- i.i.d. normal with constant variance leading to ≅ the ordinary least squares criterion MLE) and “mixing distributions” or “random effects” to account for variability in individuals—
Needs:(to carry out a careful mathematical analysis ) = P P (Q ) i) Topology on → P J P ( ) ii) Continuity of P iii) Compactness of (well-posedness) (Q) iv) Computational tools (approximations, etc.) (see Chapter 6 (by Banks, Bortz, Pinter, Potter) in Bioterrorism: Mathematical Modeling Applications in Bioterrorism: Mathematical Modeling Applications in Homeland Security (H.T.Banks and C.Castillo-Chavez,eds.) Homeland Security SIAM, Philadelphia,2003)
Let (Q,d) be a complete metric space For any closed . ⊂ ε > F Q and 0, define { } ε = ∈ < ε ∈ � � F q Q: d( , ) q q , for some q F . ρ × → + P P R Then define the Prohorov m etric : (Q) (Q) b y ρ ≡ ( , P P ) 1 2 { } ε > ≤ ε + ε ⊂ inf 0: [ P F ] P F [ ] , F closed F , Q . 1 2
RANDOM VARIABLES and ASSOCIATED METRIC SPACES { } = = P P (Q) P : P are probability measures on Q . ρ ρ P ( (Q), ) is a metric space with the Proho rov metric . It is a complete metric space and is com pact if Q is compac t. PROHOROV METRIC ∫ ∫ ρ → ⇔ → ∈ k k (P , P ) 0 gdP gdP for all g C (Q ) Q Q ⇔ convergence in expectation ⇔ → ⊂ ∂ = k P [A] P[A] for all Borel A Q w ith P ( A ) 0 For details on Prohorov m etric and an approxim ation theory, se e[1]. [1] H.T.Banks and K.L.Bihari, Modeling and estimating uncertainty in parameter estimation, CRSC-TR99-40, NCSU, Dec.,1999; Inverse Problems 17(2001),1-17.
GENERAL THEORETICAL FRAMEWORK Application here to ODE systems that include population models dx dt = ∈ Syst e m: f t x t ( , ( ), ) q q Q x (0)=x 0 → Argue that (t,x,q) f(t,x,q) is continuous from × × n n [0,T] R Q to R , locally Lipschitz in x. Then by standard continuous dependence on "parameters" → results for ODE, we obtain that q x(t;q) is continuous n fro m Q to R for each t.
∑ 2 → − E This yields P J(P)= C x t q [ ( ; ):P] d is continuous i i i ρ P 1 from (Q) to R , with respect to , the Prohorov metric, ρ P and ( (Q), ) is compact. Then the general theory of B anks-Bihari[Inverse Problems,2001] can be followed to obtain existence and stability for inverse problems (continuous dependence wrt to data of solutions of the inverse problem). Mor eover, an approximati on theor y as a basis for comp utat ional methods is obtained.
METHOD STABILITY UNDER APPROXIMATION ∪ = ⊂ M Let Q { q } Q be such that Q is dense in Q , M j M M { } ∑ M ∈ = δ ∈ ∈ ≥ P P M M (Q)= P (Q):P p , q Q ,p R, p 0 . = M M M j j M j j j 1 q j ˆ ˆ = k k Let d={d}, d {d } be sets of data(observations) such that i i → ˆ ˆ k d d. ∗ = ˆ P k k M Define P (d ) set of minimizers f or J (P) over (Q), M ∗ ˆ = P and P (d) set of minimizers for J(P) over (Q). Let dist(A,B) be the Hausdorff distance between sets A and B. ∗ ∗ ˆ ˆ → →∞ ˆ → ˆ k k Theorem : dist(P (d ),P (d )) 0 as M , d d, so that M solutions de pend continuously on data and approximate problems are "method stable".
Propagation of Uncertainty in Dynamical Systems (i) Cryptodeterministic (deterministic propagation of random IC) (ii) Stochastic differential equations (Ito diffusion processes, Fokker-Planck) (iii) Random differential equations (nonlinear, nonadditive dependence on uncertainty) (iiia) Distributions on parameters in deterministic dynamics (GRD) (iv) Individual/population modeling in hierarchial statistical framework (v) Probability distribution dependent dynamics 3
From joint efforts on modeling of variability in growth in shrimp populations with V. A. Bokil, J.L. Davis, Stacey Ernstberger, Shuhua Hu (CRSC) E. Artimovich, A. K. Dhar, R. A. Bullis, (AdvBioNutrition) C. L. Browdy (Waddell Marine Culture Center) Builds on interests and efforts since late 1980’s on uncertainty in population modeling–mosquitofish–by HTB, F. Kappel, L. Botsford, C. Wang, B. Fitzpatrick, H. Tran, Y. Zhang, L. Potter, K. Bihari, et al.,–(Brown, USC, NCSU) 4
References [1] HTB, V.A. Bokil, S. Hu, A.K. Dhar, R.A. Bullis, C.L. Browdy and F.C.T. Allnutt, Modeling shrimp biomass and viral infection for production of biological countermeasures, Mathematical Bioscience and Engineering, 3 (2006), 635–660. . [2] HTB, V.A. Bokil and S. Hu, Monotone approximation for a size and class age structured epidemic model, Nonlinear Analysis: Real World Applications , 8 (2007), 834–852. [3] HTB, Stacey Ernstberger and Shuhua Hu, Sensitivity equations for a size-structured population model, CRSC-TR07-18, NCSU, September, 2007; Quart. Appl. Math , to appear. 5
[4] HTB, J.L. Davis, S.L. Ernstberger, Shuhua Hu, E. Artimovich, A.K. Dhar and C.L. Browdy, Comparison of probabilistic and stochastic formulations in modeling growth uncertainty and variability, CRSC-TR08-03, February, 2008; J. Biological Dynamics , submitted. [5] HTB, J.L. Davis, S.L. Ernstberger, Shuhua Hu, E. Artimovich, A.K. Dhar and C.L. Browdy, Estimation of growth rate distributions in size-structured shrimp populations, in preparation, Summer, 2008. [6] HTB, J.L. Davis, Shuhua Hu, A computational comparison of alternatives in modeling growth uncertainty/stochasticity in populations, in preparation, Summer, 2008. 6
Research Motivation, Method and Procedures Motivation • Develop a stable operational platform for rapid production of large quantities of therapeutic and/or preventative countermeasures responding to bio toxic attacks on population. • Foundation in economical platform for production of complex protein therapeutics to replace mammalian cell culture production methods used in pharmaceutical industry. Method • Use shrimp as scaffold organism to produce biological countermeasures. • Recruit biochemical machinery in existing biomass for production of vaccine or antibody – infection using a virus carrying a passenger gene for desired countermeasure. 7
Procedures • Stock genetically selected specific pathogen free shrimp postlarvae and allow to grow normally in controlled greenhouse-enclosed biosecure system. • Infect with recombinant viral vector (such as Taura Sydrome Virus) expressing foreign antigen, resulting in vaccine production in live infected shrimp. 8
Penaeus Vannamei Shrimp 9
Biosecure Shrimp Production System at Waddell Marine Culture Center 10
Hybrid Model of Shrimp Biomass/Vaccine Production System Model Components • Simulating biomass production model over some time interval. • Feeding the output of biomass production model to the input of vaccine production model. 11
Biomass Production Model Production System • Size dependent characteristics. Size-Structured Population Model (Sinko-Streifer) u t + ( g ( x, t ) u ) x + m ( x, t ) u = 0 ( x, t ) ∈ (0 , x max ] × (0 , T ] , t ∈ (0 , T ] , u (0 , t ) = 0 , u ( x, 0) = u 0 ( x ) , x ∈ [0 , x max ] . where u ( x, t ) = density of individuals of size x in gms at time t (number per unit mass), g ( x, t ) = dx dt = growth rate of individuals of size x at time t (mass per unit time), m ( x, t ) = mortality rate of individuals of size x at time t (per unit time). 12
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