REDUCED BAYESIAN INVERSION Reducing Bayesian Inverse Problems with Balanced Truncation Christian Himpe (christian.himpe@uni-muenster.de) Mario Ohlberger (mario.ohlberger@uni-muenster.de) Synopsis Combined reduction enables the concurrent reduction of state and parameter space dimensions, which accelerates the integration. Here the reduction of state space is based upon experimental observations which are balanced against a models output. The parameter space is reduced based on the identiﬁability inside the model. In a Bayesian inversion setting with undetermined posterior parameter distributions, the estimation duration can be signiﬁcantly shortened. Balanced Truncation Control Systems As an underlying model to the (Bayesian) inversion, control systems are considered: ˙x = f (x, u, θ) y = g(x, u, θ). First, to introduce balanced truncation the focus is narrowed to linear control systems: ˙x = Ax + Bu y = Cx. The empirical gramians are based on the same approach, yet allow nonlinear and thus more general control systems. Controllability + Observability Controllability quantiﬁes how well a state can be driven by input or control. The controllability gramian WC characterizes the controllability of a linear system and is computed as the smallest semi-positive deﬁnite solution of the Lyapunov equation: AWC + WCAT = −BBT. Observability quantiﬁes how well a change in state is reﬂected by the outputs. The observability gramian WO characterizes the observability of a linear system and is computed as the smallest semi-positive deﬁnite solution of the Lyapunov equation: ATWO + WOA = −C TC. Balanced Truncation The Hankel operator maps past inputs to future outputs via the impulse response: H(t) = Z 0 −∞ CeA(t−τ)Bu(τ)dτ. The singular values of the Hankel operator σi indicate the importance of the associated state to the system dynamics. To determine the least controllable and least observable states, the controllability and observability gramians are balanced by a similarity transformation that balances the systems components (A, B, C). Truncating the balancing transformation U, V with VWCV T = UTWOU = σi, the reduced linear system is given by (VAU, VB, CU). Empirical Gramians Empirical Controllability Gramian The empirical controllability gramian ([1]) is computed by averaging the outer product of state time-series centered around the steady state ¯x: WC = 1 |Qu||Ru| |Qu| X h=1 |Ru| X i=1 m X j=1 1 c2 h Z ∞ 0 Ψhij(t)dt Ψhij(t) = (xhij(t) −¯x)(xhij(t) −¯x)∗∈Rn×n. xhij is a state time-series generated using the perturbed input uhij(t) = chSieju(t) + ¯u based on the perturbation sets: Eu = {ei ∈Rj; ∥ei∥= 1; eiej̸=i = 0; i = 1, . . . , m} Ru = {Si ∈Rj×j; S∗ i Si = 1; i = 1, . . . , s} Qu = {ci ∈R; ci > 0; i = 1, . . . , q}. For linear systems the empirical controllability gramian equals the classic controllability gramian, but since being based on snapshots extends also to nonlinear systems! Empirical Observability Gramian The empirical observability gramian ([1]) is computed by averaging the inner product of output time-series centered around the steady state output ¯y: WO = 1 |Qx||Rx| |Qx| X k=1 |Rx| X l=1 1 d2 k Tl Z ∞ 0 Ψkl(t)dt T ∗ l Ψkl ab = (ykla(t) −¯y)∗(yklb(t) −¯y) ∈R. ykla is an output time-series generated using the perturbed initial state xkla 0 = dkSlfa + ¯x based on the perturbation sets: Ex = {fi ∈Rn; ∥fi∥= 1; fifj̸=i = 0; i = 1, . . . , n} Rx = {Ti ∈Rn×n; T ∗ i Ti = 1; i = 1, . . . , t} Qx = {di ∈R; di > 0; i = 1, . . . , r}. For linear systems the empirical observability gramian equals the classic observability gramian, but since being based on snapshots extends also to nonlinear systems! Empirical Identiﬁability Gramian A system can be augmented by a constant state for each parameter with the parameter value as initial state: ˙xa = ˙x ˙θ  = f (x, u, θ) 0  xa(0) = x0 θ  . Then the empirical identiﬁability gramian ([2]) is given as the empirical observability gramian of this augmented system: WO,a = WO WM W ∗ M WP  .