Filtering as Gradient Descent

Transcription

1 Filtering as Gradient Descent Abhishek Halder Department of Applied Mathematics and Statistics University of California, Santa Cruz Santa Cruz, CA 95064

2 Joint work with Tryphon T. Georgiou

3 Motivation: Mars Entry-Descent-Landing Navigational uncertainty Heating uncertainty Landing footprint uncertainty Chute deployment uncertainty Image credit: NASA JPL

4 Motivation: Mars Entry-Descent-Landing Navigational uncertainty Heating uncertainty Landing footprint uncertainty Chute deployment uncertainty Image credit: NASA JPL Large number of uncertain scenarios Probability density

5 Motivation: Mars Entry-Descent-Landing higher Mach numbers result in increased aerothermal heating of parachute structure, which can reduce material strength; and (3) at Mach numbers above Mach.5, DGB parachutes exhibit an instability, known as areal oscillations, which result in multiple partial collapses and violent re-inflations. The chief concern with high Mach number deployments, for parachute deployments in regions where the heating is not a driving factor, is therefore, the increased exposure to areal oscillations. Supersonic parachute Figure 2. Mars Science Laboratory DGB parachute undergoing full-scale wind-tunnel testing. The Viking parachute system was qualified to deploy between Mach.4 and 2., and a dynamic pressure between 250 and 700 P a []. However, Mach 2. is not a hard limit for successfully operating DBG parachutes at Mars and there is very little flight test data above Mach 2. with which to quantify the amount of increased EDL system risk. Figure 3 shows the relevant flight tests and flight experience in the region of the planned MSL.parachute deploy. While parachute Table MSL Candidate Landing Sites experts agree that higher Mach numbers result in a higher probability of failure, they have different opinions on where the limit Lat. Gillis [5]Lon. Elevation should besite placed. For example, has proposed an upper bound of Mach 2 for parachute Name (deg) aerodynamic (deg) decelerators (km) at Mars. However, Cruz [3] places the upper o o Mach number Mawrth Valis between E range somewhere two N and three. deliver such a large and capable rover safely to a scientifically compelling site, which is rich in minerals likely to trap and preserve biomarkers, presents a myriad of engineering challenges. Not only is the payload mass significantly larger Gale Crater 4.49o S 37.42o E than all previous Mars missions, the delivery accuracy and This presents a challenge foroedl system odesigners, who terrain requirements are also more stringent. In August of Eberswalde Crater S E -.45 must then weigh the system performance gains and risks as202, MSL will enter the Martian atmosphere with the largest o o Holden Crater S sociated with deploying the parachute earlier,eat 37.42E) both-.94 higher aeroshell ever flown to Mars, fly the first guided lifting entry Gale Crater (4.49S, altitudes and Mach numbers, against a very real, but not well at Mars, generate a higher hypersonic lift-to-drag ratio than quantified, probability of parachute failure. It is clear that any previous Mars mission, and decelerate behind the largest deploying a DGB at Mach 2.5 or 3.0 represents a significant supersonic parachute ever deployed at Mars. The MSL EDL pose of proposing andanselecting landing sites. itwhile increase in risk over inflation possible at Mach 2.0. However, is system will also, for the first time ever, softly land Curiosity many siteshow were initially proposed the first ofby these worknot clear much additional risk isatencumbered deploydirectly on her wheels, ready to explore the planet s surface. shops, theparachute outcomeatof2.25 the 4th Landing SiteThis Workshop in 2008 ing the instead of is especially was listmars of four sites, listed large in Table. Of the trueafor EDLcandidate in light of the extremely uncertainties Parachute Decelerators for Mars

6 Problem: Uncertainty Propagation Initial conditions Parameters Process model Process noise State density r(x(t), t)

7 Problem: Uncertainty Propagation Initial conditions Parameters Process model Process noise State density r(x(t), t) Trajectory flow: dx(t) = f(x, t) dt + g(x, t) dw(t), dw(t) N (0, Qdt)

8 Problem: Uncertainty Propagation Initial conditions Parameters Process model Process noise State density r(x(t), t) Trajectory flow: dx(t) = f(x, t) dt + g(x, t) dw(t), dw(t) N (0, Qdt) Density flow: ρ t = L FP(ρ) := (ρf) + 2 n i,j= 2 x i x j ( ( gqg ) ) ρ ij

9 Problem: Filtering Initial conditions Parameters Process model Process noise Prior density r (x(t), t) + x t, t Measurement model Sensor noise Posterior density r + (x(t), t)

10 Problem: Filtering Initial conditions Parameters Process model Process noise Prior density r (x(t), t) + x t, t Measurement model Sensor noise Posterior density r + (x(t), t) Trajectory flow: dx(t) = f(x, t) dt + g(x, t) dw(t), dz(t) = h(x, t) dt + dv(t), dw(t) N (0, Qdt) dv(t) N (0, Rdt)

11 Problem: Filtering Initial conditions Parameters Process model Process noise Prior density r (x(t), t) + x t, t Measurement model Sensor noise Posterior density r + (x(t), t) Trajectory flow: dx(t) = f(x, t) dt + g(x, t) dw(t), dz(t) = h(x, t) dt + dv(t), dw(t) N (0, Qdt) dv(t) N (0, Rdt) Density flow: [ dρ + = L FP dt+ ( h(x, t) E ρ +{h(x, t)} ) R ( dz(t) E ρ +{h(x, t)}dt )] ρ +

12 Research Scope Density Flow PDE formulation Variational formulation Numerically approximate solution of PDE Recursively evaluate proximal operators

13 Research Scope Density Flow PDE formulation Variational formulation Numerically approximate solution of PDE Recursively evaluate proximal operators Density flow gradient descent in infinite dimensions

14 Gradient Descent in Finite Dimensions Problem: minimize x R n φ(x) Algorithm: x k = x k h φ(x k )

15 Gradient Descent in Finite Dimensions Problem: minimize x R n φ(x) Algorithm: x k = x k h φ(x k ) Advantage: - is a descent method: φ(x k ) φ(x k ) - convergence under very few assumptions - simple first order method - can account constraints (projected gradient descent)

16 Why does gradient descent work? φ(x) is the max-rate descending direction (why?)

17 Rate of Convergence for Gradient Descent If then φ is ( h )-smooth O( kh ) ( φ is h Lipschitz) φ is ( h )-smooth O( h exp ( hσ 2 k) ) AND σ-strongly convex

18 Gradient Descent Gradient Flow - GD is Euler discretization of GF dx dt = φ(x), x Rn - Rate matching: GD rate O( kh ) when φ is ( h )-smooth GF rate O( t ) when φ is convex

19 Gradient Descent Proximal Operator x k = x k h φ(x k ) x k = proximal hφ (x k ) := argmin x R n { 2 x x k 2 + hφ(x)}

20 Gradient Descent Proximal Operator x k = x k h φ(x k ) x k = proximal hφ (x k ) := argmin x R n This is nice because { 2 x x k 2 + hφ(x)} - argmin of φ fixed point of prox. operator - prox. is smooth even when φ is not - reveals metric structure of gradient descent

21 Gradient Descent in Infinite Dimensions D Proximal recursion: ρ k = arginf { 2 d2 (ρ, ρ k ) + hφ(ρ)} ρ D

22 Gradient Descent Summary Finite dimensions Infinite dimensions dx dt = φ(x), x Rn ρ = L(x, ρ), t x Rn, ρ D x k (h) = x k h φ(x k ) = argmin{ 2 x x k 2 +hφ(x)} x = proximal hφ (x k ) x k (h) x(t=kh), as h 0 ρ k (x, h) = argmin{ 2 d(ρ, ρ k ) 2 + hφ(ρ)} ρ = proximal d(, ) hφ (ρ k ) ρ k (x, h) ρ(x, t=kh), as h 0

23 Related Work Transport PDE ρ t = L(x, ρ) Gradient descent scheme L(x, ρ) 2 d2 (ρ, ρ k ) Φ(ρ) ρ 2 ρ ρ k 2 L 2 (R n ) 2 R n ρ 2 Heat equation (822) Squared L 2 norm of difference Dirichlet energy, CFL (928) ( U(x)ρ) + β ρ 2 W 2 (ρ, ρ k ) E ρ [ U(x) + β log ρ ] Fokker-Planck-Kolmogorov PDE (94, 7, 3) Optimal transport cost Free energy, JKO (998) ( (h Eρ [h] ) R ( dz E ρ [h]dt )) ρ D KL (ρ ρ k ) [ 2 E ρ (yk h) R (y k h) ] Kushner-Stratonovich SPDE (964, 59) Kullback-Leibler divergence Quadratic surprise, LMMR (205)

24 Related Work Transport PDE ρ t = L(x, ρ) Gradient descent scheme L(x, ρ) 2 d2 (ρ, ρ k ) Φ(ρ) ρ 2 ρ ρ k 2 L 2 (R n ) 2 R n ρ 2 Heat equation (822) Squared L 2 norm of difference Dirichlet energy, CFL (928) ( U(x)ρ) + β ρ 2 W 2 (ρ, ρ k ) E ρ [ U(x) + β log ρ ] Fokker-Planck-Kolmogorov PDE (94, 7, 3) Optimal transport cost Free energy, JKO (998) ( (h Eρ [h] ) R ( dz E ρ [h]dt )) ρ D KL (ρ ρ k ) [ 2 E ρ (yk h) R (y k h) ] Kushner-Stratonovich SPDE (964, 59) Kullback-Leibler divergence Quadratic surprise, LMMR (205) Process dynamics is stochastic gradient flow: Gibbs density dx(t) = U(x) dt + 2β dw(t), ρ (x) e βu(x)

25 Related Work Transport PDE ρ t = L(x, ρ) Gradient descent scheme L(x, ρ) 2 d2 (ρ, ρ k ) Φ(ρ) ρ 2 ρ ρ k 2 L 2 (R n ) 2 R n ρ 2 Heat equation (822) Squared L 2 norm of difference Dirichlet energy, CFL (928) ( U(x)ρ) + β ρ 2 W 2 (ρ, ρ k ) E ρ [ U(x) + β log ρ ] Fokker-Planck-Kolmogorov PDE (94, 7, 3) Optimal transport cost Free energy, JKO (998) ( (h Eρ [h] ) R ( dz E ρ [h]dt )) ρ D KL (ρ ρ k ) [ 2 E ρ (yk h) R (y k h) ] Kushner-Stratonovich SPDE (964, 59) Kullback-Leibler divergence Quadratic surprise, LMMR (205) No process dynamics, only measurement update: dx(t) = 0, dz(t) = h(x, t) dt + dv(t), dv(t) N (0, Rdt)

26 Our Contribution Transport description Gradient descent scheme PDE/SDE/ODE 2 d2 (ρ, ρ k ) Φ(ρ) [ ] Mean ODE, Lyapunov ODE 2 W 2 (ρ, ρ k ) E ρ U(x, t)+ tr(p ) n log ρ Linear Gaussian uncertainty propagation Optimal transport cost Generalized free energy Conditional mean SDE, Riccati ODE D KL (ρ ρ k ) 2 E ρ [ (yk h) R (y k h) ] Kalman-Bucy filter Kullback-Leibler divergence Quadratic surprise ditto 2 d 2 FR (ρ, ρ k ) ditto Fisher-Rao metric Kushner-Stratonovich SPDE ditto ditto Nonlinear filter Fisher-Rao metric

27 The Case for Linear Gaussian Systems Model: dx(t) = Ax(t)dt + Bdw(t), dw(t) N (0, Qdt) dz(t) = Cx(t)dt + dv(t), dv(t) N (0, Rdt) Given x(0) N (µ 0, P 0 ), want to recover: For uncertainty propagation: µ = Aµ, µ(0) = µ 0 ; Ṗ = AP + PA + BQB, P(0) = P 0. For filtering: dµ + (t) = Aµ + (t)dt+ P + CR K(t) (dz(t) Cµ + (t)dt), Ṗ + (t) =AP + (t) + P + (t)a + BQB K(t)RK(t).

28 The Case for Linear Gaussian Systems Challenge : How to actually perform the infinite dimensional optimization over D 2? Challenge 2: If and how one can apply the variational schemes for generic linear system with Hurwitz A and controllable (A, B)?

29 Addressing Challenge : How to Compute Two Step Optimization Strategy Notice that the objective is a sum: first second functional functional arginf ρ D 2 { 2 d(ρ, ρ k ) 2 + hφ(ρ) } Choose a parametrized subspace of D 2 such that the individual minimizers over that subspace match Then optimize over parameters D µ,p D 2 works!

30 Addressing Challenge 2: Generic (A, 2B) Two Successive Coordinate Transformations #. Equipartition of energy: Define thermodynamic temperature θ := n tr(p ), and inverse temperature β := θ State vector: x x ep := θp 2 x System matrices: A ep B ep A, 2B P 2 AP 2, 2θ Stationary covariance: P θi P 2 B

31 Addressing Challenge 2: Generic (A, 2B) Two Successive Coordinate Transformations #2. Symmetrization: State vector: x ep x sym := e Askew ep t x ep System matrices: F(t) G(t) A ep, 2θB ep e Askew ep Stationary covariance: θi θi t A sym ep e Askew ep t, 2θ e Askew ep t B ep Potential: U(x sym, t) := 2 x symf(t)x sym 0

32 Summary Two successive coordinate transformations bring generic linear system to JKO canonical form Can apply two step optimization strategy in x sym coordinate Recovers mean-covariance propagation, and Kalman-Bucy filter in h 0 limit Changing the distance in LMMR from D KL to 2 W2 2 gives Luenberger-type observers Future work: computation for nonlinear filtering

33 Thank You

34 Backup Slides

35 Gradient Descent with Constraints minimize x C φ(x) x k = proj C (x k h φ(x k )) x k = proximal hφ (x k ) := argmin x C { 2 x x k 2 + hφ(x)}