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Olga Mula (Eindhoven University of Technology, Netherlands) Reduced Models in Wasserstein Spaces for Forward and Inverse Problems

Jia-Jie Zhu
March 28, 2024
57

Olga Mula (Eindhoven University of Technology, Netherlands) Reduced Models in Wasserstein Spaces for Forward and Inverse Problems

WORKSHOP ON OPTIMAL TRANSPORT
FROM THEORY TO APPLICATIONS
INTERFACING DYNAMICAL SYSTEMS, OPTIMIZATION, AND MACHINE LEARNING
Venue: Humboldt University of Berlin, Dorotheenstraße 24

Berlin, Germany. March 11th - 15th, 2024

Jia-Jie Zhu

March 28, 2024
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  1. Reduced Models in Wasserstein Spaces for State Estimation Olga Mula

    (TU Eindhoven) Workshop on Optimal Transport Theory and Applications Berlin, 2024/03/11-15 Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 1/ 30
  2. Setting: Forward and Inverse Problems in PDEs Let (V ,

    d) be a metric space over a compact domain D ⊂ Rd . Parametrized PDE: Given parameters θ ∈ Θ, solve B θ (u) = 0 ⇝ u(θ) ∈ V Solution set: M := {u(θ) ∈ V : θ ∈ Θ} Observation operator: Given {ℓi }m i=1 linear functionals in V ′, M : V → Rm u → M(u) = (ℓ1(u), . . . , ℓm(u))T = z Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 2/ 30
  3. Setting: Forward and Inverse Problems in PDEs Let (V ,

    d) be a metric space over a compact domain D ⊂ Rd . Parametrized PDE: Given parameters θ ∈ Θ, solve B θ (u) = 0 ⇝ u(θ) ∈ V Solution set: M := {u(θ) ∈ V : θ ∈ Θ} Observation operator: Given {ℓi }m i=1 linear functionals in V ′, M : V → Rm u → M(u) = (ℓ1(u), . . . , ℓm(u))T = z Tasks: (i) Forward Problem θ → u(θ) (ii) Forward Model Reduction M ≈ Vn ⊂ V (iii) Inverse state estimation (z, M) → u ≈ M−1(z) ∩ M (iv) Inverse parameter estimation (z, M) → θ† Goal: Numerical solvers for (ii) and (iii) when V = (P2(D), W2) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 2/ 30
  4. State estimation Mz = M−1(z) ∩ M Task: Build A

    : Rm → V s.t. A(z) ≈ u for all u ∈ M. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 3/ 30
  5. Agenda 1 Optimal Algorithms for State Estimation in Hilbert spaces.

    2 Developments in Wasserstein spaces: 1 Forward Reduced Modeling 2 State Estimation Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 4/ 30
  6. Part I Optimal algorithms for State Estimation in Hilbert spaces

    [CDMN22] Nonlinear reduced models for state and parameter estimation (SIAM UQ, 2020) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 5/ 30
  7. An example: pollution in the city 250000 252500 255000 257500

    260000 262500 265000 267500 6.246 6.248 6.250 6.252 6.254 6.256 1e6 Station locations V = RKHS B θ (u) = div(A∇u + vu) + c θ = (A, v, c) ∈ Θ zi = ℓi (u) = δxi (u) = ⟨ωi , u⟩V When V is a Hilbert space, we can define the observation space W = span{ωi }m i=1 , where the ωi are the Riesz representers of the ℓi , and z = (zi )m i=1 , zi = ℓi (u) = ⟨ωi , u⟩V ⇔ ω= PWm u Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 6/ 30
  8. State estimation Mω = M−1(ω) ∩ M = (ω +

    W ⊥) ∩ M Task: Build A : W → V s.t. A(PW u) ≈ u for all u ∈ M. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 7/ 30
  9. Optimal reconstruction algorithms Task: Build A : Rm → V

    s.t. A(M(u)) ≈ u for all u ∈ M. Quality of A : Rm → V : E(A, M) = max u∈M ||u − A(M(u))|| Optimal performance among all algorithms: E∗(M) = min A:W →V E(A, M). Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 8/ 30
  10. An optimal algorithm A∗ in V=Hilbert space [BCD+17] Practical issue:

    A∗ is not easily computable since M may have a complicated geometry which is in general not given explicitly. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 9/ 30
  11. Affine reconstruction algorithms (V=Hilbert space) Definition: Let V n =

    ¯ u + Vn be an affine subspace with 1 ≤ n ≤ m. The mapping A : W → V ω → A(ω) := arg min v∈ω+W ⊥ dist(v, V n) is a called an affine reconstruction algorithm. This name is because: Im(A) = ¯ u + Vn ⊕ (W ∩ V ⊥ n ) A(· − PW ¯ u) ∈ L(W , Vn ⊕ (W ∩ V ⊥ n )) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 10/ 30
  12. Affine reconstruction algorithms (V=Hilbert space) Easy to compute: Linear least-squares

    problem of size n × m. Characterization: The family of affine algorithms is generated by all affine subspaces V n with 1 ≤ n ≤ m. [CDD+20] Performance: dm+1(M) ≤ E(A, M) ≤ β−1 n,m εn dm+1(M) := min Vn ⊆V 1≤n≤m+1 max u∈M dist(u, Vn) (Kolmogorov m + 1 width) εn:= max u∈M dist(u, V n), βn,m := inf v∈Vn ∥PWm v∥ ∥v∥ = cos(θVn,Wm ) ∈ (0, 1] Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 10/ 30
  13. Affine reconstruction algorithms (V=Hilbert space) Easy to compute: Linear least-squares

    problem of size n × m. Characterization: The family of affine algorithms is generated by all affine subspaces V n with 1 ≤ n ≤ m. [CDD+20] Performance: dm+1(M) ≤ E(A, M) ≤ β−1 n,m εn dm+1(M) := min Vn ⊆V 1≤n≤m+1 max u∈M dist(u, Vn) (Kolmogorov m + 1 width) εn:= max u∈M dist(u, V n), βn,m := inf v∈Vn ∥PWm v∥ ∥v∥ = cos(θVn,Wm ) ∈ (0, 1] Good strategy for elliptic problems: dn(M) ≲ e−nα [CD16] We can build V n such that εn ≲ e−nα [BCD+11] Need to pay attention to βn,m (sensor placement, [BCMN18]) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 10/ 30
  14. Piecewise-affine algorithms [CDMN22] Consider a partition of the parameter domain

    in K subsets: Θ = Θ1 ∪ · · · ∪ ΘK ⇝ M = M1 ∪ · · · ∪ MK . We omit the details on how to build the partition and select the index ˆ k. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 11/ 30
  15. Piece-wise affine algorithm is near-optimal Theorem 1 (Cohen, Dahmen, Mula,

    Nichols, 2021) For a given target tolerance σ > 0, we can build a partition of M s.t. E∗(M) ≤ E(Aˆ k , M) ≤ E∗(Mσ ) where ˆ k comes from our model selection on the residual. σ ≥ σ0 > 0 in presence of noise and model error. σ → σ0 as K ↗ Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 12/ 30
  16. Piece-wise affine algorithm is near-optimal Theorem 1 (Cohen, Dahmen, Mula,

    Nichols, 2021) For a given target tolerance σ > 0, we can build a partition of M s.t. E∗(M) ≤ E(Aˆ k , M) ≤ E∗(Mσ ) where ˆ k comes from our model selection on the residual. σ ≥ σ0 > 0 in presence of noise and model error. σ → σ0 as K ↗ Merits and Limitations: ✓ Very general algorithm. ✓ Elliptic, parabolic pbs with possibly weak coercivity: few partitions expected ✗ Transport-dominated problems: many partitions expected. ✗ Only for V Hilbert space. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 12/ 30
  17. Shape reconstruction to subcell resolution [CMS24] Task: Reconstruct continuous 2d

    line. ℓi (u) = “cell averages” Vn = “Parametric curves on each cell” (nonlinear set) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 13/ 30
  18. Shape reconstruction to subcell resolution [CMS24] Task: Reconstruct continuous 2d

    line. ℓi (u) = “cell averages” Vn = “Parametric curves on each cell” Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 14/ 30
  19. Part II State Estimation in V = (P2(D), W2) Forward

    Model Reduction to build Vn [ELMV20] Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces (M2AN, 2020) [DFM22] Approximation and Structured Prediction with Sparse Wasserstein Barycenters (preprint) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 15/ 30
  20. Forward Model reduction in V = (P2 (D), W2 )

    We consider conservative PDEs, or Wasserstein gradient flows, and we view solutions in P2(D). We want to approximate u : Θ → V θ → u(θ) with ˆ un : Θ → Vn θ → ˆ un(θ) and the approximation is efficient if the error en = max θ∈Θ W2(u(θ), ˆ un(θ)) decays fast with n. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 16/ 30
  21. Wasserstein Barycenters Given barycentric weights Λn ∈ Σn := {λ

    ∈ Rn : n ∑ i=1 λi = 1, λi ≥ 0} and a set of measures Un = {u1, . . . , un} ⊂ P2(D), the barycenter associated to (Λn, Un) is Bar(Λn, Un) = arg min v∈P2(Ω) n ∑ i=1 λi W 2 2 (v, ui ) We want to identify a good Un and work with Vn = Bar(Σn, Un) := {Bar(Λn, Un) : Λn ∈ Σn} Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 17/ 30
  22. Greedy algorithm to find Un [ELMV20] Compute a finite training

    set of N ≫ 1 snapshots M = {u(θ1), . . . , u(θN )} = {u(θ) : θ ∈ Θ} We start by finding (u1, u2) ∈ arg max (v1,v2)∈M×M W2(v1, v2) and set U2 = {u1, u2}, V2 = Bar(Σ2, U2) For n > 2, given Un−1 and Vn−1, find un ∈ arg max v∈M W2(v, Vn−1) and set Un = Un−1 ∪ {un}, Vn = Bar(Σn, Un). Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 18/ 30
  23. Building ˆ un : Θ → Vn [ELMV20] For all

    u ∈ M, we compute the best barycentric weigths min v∈Vn W2(u(θ), v) = min Λn∈Σn W2(u(θ), Bar(Λn, Un)) ⇝ Λ∗ n (θ), ∀θ ∈ Θ. We build an interpolation operator Λn : Θ → Σn such that Λn(θ) = Λ∗ n (θ), ∀θ ∈ Θ. and we define ˆ un(θ) := Bar(Λn(θ), Un). Practical computation: 1d: Leverage close-form expressions for W2 and Bar(·, Un) > 1d: Entropic regularization ([DFM23]) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 19/ 30
  24. Examples in 1D with n = 5 Viscous Burger’s Camassa-Holm

    KdV Viscous Burger’s equation: ∂tu + 1 2 ∂x (u2) + θ∂xx u = 0 Camassa-Holm: ∂tm + u∂x m + 2m∂x u = 0, with m = u − α2 ∂xx u. Korteveg-de-Vries: ∂tu + 6u∂x u + ∂3 x u = 0 Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 20/ 30
  25. Open questions Decay rate of “barycentric Kolomogorov n-width”? dn(M) :=

    inf Vn=Bar(Σn,Un) Un⊂P2(D) sup u∈M W2(u, Vn) Decay rate with the Vn from the greedy algorithm en = max θ∈Θ W2(u(θ), ˆ un(θ)) Can we prove that en ≤ C dn(M) for some C > 0? Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 21/ 30
  26. Adaptive Vn (θ) = Bar(Σn , Un (θ)) [DFM23] Best

    n-term barycentric approximation using M: For a given θ ∈ Θ, we define Λn N (θ) ∈ arg min ΛN ∈ ΣN ∩∆n N W 2 2 u(θ), Bar(ΛN, M) where ∆n N := {v ∈ RN : ∥v∥0 ≤ n} ΣN := {v ∈ RN : vi ≥ 0, ∑N i=1 vi = 1} We obtain a sparse vector Λn N (θ) = (0, . . . , 0, λiθ 1 , 0, . . . , 0, λiθ n , 0, . . . , 0)T ∈ ΣN We next define Λn(θ) = (λiθ 1 , . . . , λiθ n ) ∈ Σn, Un(θ) = {u(θiθ 1 ), . . . , u(θiθ n )}. The best n-term barycentric approximation of u(θ) is Bar(Λn(θ), Un(θ)) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 22/ 30
  27. Part III State Estimation in V = (P2(D), W2) Building

    A : Rm → Vn [MR24] State Estimation in the Wasserstein space (with P. Rai, in progress) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 23/ 30
  28. Back to State Estimation (joint work with P. Rai) Taking

    Vn = Bar(Σn, Un), we consider the reconstruction algorithm A : (Rm, ∥ · ∥Z ) → P2(D) z → A(z) = arg min v∈Vn ∥z − M(v)∥Z where M : V → Rm u → M(u) = D ωi (x)u(dx) m i=1 = z for some given ωi ∈ C1(D). Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 24/ 30
  29. Reconstruction error Assuming that CL := sup (u,v)∈P2(D)2 ∥M(u) −

    M(v)∥Z W2(u, v) < ∞ (Lip. cont.) CS := sup (u,v)∈V 2 n W2(u, v) ∥M(u) − M(v)∥Z < ∞ (stability) we have W2(u, A(M(u))) ≤ (1 + 2CLCS ) inf v∈Vn W2(u, v), ∀u ∈ P2(D), and therefore E(A, M) = max u∈M W2(u, A(M(u))) ≤ (1 + 2 CLCS :=µZ ) max u∈M inf v∈Vn W2(u, v) =en . Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 25/ 30
  30. Reconstruction error: remarks on the norms µZ = CLCS minimized

    for ∥z∥Z = inf{W2(u, v) : M(u − v) = z} For all ¯ u ∈ Vn, CS = sup (u,v)∈V 2 n W2(u, v) ∥M(u) − M(v)∥Z ≥ γ( ¯ u) := max v∈T ¯ uVn ∥v∥L2 ¯ u (D) ∥(dM)(v)∥Z which shows that CS < ∞ if n ≤ m. If ∥z∥Z = zT G−1( ¯ u)z, with G(u) = (⟨∇ωi , ∇ωj ⟩ L2 ¯ u (D) )1≤i,j≤m. then γ(u) = max v∈T ¯ uP2(D) ∥PW v∥L2 ¯ u (D) ∥v∥L2 ¯ u (D) = β−1 n,m , with W = span{∇ω1, . . . , ∇ωi }. This means that M : P2(D) → (Rm, G−1) is a submersion. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 26/ 30
  31. Sensor placement The stability factor CS may behave very badly.

    Assuming that we can choose sensor locations, we can view ωi (·) = ωi (ri , ·), 1 ≤ i ≤ m, Thus M(u, r) = D ωi (ri , x)u(dx) m i=1 and we can optimize locations by computing Copt S = min r=(r1,...,rm) CS (r), with CS (r) = sup (u,v)∈V 2 n W2(u, v) ∥M(u, r) − M(v, r)∥Z . In practice, we use consensus-based optimization. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 27/ 30
  32. A first numerical result Figure: KdV problem. Olga MULA (TU

    Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 28/ 30
  33. Line completion as an inverse problem in SE(2), [BPB+24] (with

    Daan Bon, Gautam Pai, Gijs Bellard, Remco Duits) V = (P2(SE(2)), W2(SE(2))) We lift images to SE(2), and we want to do parameter and state estimation on that space. Figure: Top: Gradient flow in SE(2). Bottom: Gradient flow in R2. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 29/ 30
  34. Conclusions Theoretical foundations for optimal algorithms for state estimation. Reduced

    Models are crucial to develop viable algorithms. A first step on the Wasserstein space with an algorithm based on barycenters. Plenty of approximation theory results are still missing. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 30/ 30
  35. References I P. Binev, A. Cohen, W. Dahmen, R. DeVore,

    G. Petrova, and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM Journal on Mathematical Analysis 43 (2011), no. 3, 1457–1472. , Data assimilation in reduced modeling, SIAM/ASA Journal on Uncertainty Quantification 5 (2017), no. 1, 1–29. P. Binev, A. Cohen, O. Mula, and J. Nichols, Greedy algorithms for optimal measurements selection in state estimation using reduced models, SIAM/ASA Journal on Uncertainty Quantification 6 (2018), no. 3, 1101–1126. D. Bon, G. Pai, G. Bellard, O. Mula, and R. Duits, Optimal transport in the lie group of roto-translations se(2), submitted (2024). S. Bahmani, B. Raj, and P. T. Boufounos, Greedy sparsity-constrained optimization, Journal of Machine Learning Research 14 (2013), no. Mar, 807–841. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 31/ 30
  36. References II A. Cohen and R. DeVore, Kolmogorov widths under

    holomorphic mappings, IMA Journal of Numerical Analysis 36 (2016), no. 1, 1–12. A. Cohen, W. Dahmen, R. DeVore, J. Fadili, O. Mula, and J. Nichols, Optimal reduced model algorithms for data-based state estimation, SIAM Journal on Numerical Analysis 58 (2020), no. 6, 3355–3381. A. Cohen, W. Dahmen, O. Mula, and J. Nichols, Nonlinear reduced models for state and parameter estimation, SIAM/ASA Journal on Uncertainty Quantification 10 (2022), no. 1, 227–267. A. Cohen, O. Mula, and A. Somacal, High order recovery of geometric interfaces from cell-average data, arXiv:2402.00946 (2024). H. Do, J. Feydy, and O. Mula, Approximation and Structured Prediction with Sparse Wasserstein Barycenters, 2022. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 32/ 30
  37. References III M.-H. Do, J. Feydy, and O. Mula, Approximation

    and structured prediction with sparse wasserstein barycenters, arXiv preprint arXiv:2302.05356 (2023). V. Ehrlacher, D. Lombardi, O. Mula, and F.-X. Vialard, Nonlinear model reduction on metric spaces. application to one-dimensional conservative pdes in wasserstein spaces, ESAIM M2AN 54 (2020), no. 6, 2159–2197. A. Kyrillidis, S. Becker, V. Cevher, and C. Koch, Sparse projections onto the simplex, International Conference on Machine Learning, PMLR, 2013, pp. 235–243. O. Mula and P. Rai, State estimation in the Wasserstein space. D. Needell and J. A. Tropp, Cosamp: Iterative signal recovery from incomplete and inaccurate samples, Applied and computational harmonic analysis 26 (2009), no. 3, 301–321. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 33/ 30
  38. Adaptive Vn (θ) [DFM23] Some remarks on the optimization problem:

    Λn N (u) ∈ arg min ΛN ∈ ΣN ∩∆n N W 2 2 u(θ), Bar(ΛN, M) Reminiscent of Compressed Sensing problems posed on V = RN of the type: min z∈RN ∥z∥0 s.t. Az = yobs Difficult optimization problem: Nonconvex Possibly plenty of local minima, non-unique minimizer. ℓ1-regularization not possible because in conflict with ΣN . We have built: a minimization algorithm that delivers satisfactory results, a surrogate version that involves θ instead of u(θ) in the loss function. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 35/ 30
  39. Best n-term barycentric approximation: Proximal Algorithm Algorithm 1: Sparse projection

    on the simplex Data: Target u ∈ P2(Ω), Training set M, sparsity degree n Result: Λn N (u) ∈ ΣN ∩ ∆n N . Initialize Λ ∈ ΣN; repeat Gradient step: Λ ← Λ − τ∇ΛW 2 2 (u, Bar(Λ, M)); Projection into ∆n N ∩ ΣN: Λ ← P∆n N ∩ΣN (Λ) until convergence; Algorithm is a generalization of CoSamp and refinements such as GSSP (see [NT09, KBCK13, BRB13]). We can build variants in which we learn adaptively the sparsity degree n. Rigorous convergence analysis missing. Difficult due to nonconvexity. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 36/ 30
  40. Experiment with functions from a Burgers’ equation N = 100

    and Ytrain N = {u(θi )}N i=1 contains solutions of a Burgers’ equation. Target function is a barycenter with support 2 from training set. 2 4 6 8 ref 2 4 6 8 2 4 6 8 fit 0.0000 0.0008 0.0016 0.0024 0.0032 0.0040 0.0048 0.0056 0.0064 Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 37/ 30
  41. Experiment with functions from a Burgers’ equation Target function is

    a barycenter with support 2 from training set. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 38/ 30
  42. Experiment with functions from a Burgers’ equation Target function is

    a barycenter with support 10 from training set. Some functions of the target barycenter are probably redundant. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 39/ 30
  43. Back to Model Reduction For a given θ ∈ Θ,

    computing Λn N (θ) is based on solving Λn N (θ) ∈ arg min ΛN ∈ ΣN ∩∆n N W 2 2 u(θ), Bar(ΛN, Ytrain N ) The computation requires u(θ) so we can only use it for θ ∈ ΘN. For a general θ ∈ Θ, we cannot assume that u(θ) is given. We need an extra approximation step. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 40/ 30
  44. Current strategy Offline: For θ ∈ ΘN: Compute Λn N

    (θ). Find local Euclidean embedding: M(θ) ∈ arg min M≥0 ∑ y(α)∈Nn(y(θ)) |(θ − α)T M(θ − α) − W 2 2 (u(θ), u(α))|2 Online: Given θ ∈ Θ: Find the n nearest neighbors of u(θ) by using the Euclidean embedding. For this, evaluate (α − θ)T M(α)(α − θ) ≈ W 2 2 (y(θ), y(α)), ∀α ∈ ΘN and pick the n smallest values to define the neighbors. For the neighbors u(α) ∈ Nn(u(θ)), we have Λz n , Xz n , Yz n . We want to use this information to define an interpolation strategy for the weights. Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 41/ 30
  45. Interpolation strategy ∑ u(α)∈Nn(u(θ)) |W 2 2 (u(θ), u(α)) −

    W 2 2 (b, u(α))|2 ≈ ∑ u(α)∈Nn(u(θ)) |(θ − α)T M(α)(θ − α) − W 2 2 (b, u(α))|2 So we look for min Λn∈Σn ∑ u(α)∈Nn(u(θ)) |(θ − α)T M(α)(θ − α) −W 2 2 (Bar(Λn, Nn(u(θ))), u(α))|2 ⇝ Λn N (θ) Olga MULA (TU Eindhoven) Reduced Models in Wasserstein Spaces for State Estimation 42/ 30