Parametric Uncertainty Computations with Tensor Product Representations Hermann G. Matthies A. Litvinenko, M. Meyer, O. Pajonk, B. Rosi´ c, E. Zander Institute of Scientific Computing, TU Braunschweig Brunswick, Germany wire@tu-bs.de http://www.wire.tu-bs.de $Id: 10_Paris.tex,v 7.4 2011/08/02 03:21:18 hgm Exp $
2 Overview 1. Parameter dependent problems 2. Decompositions and factorisations 3. Formulation in tensor product spaces 4. Examples 5. Model reduction and sparse representation 6. Bayesian updating, inverse problems 7. Examples and Conclusion C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
3 Mathematical formulation I Consider operator equation, physical system modelled by A , depending on quantity q : A ( q ; u ) = f u ∈ V , f ∈ F , ⇔ ∀ v ∈ V : a ( q ; u ; v ) = � A ( q ; u ) , v � = � f, v � , V — space of states, F = V ∗ — dual space of actions / forcings. Variant: A ( ς ( q ); u ) = f , dependence on a function ς ( q ) , such that parameter p ∈ P may be p = q | p = ( q, f ) | p = ( q, f, u 0 ) | p = ( ς ( q ) , . . . ) . . . General formulation—non-linear operator, semi-linear form: ⇔ ∀ v ∈ V : a ( p ; u ; v ) = � A ( p ; u ) , v � = � f, v � . A ( p ; u ) = f Want to describe A ( p, · ) , f ( p ) , ς ( p ) , or u ( p ) − → r ( p ) . In the end desired quantities of interest (QoI) Ψ ι ( p, u ( p )) . C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
4 Problem with parameters—diffusion SPDE Geometry 2 1.5 0 1 0.5 0.5 1 1.5 0 2 2D model domain G Aquifer Simple stationary model of groundwater flow with parameters x ∈ G ⊂ R d −∇ · ( κ ( x ) · ∇ u ( x )) = f ( x ) & b.c. Parameters from modelling epistemic / aleatoric uncertainty or design. Specific values of parameter p are realisations of κ , f , or b.c. This involves an infinite (at first sight uncountable) real functions (random variables—RVs) C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
5 Realisation of κ C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
6 Parametric problems For each p in a parameter set P , let r ( p ) be an element in a Hilbert space Z (for simplicity). With r : P → Z , denote U = span r ( P ) = span im r . What we are after: other representations of r or U = span im r . To each function r : P → U corresponds a linear map R : U → ˜ R : R : U ∋ u �→ � r ( · ) | u � U ∈ ˜ R = im R ⊂ R P . By construction R is injective. Use this to make ˜ R a pre-Hilbert space: ∀ φ, ψ ∈ ˜ R : � φ | ψ � R := � R − 1 φ | R − 1 ψ � U . R − 1 is unitary on completion R . C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
7 RKHS and classification R is a reproducing kernel Hilbert space —RKHS— with symmetric kernel κ ( p 1 , p 2 ) = � r ( p 1 ) | r ( p 2 ) � U ∈ R P×P ; ∀ p ∈ P : κ ( p, · ) ∈ R , and span { κ ( · , p ) | p ∈ P} = R . Reproducing property: ∀ φ ∈ R : � κ ( p, · ) | φ ( · ) � R = φ ( p ) . In other settings (classification, machine learning, SVM), when different subsets of P have to be classified, the space U and the map r : P → U is not given, but can be freely chosen. It is then called the feature map. The whole procedure is called the kernel trick. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
8 Examples The function r ( p ) may be • function(s) describing some system— A ( p ; u ) = f . • a random field / process as input to some system— A ( ς ( p ); u ) = f . • the solution / state of some system depending on the above— u ( p ) . • the (non-linear) operator A ( p ; · ) / bi-linear / semi-linear form a ( p ; · ; · ) determining some system. One special case is when the parameter is a random quantity. Methods can be inspired from this model. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
9 Representation on RKHS In contrast to P (just some set), R is a vector space. Assume that R is separable, choose complete orthonormal system (CONS) { y m } m such that span { y 1 , y 2 , . . . } = R . Set u m = R − 1 y m ∈ U then r ( p ) = � m y m ( p ) u m (linear in y m ). We find that r ∈ U ⊗ R , and m u m ⊗ y m and R − 1 = � R = � m y m ⊗ u m . But choice of CONS is arbitrary. Let Q R : ℓ 2 ∋ a = ( a 1 , a 2 , . . . ) �→ � m a m y m ∈ R —a unitary map. Then R − 1 ◦ Q R (unitary) represents U , linear in a − → r ∈ U ⊗ ℓ 2 . We are looking for representations on other vector spaces. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
10 ‘Correlation’ If there is another inner product �·|·� Q on a subspace Q ⊂ R P , (e.g. if ( P , µ ) is measure space, set Q := L 2 ( P , µ ) ) a linear map C := R ∗ R —the ‘correlation’ operator—is defined by R ∗ w.r.t. Q . ∀ u, v ∈ U ; � Cu, v � U ′ ×U = � Ru | Rv � Q ; � � � In case Q = L 2 ( P , µ ) : C = P r ( p ) ⊗ r ( p ) µ (d p ) It is self-adjoint and positive definite → has spectrum σ ( C ) ⊆ R + . Spectral decomposition with projectors E λ on λ ∈ σ ( C ) = σ p ( C ) ∪ σ c ( C ) � ∞ � � Cu = λ d E λ u = λ m � v m | u � U v m + λ d E λ u. 0 σ c ( C ) λ m ∈ σ p ( C ) (Assume simple spectrum for simplicity ;-) C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
11 Spectral decomposition Often C has a pure point spectrum (e.g. C or C − 1 compact) ⇒ last integral vanishes, i.e. σ ( C ) = σ p ( C ) : � � λ m � v m | u � v m = λ m ( v m ⊗ v m ) u. Cu = m m If σ ( C ) c � = ∅ need generalised eigenvectors v λ and Gel’fand triplets (rigged Hilbert spaces) for the continuous spectrum: � � λ ( v λ ⊗ v λ ) u ̺ (d λ ) . λ d E λ u = σ c ( C ) σ c ( C ) � � ⇒ λ m ( v m ⊗ v m ) u + λ ( v λ ⊗ v λ ) u ̺ (d λ ) . Cu = σ c ( C ) λ m ∈ σ p ( C ) Representation as sum / integral of rank-1 operators. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
12 Singular value decomposition Another spectral decomposition : C unitarily equiv. to multiplication operator M k on L 2 ( X ) C = V M k V ∗ = ( V M 1 / 2 )( V M 1 / 2 ) ∗ , with M 1 / 2 = M √ k , k k k spectrum σ ( C ) is (ess.) range of k : X → R , hence k ( x ) ≥ 0 a.e. x ∈ X . This connects to the singular value decomposition (SVD) of R = SM 1 / 2 V ∗ , with a (here) unitary S − → r ∈ U ⊗ L 2 ( X ) . k � � � λ m ( v m ⊗ s m ) . With λ m s m := Rv m : R = m A sum / integral of rank-1 operators. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
13 Model reduction For purely discrete spectrum we get r ∈ U ⊗ Q � � r ( p ) = λ m s m ( p ) v m . m eve-expansion, due to SVD − → r ∈ U ⊗ L 2 ( σ ( C )) . This is Karhunen-Lo` A sum of rank-1 operators / tensors. Corresponds to R ∗ = � � λ m ( s m ⊗ v m ) . m √ Observe that r is linear in the “coordinates” λ m s m , e.g. necessary for offline part in reduced basis method (RBM). A representation of r , model reduction possible by truncation of sum, weighted by singular values √ λ m . C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
14 Factorisations / re-parametrisations R ∗ serves as representation. This is a factorisation of C = R ∗ R . Some other possible ones: ) ∗ = C 1 / 2 C 1 / 2 = B ∗ B, C = R ∗ R = ( V M 1 / 2 )( V M 1 / 2 k k where C = B ∗ B is an arbitrary one. Each factorisation leads to a representation—all unitarily equivalent. (When C is a matrix, a favourite is Cholesky: C = LL ∗ ). Assume that C = B ∗ B and B : U → H − → r ∈ U ⊗ H . Analogous results / factorisations / representations follow from C := RR ∗ : Q → Q . considering ˆ Also known as kernel decompositions, usually integral transforms. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
15 Representations We have seen several ways to represent the solution space by a—hopefully—simpler space. These can all be used for model reduction, choosing a smaller subspace. • The RKHS-representation on R together with R − 1 . eve expansion on Q via R ∗ (SVD). • The Karhunen-Lo` • The spectral decomposition over L 2 ( σ ( C )) or via V M 1 / 2 on L 2 ( X ) . k • Other multiplicative decompositions, such as C = B ∗ B on H . • Analogous: The kernel decompositions and representation based on C = RR ∗ lead to integral transforms. kernel κ or ˆ Choice depends on what is wanted / needed. Notion of measure / probability measure on P was not needed. C C Scientifi omputing TU Braunschweig Institute of Scientific Computing
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