Introduction Interpolation Non-adaptive Adaptive Conclusion Interpolation in high dimensions: Non-intrusive reduced order modeling Akil Narayan 1 1 Department of Mathematics University of Massachusetts Dartmouth June 7, 2013 GR-ROM @ Caltech Pasadena, CA A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Parameterized functions Problems of interest are often functions that depend both on space x and a parametric variable µ . Let x ∈ D ⊆ R p be a space-like variable ( p = 1 , 2 , 3) and µ ∈ Ω ⊆ R d a parameter ( d ≥ 1). If u = u ( x ; µ ) , an approximation u N ≃ u is usually formed via some combined spatial ( x ) discretization and parametric ( µ ) discretization. The whole game: compute u N . For each µ evaluating u ( x ; µ ) is expensive. The main goal: approximate u ( x ; µ ) with as µ 1 few parametric degrees of freedom as possible. µ 2 Ω In particular, use only µ -point-evaluation information. A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion The main ideas Why is this important? need u ( x ; µ ) for numerous values of µ for a given µ , need fast query of u ( x ; µ ) want µ -moment information about u ( x ; µ ) The major points and discussions in this talk: Interpolatory ("non-intrusive") methods can perform on par with projective ("intrusive") methods Non-adaptive interpolatory methods: single-dimension fundamentals and high-dimensional techniques Adaptive interpolatory methods: optimal approximation spaces and reconstructions Themes throughout: greedy schemes, pivoted linear algebra routines A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion General setup We are concerned with the standard linear approximation techniques: N N ∑ ∑ u N ( x ; µ ) = c n b n ( µ ) v n ( x ) = C n ( µ ) v n ( x ) m , n = 1 n = 1 The coefficients C n ( µ ) and the basis v n determine the approximation. Some notation throughout: V approximates x V n subspace of V with basis v n B approximates µ B n subspace of B with basis b n Generally, simulation tools are developed to evaluate the following map: µ �→ u ( x ; µ ) , µ fixed (1) This limited information about u ( x ; µ ) constrains our knowledge. A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Intrusive methods One approach: with some preconceived basis v n ( x ) , b n ( µ ) in a Hilbert space V × B , construct N ∑ u N ( x ; µ ) = c m , n b m ( µ ) v n ( x ) , m , n = 1 and ask that u N = proj V N × B N u . (Or appropriate residual formulation for DE.) Determining the approximation coefficients c m , n requires information ⟨ u ( x ; µ ) , v n ( x ) b m ( µ ) ⟩ V × B , but we can only evaluate the map u ( x ; µ ) for a fixed µ . Thus we require data beyond what (1) can provide, so a rewrite of existing simulation tools is necessary: intrusive. A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Non-intrusive methods A second approach: with some preconceived basis v n ( x ) , b n ( µ ) , construct N ∑ u N ( x ; µ ) = c m , n b m ( µ ) v n ( x ) , m , n = 1 and ask that u N ( · ; µ n ) = proj V N u ( · ; µ n ) for some chosen nodes µ n . Note: In principle no Hilbertian structure on V necessary. This is an interpolatory approach; the only data we need is u ( x ; µ n ) at the sites µ n . Thus we can use the existing simulation tools: non-intrusive A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion A short, sweet example For concreteness, consider an elliptic problem ( ) − d κ ( x , µ ) d u ( x ; µ ) = f ( x ; µ ) , d x d x with x ⊂ R and µ ∈ [ − 1 , 1 ] 8 ⊂ R 8 . The diffusion coefficient is given by 8 ∑ 1 κ ( x ; µ ) = 1 + π j 2 cos ( 2 π jx ) µ j , j = 1 We seek to approximate u ( x ; µ ) : In this case, a non-intrusive method can perform comparably to an intrusive method. A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion A classical ("intrusive") approach Consider a single variable ( x , µ ) and perform a Galerkin (FEM-like) approximation: Find u N ∈ V N × B N such that ⟨ ⟩ ( ) ′ , v ( x ; µ ) κ ( x ; µ ) u N ′ ( x ; µ ) = ⟨ f ( x ; µ ) , v ( x ; µ ) ⟩ , − ∀ v ∈ V N × B N . This is intrusive: we require projective, non-interpolatory information about u ( µ ) . The non-intrusive interpolatory approach: select some parametric locations µ n , n = 1 , . . . , N , and find u N ∈ V N × B N such that u N ( x ; µ n ) = u ( x ; µ n ) , ∀ µ n A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Convergence Intrusive methods are generally more accurate, but are more expensive. 10 0 Intrusive Galerkin Non-intrusive interpolation Galerkin-FEM intrusive k = 6 10 − 4 solution requires a linear solve of size ∼ 10 5 L 2 error 10 − 8 The non-intrusive interpolatory k = 6 10 − 12 approach requires ∼ 3000 linear solves of size ∼ 30. 10 − 16 1 2 3 4 5 6 Polynomial order k A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Non-adaptive interpolation topics Interpolation: u ( · ; µ n ) = u N ( · ; µ n ) , n = 1 , . . . , N . Choice of µ n is the next subject under discussion. Lagrange interpolation and Lebesgue constants polynomials: one-dimensional grids higher dimensions: tensorizations, sparse grids greedy, 'unstructured' methods: Fekete and Leja points A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Non-adaptive interpolation For non-adaptive interpolation, both the basis and the coefficients are chosen independent of the function u : N ∑ u ( x ; µ ) ≃ u N = C n ( µ ) u ( x ; µ n ) n = 1 Both the parametric locations µ n and the parametric dependence C n ( µ ) are free to be chosen. In order to intelligently choose µ n , we specify a basis for C n : N ∑ C n ( µ ) = c n , m b m ( µ ) , n = 1 , . . . , N . m = 1 Realistically, the b n are selected from a standard µ -approximation set, e.g. polynomials, trigonometric functions, wavelets, splines, etc. A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Lagrange interpolation A clearer way to see what is happening: solve for c n , m so that u N interpolates u at µ n . Then: N ∑ C n ( µ ) = c n , m b m ( µ ) = ℓ n ( µ ) , m = 1 with ℓ n the cardinal Lagrange interpolant of the b n at the sites µ n . I.e.: n = 1 , . . . , N ℓ n ( µ m ) = δ n , m , ℓ n ( µ ) determines "how much" information from u ( · ; µ n ) contributes to reconstruction at µ . These can be constructed without the data from u . This process is 100% independent from u . A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Lagrange interpolation 1 1 0 . 8 0 . 5 0 . 6 0 . 4 µ 2 0 µ 0 . 2 − 0 . 5 0 − 0 . 2 − 1 -1 -0.5 0 0.5 1 µ 1 µ A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Interpolation Error Error estimates from classical spatial interpolation may be augmented here. � � sup ∥ u ( · ; µ ) − u N ( · ; µ ) ∥ V N ≤ sup � u ( · ; µ ) − proj V N u ( · ; µ ) � µ ∈ Ω µ ∈ Ω [ ] + ( 1 + Λ) d proj V N u ( · ; µ ) , V N × B N Some of the terms are optimal so we cannot do better. They depend only on the approximation spaces V N and B N . But there is a penalty for interpolation; the "Lebesgue constant". With the approximation space fixed, it depends only on the choice of interpolation nodes. One central question in the formulation of non-adaptive interpolation methods: how to choose µ n to minimize Λ ? A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
Introduction Interpolation Non-adaptive Adaptive Conclusion Polynomial interpolation For concreteness, consider polynomials. Some one-dimensional intuition: the Lebesgue constant for any nodal array is unbounded in N equispaced nodes are bad -- exponentially growing Λ arcsine-distributed nodes are good -- logarithmically growing Λ Bad Good A. Narayan (University of Massachusetts Dartmouth) Nonadaptive and adaptive interpolation
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