Reduced order modeling and numerical linear algebra Akil Narayan 1 1 Department of Mathematics, and Scientific Computing and Imaging (SCI) Institute University of Utah February 7, 2020 ICERM A. Narayan (U. Utah – SCI) NLA and ROM
Continuous Ø discrete analogies Most standard techniques for reduced basis methods can be understood from numerical linear algebra. Kolmogorov n widths Ø Singular value decompositions Reduced basis methods Ø QR decompositions Empirical interpolation methods Ø LU decompositions A. Narayan (U. Utah – SCI) NLA and ROM
Kolmogorov n widths are (essentially) singular values A. Narayan (U. Utah – SCI) NLA and ROM
Singular value decompositions Let A P ❘ M ˆ N , with M " N . We will think of the columns of A as snapshots. ¨ ˛ ˝ a 1 A : “ a 2 ¨ ¨ ¨ a N ‚ The SVD of A is A “ U Σ V T , where U and V are orthogonal M ˆ M and N ˆ N matrices, respectively. Σ is a diagonal matrix with non-negative entries. We’ll use the following non-standard notation for the entries in Σ : σ 0 ě σ 1 ě ¨ ¨ ¨ ě σ N ´ 1 . A. Narayan (U. Utah – SCI) NLA and ROM
Low-rank approximations Among the nice properties of the SVD is its ability to form low-rank approximations, A k : “ U k Σ k V T 1 ď k ď N, k , where U k and V k are k -column truncations, and Σ k is a k ˆ k principcal submatrix truncation. With rank p A k q “ k , then A k “ arg min } A ´ B } ˚ , rank p B qď k for ˚ “ 2 , F . A. Narayan (U. Utah – SCI) NLA and ROM
Low-rank approximations Among the nice properties of the SVD is its ability to form low-rank approximations, A k : “ U k Σ k V T 1 ď k ď N, k , where U k and V k are k -column truncations, and Σ k is a k ˆ k principcal submatrix truncation. With rank p A k q “ k , then A k “ arg min } A ´ B } ˚ , rank p B qď k for ˚ “ 2 , F . Equivalently, A k is the projection of the columns of A onto R p U k q : ¨ ˛ ˝ P R p U k q a 1 A k “ P R p U k q a 2 ¨ ¨ ¨ P R p U k q a N ‚ A. Narayan (U. Utah – SCI) NLA and ROM
Projections onto arbitrary spaces What if we project A onto other spaces? If V Ă ❘ M is any subspace, we could consider ¨ ˛ ˝ P V a 1 P V A : “ P V a 2 ¨ ¨ ¨ P V a N ‚ A. Narayan (U. Utah – SCI) NLA and ROM
Projections onto arbitrary spaces What if we project A onto other spaces? If V Ă ❘ M is any subspace, we could consider ¨ ˛ ˝ P V a 1 P V A : “ P V a 2 ¨ ¨ ¨ P V a N ‚ And we could ask about a certain type of error committed by this approximation E p V q : “ max } x } 2 “ 1 } A x ´ P V A x } 2 We know V “ R p U k q does a pretty good job. What about other spaces? A. Narayan (U. Utah – SCI) NLA and ROM
❘ ❘ Optimal projections For a given rank k , an “optimal” projection commits the smallest error: E k : “ V Ă ❘ M E p V q min A. Narayan (U. Utah – SCI) NLA and ROM
❘ Optimal projections For a given rank k , an “optimal” projection commits the smallest error: E k : “ V Ă ❘ M E p V q min So an extremal characterization of an SVD-based low rank approximation is R p U k q “ arg min } x } 2 “ 1 } A x ´ P A x } 2 . max V Ă ❘ N A. Narayan (U. Utah – SCI) NLA and ROM
Optimal projections For a given rank k , an “optimal” projection commits the smallest error: E k : “ V Ă ❘ M E p V q min So an extremal characterization of an SVD-based low rank approximation is R p U k q “ arg min } x } 2 “ 1 } A x ´ P A x } 2 . max V Ă ❘ N Or, an (unnecessarily?) pedantic alternative: E k “ σ k p A q “ min v P V } Ax ´ v } 2 } x } 2 “ 1 min max V Ă ❘ N A. Narayan (U. Utah – SCI) NLA and ROM
❘ ❘ ❘ ❘ SVD projections Given A P ❘ M ˆ N , the success of a low-rank projection is dictated by the approximation numbers σ k p A q “ min } x } 2 “ 1 min max v P V } Ax ´ v } 2 . V Ă ❘ N More precisely, it is dictated by fast decay of these numbers as k increases. A. Narayan (U. Utah – SCI) NLA and ROM
SVD projections Given A P ❘ M ˆ N , the success of a low-rank projection is dictated by the approximation numbers σ k p A q “ min } x } 2 “ 1 min max v P V } Ax ´ v } 2 . V Ă ❘ N More precisely, it is dictated by fast decay of these numbers as k increases. These numbers are defined by our choice of metric on “output” space ❘ M , and our choice of metric on “measurement” space ❘ N . I.e., a generalization might look like ´ A ; ℓ p ´ ❘ M ¯ , ℓ q ´ ❘ N ¯¯ “ v P V } Ax ´ v } p . σ k min } x } q “ 1 min max dim V ď k A. Narayan (U. Utah – SCI) NLA and ROM
Kolmogorov n widths ´ A ; ℓ p ´ ❘ M ¯ , ℓ q ´ ❘ N ¯¯ σ n “ min } x } q “ 1 min max v P V } Ax ´ v } p . dim V ď n These numbers tell us how well the columns of A are ℓ p -approximated by a linear space using ℓ q measurements. Another definition might be the maximum column norm error: ´ A ; ℓ p ´ ❘ M ¯¯ σ n “ dim V ď n max min i Pr N s min v P V } Ae i ´ v } p . Great. How do we do all this with functions? A. Narayan (U. Utah – SCI) NLA and ROM
Kolmogorov n widths ´ A ; ℓ p ´ ❘ M ¯ , ℓ q ´ ❘ N ¯¯ σ n “ min } x } q “ 1 min max v P V } Ax ´ v } p . dim V ď n These numbers tell us how well the columns of A are ℓ p -approximated by a linear space using ℓ q measurements. Another definition might be the maximum column norm error: ´ A ; ℓ p ´ ❘ M ¯¯ σ n “ dim V ď n max min i Pr N s min v P V } Ae i ´ v } p . Great. How do we do all this with functions? Let A be a collection of functions in a Hilbert space H . Then one way to talk about similar concepts to ( ℓ 2 ) singular values is σ n p A ; H q “ v P V } a ´ v } dim V ď n sup inf inf a P A This is called the Kolmogorov n width of A (with respect to H ). A. Narayan (U. Utah – SCI) NLA and ROM
Reduced basis methods (essentially) perform QR decompositions A. Narayan (U. Utah – SCI) NLA and ROM
Interpolative decompositions One disadvantage of SVD-based low rank approximations, ¨ ˛ ‚ “ U Σ V T , ˝ a 1 A “ ¨ ¨ ¨ a 2 a N is that we need information from all columns of A to define U . A. Narayan (U. Utah – SCI) NLA and ROM
Interpolative decompositions One disadvantage of SVD-based low rank approximations, ¨ ˛ ‚ “ U Σ V T , ˝ a 1 A “ ¨ ¨ ¨ a 2 a N is that we need information from all columns of A to define U . One alternative: Interpolative decompositions, or matrix skeletonizations. Basic idea: project all columns of A onto a subspace spanned by a few columns. A rank- n column skeletonization of A is ¨ ˛ ¯ : ´ A T A T ˝ e s 1 B “ A S S A S A , A S : “ A e s 2 ¨ ¨ ¨ e s n ‚ , S l jh n P R p A S q with S “ t s 1 , . . . s n u Ă r N s . A. Narayan (U. Utah – SCI) NLA and ROM
Choosing the columns S The problem of choosing S that is optimal in some metric is the column subset selection problem. For metrics of interest, it’s NP-hard. A. Narayan (U. Utah – SCI) NLA and ROM
Choosing the columns S The problem of choosing S that is optimal in some metric is the column subset selection problem. For metrics of interest, it’s NP-hard. So let’s do something else: Let’s pick columns greedily: Given S Ă r N s of size n , we’ll add a column index via the procedure › › s n ` 1 “ arg max › a j ´ P R p A S q a j 2 . › j Pr N s This is much cheaper since I need only to evaluate N vector norms at each step. A. Narayan (U. Utah – SCI) NLA and ROM
Choosing the columns S The problem of choosing S that is optimal in some metric is the column subset selection problem. For metrics of interest, it’s NP-hard. So let’s do something else: Let’s pick columns greedily: Given S Ă r N s of size n , we’ll add a column index via the procedure › › s n ` 1 “ arg max › a j ´ P R p A S q a j 2 . › j Pr N s This is much cheaper since I need only to evaluate N vector norms at each step. There’s already a well-polished algorithm that does this: the QR decomposition. A. Narayan (U. Utah – SCI) NLA and ROM
The QR decomposition (1/2) The column-pivoted QR decomposition iteratively computes orthonormal vectors in the range of A . At step j , the next column is identified as the one whose projected residual is largest. P j ´ 1 : “ Q j ´ 1 Q T j ´ 1 s j “ arg max } a j ´ P j ´ 1 a j } 2 j Pr N s a s j “ ‰ q j : “ } a s j } 2 , Q j “ Q j ´ 1 , q j A. Narayan (U. Utah – SCI) NLA and ROM
The QR decomposition (1/2) The column-pivoted QR decomposition iteratively computes orthonormal vectors in the range of A . At step j , the next column is identified as the one whose projected residual is largest. P j ´ 1 : “ Q j ´ 1 Q T j ´ 1 s j “ arg max } a j ´ P j ´ 1 a j } 2 j Pr N s a s j “ ‰ q j : “ } a s j } 2 , Q j “ Q j ´ 1 , q j The residual › › r j ´ 1 : “ › a s j ´ P j ´ 1 a s j 2 , › is the largest ( ℓ 2 -norm) column mistake we make by choosing S “ t s 1 , . . . , s j ´ 1 u , i.e., by replacing A Ð P V A , V : “ span t a s 1 , . . . , a s j ´ 1 u . A. Narayan (U. Utah – SCI) NLA and ROM
The QR decomposition (2/2) This algorithm is a greedy algorithm: instead of all-at-once optimization, we optimize one at a time. Clearly, we don’t expect this to perform as well as the optimal SVD-based subspace. But how well does this greedy procedure work in practice? A. Narayan (U. Utah – SCI) NLA and ROM
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