TY - JOUR

T1 - Optimal numerical approximation of a linear operator

AU - Weinberger, H. F.

N1 - Funding Information:
through grant NSF MCS

PY - 1983/7

Y1 - 1983/7

N2 - Let S be a bounded linear transformation from a. Hilbert space B to a Hilbert space Σ. Then Su can be thought of as the solution of a linear differential equation with right-hand side, initial data, or boundary data u. Given the incomplete information Nu = v, ∥u∥B ≤ 1 about the data, where N is a linear operator from B to a Euclidean space En, and a linear interpolation M from Em to Σ, one defines the optimal approximation to Su to be the point Mâ(v) in the range of M which is the center of the smallest ball containing all points of the form Su with Nu = v and ∥u∥B ≤ 1 and centered in M. A characterization is given for the optimal approximation Mâ(v). It is shown to be unique and, in general, nonlinear. Simpler approximations and relations with other concepts of optimality are investigated.

AB - Let S be a bounded linear transformation from a. Hilbert space B to a Hilbert space Σ. Then Su can be thought of as the solution of a linear differential equation with right-hand side, initial data, or boundary data u. Given the incomplete information Nu = v, ∥u∥B ≤ 1 about the data, where N is a linear operator from B to a Euclidean space En, and a linear interpolation M from Em to Σ, one defines the optimal approximation to Su to be the point Mâ(v) in the range of M which is the center of the smallest ball containing all points of the form Su with Nu = v and ∥u∥B ≤ 1 and centered in M. A characterization is given for the optimal approximation Mâ(v). It is shown to be unique and, in general, nonlinear. Simpler approximations and relations with other concepts of optimality are investigated.

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U2 - 10.1016/0024-3795(83)80045-7

DO - 10.1016/0024-3795(83)80045-7

M3 - Article

AN - SCOPUS:48749145473

VL - 52-53

SP - 717

EP - 737

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - C

ER -