Efficient iterative solutions to complex-valued nonlinear least-squares problems with mixed linear and antilinear operators
We consider a setting in which it is desired to find an optimal complex vector x ∈ C N that satisfies A ( x ) ≈ b in a least-squares sense, where b ∈ C M is a data vector (possibly noise-corrupted), and A ( · ) : C N → C M is a measurement operator. If A ( · ) were linear, this reduces to the classi...
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| Vydáno v: | Optimization and engineering Ročník 23; číslo 2; s. 749 - 768 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.06.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 1389-4420, 1573-2924, 1573-2924 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We consider a setting in which it is desired to find an optimal complex vector
x
∈
C
N
that satisfies
A
(
x
)
≈
b
in a least-squares sense, where
b
∈
C
M
is a data vector (possibly noise-corrupted), and
A
(
·
)
:
C
N
→
C
M
is a measurement operator. If
A
(
·
)
were linear, this reduces to the classical linear least-squares problem, which has a well-known analytic solution as well as powerful iterative solution algorithms. However, instead of linear least-squares, this work considers the more complicated scenario where
A
(
·
)
is nonlinear, but can be represented as the summation and/or composition of some operators that are linear and some operators that are antilinear. Some common nonlinear operations that have this structure include complex conjugation or taking the real-part or imaginary-part of a complex vector. Previous literature has shown that this kind of mixed linear/antilinear least-squares problem can be mapped into a linear least-squares problem by considering
x
as a vector in
R
2
N
instead of
C
N
. While this approach is valid, the replacement of the original complex-valued optimization problem with a real-valued optimization problem can be complicated to implement, and can also be associated with increased computational complexity. In this work, we describe theory and computational methods that enable mixed linear/antilinear least-squares problems to be solved iteratively using standard linear least-squares tools, while retaining all of the complex-valued structure of the original inverse problem. An illustration is provided to demonstrate that this approach can simplify the implementation and reduce the computational complexity of iterative solution algorithms. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ISSN: | 1389-4420 1573-2924 1573-2924 |
| DOI: | 10.1007/s11081-021-09604-4 |