Positive solutions to linear inverse problems

Abstract

We discuss two methods for incorporating the prior knowledge that the solution is positive into the truncated singular value decomposition method for solving linear inverse problems. The methods are based on mathematical programming techniques. One method can be viewed as a primal method and the other as its dual. Provided the singular functions are analytic these methods both deliver the same solution-namely the positive solution of minimum 2-norm which agrees with the truncated singular function expansion in its known terms-and this solution also appears to possess higher resolution. In the presence of noise both methods can sometimes fail to converge and in these situations we give simple remedies which yield approximate solutions. While there may be no reason to suppose the unknown object should be that of minimum 2-norm, our method has the advantage over other nonlinear methods that for a noiseless band-limited object the exact solution is delivered.

The BCS-BEC Crossover and the Unitary Fermi Gas

“…We use this flexibility of choosing the expansion coefficients to produce a solution satisfying constraints (9.55) [78]. The relative advantages of each method will be discussed elsewhere [79].…”

Conductance of the single-electron transistor: A comparison of experimental data with Monte Carlo calculations

“…Taking into account that for an ill-posed problem such as (8) the singular values σ j vanish overexponentially with increasing j it becomes obvious that only the first few terms in the expansion (12) contain meaningful information whereas the higher terms will just corrupt the result by amplifying the noise of the data. This idea is used in the truncated SVD approach 19,20 which truncates the summation in (12) by neglecting all terms for which σ j /σ 0 is smaller than the statistical error of the correlation function C(τ ).…”

“…Previous studies 19,21 have shown that this implementation of positiveness reduces the resolution of the method. Recent work 20 has shown how to use supplementary information of positiveness to enhance the resolution of the singular value decomposition method. The idea is to determine additional expansion coefficients which cannot be inferred from the inverse problem.…”

Probabilistic regularization in inverse optical imaging

“…(30) can be different from the set I of Eq. (18). The former is actually the result of an algorithm acting on a given set of data and can be thought of as a numerical realization of the theoretical set I of (18).…”

“…(18). The former is actually the result of an algorithm acting on a given set of data and can be thought of as a numerical realization of the theoretical set I of (18). A similar role is played by the numerical approximation f I (x) with respect to the theoretical approximation Bḡ of Eq.…”