On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

Jin, Bangti; Zhou, Zehui; Zou, Jun

doi:10.48550/arxiv.1907.03132

Cited by 3 publications

(2 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this situation, the forward model is the classical elliptic type equation, and mathematically, recovering the optical parameter amounts to reconstructing the diffusion coefficient in the elliptic equation using the Dirichlet-to-Neumann map [42], and is proven mathematically to be logarithmically unstable [1]. Multiple strategies are adopted to 'stabilize' the problem [36,40], both by adjusting the experimental modalities upfront, or introducing image deblurring techniques as a post-processing [18,29,39,41].…”

Section: Introductionmentioning

confidence: 99%

On diffusive scaling in acousto-optic imaging

Chung¹,

Lai²,

Li³

2020

Inverse Problems

View full text Add to dashboard Cite

Acousto-optic imaging (AOI) is a hybrid imaging process. By perturbing the to-be-reconstructed tissues with acoustic waves, one introduces the interaction between the acoustic and optical waves, leading to a more stable reconstruction of the optical properties. The mathematical model was described in [27], with the radiative transfer equation serving as the forward model for the optical transport. In this paper we investigate the stability of the reconstruction. In particular, we are interested in how the stability depends on the Knudsen number, Kn, a quantity that measures the intensity of the scattering effect of photon particles in a media. Our analysis shows that as Kn decreases to zero, photons scatter more frequently, and since information is lost, the reconstruction becomes harder. To counter this effect, devices need to be constructed so that laser beam is highly concentrated. We will give a quantitative error bound, and explicitly show that such concentration has an exponential dependence on Kn. Numerical evidence will be provided to verify the proof.

show abstract

Section: Introductionmentioning

confidence: 99%

On diffusive scaling in acousto-optic imaging

Chung¹,

Lai²,

Li³

2020

Inverse Problems

View full text Add to dashboard Cite

show abstract

“…In this situation, the forward model is the classical elliptic type equation, and mathematically, recovering the optical parameter amounts to reconstructing the diffusion coefficient in the elliptic equation using the Dirichlet-to-Neumann map [39], and is proven mathematically to be logarithmically unstable [1]. Multiple strategies are adopted to "stabilize" the problem [33,37], both by adjusting the experimental modalities upfront, or introducing image deblurring techniques as a post-processing [16,27,36,38].…”

Section: Introductionmentioning

confidence: 99%

On diffusive scaling in acousto-optic imaging

Chung

Lai²,

Qin³

2020

Preprint

View full text Add to dashboard Cite

Acousto-optic imaging (AOI) is a hybrid imaging process. By perturbing the to-bereconstructed tissues with acoustic waves, one introduces the interaction between the acoustic and optical waves, leading to a more stable reconstruction of the optical properties. The mathematical model was described in [25], with the radiative transfer equation serving as the forward model for the optical transport. In this paper we investigate the stability of the reconstruction. In particular, we are interested in how the stability depends on the Knudsen number, Kn, a quantity that measures the intensity of the scattering effect of photon particles in a media. Our analysis shows that as Kn decreases to zero, photons scatter more frequently, and since information is lost, the reconstruction becomes harder. To counter this effect, devices need to be constructed so that laser beam is highly concentrated. We will give a quantitative error bound, and explicitly show that such concentration has an exponential dependence on Kn. Numerical evidence will be provided to verify the proof.

show abstract

On sampling Kaczmarz–Motzkin methods for solving large-scale nonlinear systems

Zhang

Bao

et al. 2023

Comp. Appl. Math.

View full text Add to dashboard Cite

On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

Cited by 3 publications

References 26 publications

On diffusive scaling in acousto-optic imaging

On diffusive scaling in acousto-optic imaging

On diffusive scaling in acousto-optic imaging

On sampling Kaczmarz–Motzkin methods for solving large-scale nonlinear systems

Contact Info

Product

Resources

About