# Registered Data

## [02277] New regularizing algorithms for solving inverse and ill-posed problems

**Session Date & Time**:- 02277 (1/6) : 3E (Aug.23, 17:40-19:20)
- 02277 (2/6) : 4C (Aug.24, 13:20-15:00)
- 02277 (3/6) : 4D (Aug.24, 15:30-17:10)
- 02277 (4/6) : 4E (Aug.24, 17:40-19:20)
- 02277 (5/6) : 5B (Aug.25, 10:40-12:20)
- 02277 (6/6) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Regularization methods are an important numerical technique in the robust solution of ill-posed inverse problems. In the recent years, many new regularization methods had been developed in various areas of applied mathematics and data science. This minisymposium focuses on the regularization theory of abstract (linear and nonlinear) operator equations and some efficient regularizing algorithms for solving parameter identification problems in (local and nonlocal) PDEs. Such regularizing algorithms includes stochastic asymptotic regularization for general operator equation, asymptotic expansion regularization for inverse problems in singularly perturbed PDEs, machine learning based approaches, etc. The speakers mainly from Russian and China.**Organizer(s)**: Ye Zhang, Maxim Shishlenin, Anatoly Yagola and Sergey Kabanikhin**Classification**:__35R30__,__47A52__,__65R30__,__65N30__,__35N25__**Speakers Info**:- Anatoly Yagola (Moscow State University)
- Vladimir Vasin (Krasovskii Institute of Mathematics and Mechanics)
- Michail Belishev (Steklov Mathematical Institute)
- Dmitry Lukyanenko (Moscow State University)
- Michail Kokurin (Mari State University)
- Denis Sidorov (Energy Systems Institute)
- Nikita Novikov (Sobolev Institute of Mathematics)
- Olga Krivorotko (Sobolev Institute of Mathematics)
- Rongfang Gong (Nanjing University of Aeronuatics and Astronautics)
- Wei Wang (Jiaxing university)
- Wenlong Zhang (Southern University of Science and Technology(SUSTech))
- Chunlong Sun (Nanjing University of Aeronautics and Astronautics)
- Long Li (Harbin Institute of Technology)
- Chen Xu (Shenzhen MSU-BIT University)
- Haie Long (Shenzhen MSU-BIT University)
- Dmitrii Chaikovskii (Shenzhen MSU-BIT University)
- Yue Qiu (Chongqing University)
- Ning Zheng (Tongji University)
- Zhigang Yao (National University of Singapore)
- Zuurakan Kadenova (National Academy of Sciences of the Republic of Kyrgyzstan)
- Shuang Liu (Novosibirsk State University)

**Talks in Minisymposium**:**[02803] Uniqueness and numerical inversion in the time-domain fluorescence diffuse optical tomography****Author(s)**:**Chunlong Sun**(Nanjing University of Aeronautics and Astronautics)

**Abstract**: This work considers the time-domain fluorescence diffuse optical tomography. We recover the distribution of fluorophores in biological tissue by the boundary measurements. With the Laplace transform and the knowledge of complex analysis, we build the uniqueness theorem of this inverse problem. The numerical inversions are considered. We introduce an iterative inversion algorithm under the framework of regularizing scheme.

**[02806] Data-Driven Regularization in Variational Data Assimilation from An Ocean Perspective****Author(s)**:**Long Li**(Harbin Institute of Technology)- Jianwei Ma (Peking University)

**Abstract**: Current machine learning-driven methods make a positive difference to outlook on data assimilation. However, its reliability remains to be studied since little deterministic information from physical laws is involved. In this talk, we will introduce our recent work about the efficient assimilation by the approximation of a deep neural network. The sparsity regularization is employed to improve the well-posedness. Results show the technique is robust for reconstructing the velocity of a fluid with vortex structures.

**[02839] Stochastic asymptotical regularization for nonlinear ill-posed problems****Author(s)**:**Haie Long**(Shenzhen SMU-BIT University)

**Abstract**: In this paper, we establish an initial theory regarding the stochastic asymptotical regularization (SAR) for the uncertainty quantification of the stable approximate solution of ill-posed nonlinear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. By combining techniques from classical regularization theory and stochastic analysis, we prove the regularizing properties of SAR with regard to mean-square convergence. The convergence rate results under the canonical sourcewise condition are also studied. Several numerical examples are used to show the accuracy and advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of a solution and escape local minimums for nonlinear problems.

**[03062] A new framework to quantify the uncertainty in inverse problems****Author(s)**:**Wenlong Zhang**(Southern University of Science and Technology)

**Abstract**: In this work, we investigate the regularized solutions and their finite element solutions to the inverse problems governed by partial differential equations, and establish the stochastic convergence and optimal finite element convergence rates of these solutions, under point wise measurement data with random noise. The regularization error estimates and the finite element error estimates are derived with explicit dependence on the noise level, regularization parameter, mesh size, and time step size, which can guide practical choices among these key parameters in real applications. The error estimates also suggest an iterative algorithm for determining an optimal regularization parameter.

**[03104] Numerical algorithms for solving the nonlinear Schrödinger equation****Author(s)**:**Shuang Liu**(Novosibirsk State University)

**Abstract**: In this paper, the Physical Information Neural Networks algorithm is used to solve the nonlinear Schrödinger equation in a dispersed medium. Adaptive activation functions are used to accelerate PINN convergence, and this approach uses very little data to obtain an exact solution. Due to the approximation capability of the neural network, the results are used in semiconductor optical amplifier fiber lasers where nonlinear effects allow spectral tuning of the generated pulses.

**[03354] Multidimensional Ill-Posed Problems in Applications****Author(s)**:**Anatoly Yagola**(Lomonosov Moscow State University)

**Abstract**: The report will consider applied multidimensional inverse problems of geophysics (magnetometry and gravimetry) and electron microscopy (electron backscattering), regularizing algorithms for their solution and the results of experimental data processing.

**[04352] Alternating modulus iterations for constrained Tikhonov regularization problem****Author(s)**:**Ning Zheng**(The Institute of Statistical Mathematics)

**Abstract**: The linear discrete ill-posed problem with total variation Tikhonov model is considered for preserving sharp attributes in images. Meanwhile, the restored images from TV-based methods are constrained in a given dynamic range. We propose using the alternating direction method of multipliers to solve the constrained models and the constrained subproblems are solved by modulus-based iteration methods. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, and the proposed algorithm is better than state-of-the-art unconstrained algorithms in terms of both accuracy and robustness with respect to the regularization parameter.

**[04375] Solution of inverse problems in three-dimensional singularly perturbed PDEs****Author(s)**:**Dmitrii Chaikovskii**(Shenzhen MSU-BIT University)- Ye Zhang (Beijing Institute of Technology)

**Abstract**: We present an efficient asymptotic expansion method for solving forward and inverse problems in a nonlinear, time-dependent, singularly perturbed reaction-diffusion-advection equation. We prove the existence and uniqueness of a smooth solution in 3D PDEs using asymptotic expansion. A simplified equation for the inverse source problem is derived, maintaining accuracy even with noisy data. We propose an asymptotic expansion regularization algorithm for the 3D inverse source problem and demonstrate its feasibility through a model problem.

**[05101] The coupled complex boundary methods for inverse problems of partial differential equations****Author(s)**:**Rongfang Gong**(Nanjing University of Aeronautics and Astronautics)

**Abstract**: In this talk, a coupled complex boundary method (CCBM) is proposed for an inverse source problem. With the introduction of imaginary unit, the CCBM transfers the original real problem to a complex one. The CCBM has several merits and is further improved. Also, the applications of the CCBM to bioluminescence tomography, inverse Cauchy problem, chromatography etc. are delivered.