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
[05628] Regularization of linear inverse problems and neural networks
Format : Online Talk on Zoom
Author(s) :
Sergey Kabanikhin (Sobolev Institute of Mathematics)
Abstract : The report will consider methods of numerical regularization of linear inverse problems and neural networks in applications.
[05627] Applied inverse problems for parabolic equations
Format : Online Talk on Zoom
Author(s) :
Maxim Shishlenin (Sobolev Institute of Mathematics)
Abstract : The report will consider inverse problems for parabolic equations in applications such as pharmacodynamics and financial mathematics.
[02806] Data-Driven Regularization in Variational Data Assimilation from An Ocean Perspective
Format : Talk at Waseda University
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.
[02803] Uniqueness and numerical inversion in the time-domain fluorescence diffuse optical tomography
Format : Talk at Waseda University
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.
[02839] Stochastic asymptotical regularization for nonlinear ill-posed problems
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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.
[04375] Solution of inverse problems in three-dimensional singularly perturbed PDEs
Format : Talk at Waseda University
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
Format : Talk at Waseda University
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.
[05498] Physics-informed invertible neural network for the Koopman operator learning
Format : Talk at Waseda University
Author(s) :
Yue Qiu (Chongqing University)
Abstract : The Koopman operator is used to embed a nonlinear system into an infinite, yet linear system with a set of observable functions. However, manually selecting observable functions that span the invariant subspace of the Koopman operator based on prior knowledge is inefficient and challenging, particularly when little or no information is available about the underlying systems. Furthermore, current methodologies tend to disregard the importance of the invertibility of observable functions, which leads to inaccurate results. To address these challenges, we propose the so-called FlowDMD, a Flow-based Dynamic Mode Decomposition that utilizes the Coupling Flow Invertible Neural Network (CF-INN) framework. FlowDMD leverages the intrinsically invertible characteristics of the CF-INN to learn the invariant subspaces of the Koopman operator and accurately reconstruct state variables. Numerical experiments demonstrate the superior performance of our algorithm compared to state-of-the-art methodologies.
