We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and derive conditions to ensure the solvability of the problem. We then present a set of numerical approaches, including Galerkin type algorithm and collocation type algorithm. Analysis of the algorithms are presented, along with their implementation detail. The Galerkin algorithm is more suitable for practical situations, particularly those with noisy data, as it avoids using derivative/gradient data. Various numerical examples are then presented to demonstrate the performance and properties of the numerical methods.
In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and time-fractional diffusion equations, from the flux measurement observed at one point on the boundary. The mathematical model for time-fractional diffusion equations involves a Djrbashian-Caputo fractional derivative in time. We prove a uniqueness result in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition and source), i.e., the recovery of the order of derivation in a diffusion p…
In this work, we investigate an inverse problem of recovering multiple orders in a time-fractional diffusion model from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g., diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Further, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear lea…
We consider an LQR optimal control problem with partially unknown dynamics. We propose a new model-based online algorithm to obtain an approximation of the dynamics $and$ the control at the same time during a single simulation.
This paper is concerned with an inverse problem of recovering a potential term and fractional order in a one-dimensional subdiffusion problem, which involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time, from the lateral Cauchy data. In the model, we do not assume a full knowledge of the initial data and the source term, since they might be unavailable in some practical applications. We prove the unique recovery of the spatially-dependent potential coefficient and the order $α$ of the derivation simultaneously from the measured trace data at one end point, when the mo…
We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations to the unknown load. Numerical evidences of the convergence for not too large loads are also given.
Universidad Carlos III de Madrid（马德里卡洛斯三世大学）；Pennsylvania State University（宾夕法尼亚州立大学）；Stanford University（斯坦福大学）；University of California（加利福尼亚大学）
We present a novel approach for recovering a sparse signal from cross-correlated data. Cross-correlations naturally arise in many fields of imaging, such as optics, holography and seismic interferometry. Compared to the sparse signal recovery problem that uses linear measurements, the unknown is now a matrix formed by the cross correlation of the unknown signal. Hence, the bottleneck for inversion is the number of unknowns that grows quadratically. The main idea of our proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal of the unknown matrix, whose …
The Ohio State University（俄亥俄州立大学）；Southern University of Science and Technology（南方科技大学）
We present a numerical framework for deep neural network (DNN) modeling of unknown time-dependent partial differential equations (PDE) using their trajectory data. Unlike the recent work of [Wu and Xiu, J. Comput. Phys. 2020], where the learning takes place in modal/Fourier space, the current method conducts the learning and modeling in physical space and uses measurement data as nodal values. We present a DNN structure that has a direct correspondence to the evolution operator of the underlying PDE, thus establishing the existence of the DNN model. The DNN model also does not require any geom…
We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when $m \asymp r s^γ\ln(en/s)$ for some exponent $γ> 0$. We show that this is feasible for $γ= 2$, and that the propo…
Johannes Gutenberg-Universität Mainz（美因茨约翰内斯·古腾堡大学）；Helmholtz-Institut Mainz, GSI Helmholtzzentrum für Schwerionenforschung GmbH（亥姆霍兹重离子研究中心有限责任公司美因茨亥姆霍兹研究所）；University of California – Berkeley（加州大学伯克利分校）
Deep neural networks (DNNs) are widely used in pattern-recognition tasks for which a human comprehensible, quantitative description of the data-generating process, e.g., in the form of equations, cannot be achieved. While doing so, DNNs often produce an abstract (entangled and non-interpretable) representation of the data-generating process. This is one of the reasons why DNNs are not extensively used in physics-signal processing: physicists generally require their analyses to yield quantitative information about the studied systems. In this article we use DNNs to disentangle components of osc…
Multi-dimensional imaging data recovery via minimizing the partial sum of tubal nuclear norm
School of Mathematical Sciences/Research Center for Image and Vision Computing（数学科学学院/图像与视觉计算研究中心）；University of Electronic Science and Technology of China（电子科技大学）；FinTech Innovation Center（金融科技创新中心）；Financial Intelligence and Financial Engineering Research Key Laboratory of Sichuan province（四川省金融 intelligence 和金融工程研究重点实验室）；School of Economic Information Engineering（经济信息工程学院）；Southwestern University of Finance and Economics（西南财经大学）
