Noisy Boundary Research Articles

FDM-based alternating iterative algorithms for inverse BVPs in 2D steady-state anisotropic heat conduction with heat sources

In the framework of steady-state inhomogeneous anisotropic heat conduction with heat sources, we study the convergent and stable numerical reconstruction of the missing boundary conditions (temperatures and normal heat fluxes) on an inaccessible boundary and the thermal field in a 2D solid from the knowledge of over-prescribed data on an accessible portion of the boundary and temperature data on the remaining boundary. This inverse boundary value problem is approached by employing the alternating iterative algorithms of Kozlov et al. (1991), in conjunction with the standard finite-difference method and a novel modified one, for both exact and perturbed data. For noisy data available on the over-prescribed boundary, the numerical solution is retrieved by using three regularising/stabilising stopping criteria. Numerical results are presented for 2D simply and doubly connected domains bounded by a (piecewise) smooth curve, as well as exact and noisy boundary data, whilst appropriate errors in the fractional Sobolev spaces corresponding to the boundary temperatures and normal heat fluxes, respectively, are computed efficiently.

Some Inverse Problems of Two-Dimensional Stokes Flows by the Method of Fundamental Solutions and Kalman Filter

Some inverse problems of Stokes flow, including noisy boundary conditions, unknown angular velocity, and dynamic viscous constant identification are studied in this paper. The interpolation equations for those inverse problems are constructed using the method of fundamental solutions (MFS). Based on the noise addition technique, the inverse problems are solved using MFS and a Kalman filter. It is seen from numerical experiments that these approaches and algorithms are valid and have strong robustness and high accuracy in solving inverse Stokes problems.

Physics-informed neural networks for characterization of structural dynamic boundary conditions

Structural dynamics simulations are often faced with challenges arising from unknown boundary conditions, leading to considerable prediction uncertainties. Direct measurement of these boundary conditions can be impractical for certain mounting scenarios, such as joints or screw connections. In addition, conventional inverse methods face limitations in integrating measured data and solving inverse problems when the forward model is computationally expensive. In this study, we explore the potential of physics-informed neural networks that incorporate the residual of a partial differential equation into the loss function of a neural network to ensure physically consistent predictions. We train the neural network using noisy boundary displacement data of a structure from a finite element reference solution. The network learns to predict the displacement field within the structure while satisfying the Navier–Lamé equations in the frequency domain. Our results show that physics-informed neural networks accurately predict the displacement field within a three-dimensional structure using only boundary training data. Additionally, differentiating the trained network allows precise characterization of previously unknown boundary conditions and facilitates the assessment of non-measurable quantities, such as the stress tensor.

A Robust SLIC Based Approach for Segmentation Using Canny Edge Detector

An accurate image segmentation in noisy environment is complex and challenging. Unlike existing state-of-the-art methods that use superpixels for successful segmentation, we propose a new approach for noise-robust SLIC (Simple Linear Iterative Clustering) segmentation that incorporates a Canny edge detector. By leveraging Canny edge information, the proposed method modifies the pixel intensity distance measurement to overcome boundary adherence challenge. Furthermore, we adopt a selective approach to update cluster centers, focusing on pixels that contribute less to the noise. Extensive experiments on synthetic noisy images demonstrate the effectiveness of our approach. It significantly improves SLIC's performance in noisy image segmentation and boundary adherence, making it a promising technique for vision processing tasks.

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

Periodic-net: an end-to-end data driven framework for diffuse optical imaging of breast cancer from noisy boundary data.

The machine learning (ML) approach plays a critical role in assessing biomedical imaging processes especially optical imaging (OI) including segmentation, classification, and reconstruction, intending to achieve higher accuracy efficiently. This research aims to develop an end-to-end deep learning framework for diffuse optical imaging (DOI) with multiple datasets to detect breast cancer and reconstruct its optical properties in the early stages. The proposed Periodic-net is a nondestructive deep learning (DL) algorithm for the reconstruction and evaluation of inhomogeneities in an inverse model with high accuracy, while boundary measurements are calculated by solving a forward problem with sources/detectors arranged uniformly around a circular domain in various combinations, including , , and boundary measurement setups. The results of image reconstruction on numerical and phantom datasets demonstrate that the proposed network provides higher-quality images with a greater amount of small details, superior immunity to noise, and sharper edges with a reduction in image artifacts than other state-of-the-art competitors. The network is highly effective at the simultaneous reconstruction of optical properties, i.e., absorption and reduced scattering coefficients, by optimizing the imaging time without degrading inclusions localization and image quality.

Constructive finite-dimensional boundary control of stochastic 1D parabolic PDEs

Recently, a constructive method for the finite-dimensional observer-based control of deterministic parabolic PDEs was suggested by employing a modal decomposition approach. In this paper, for the first time we extend this method to the stochastic 1D heat equation with nonlinear multiplicative noise. We consider the Neumann actuation and study the observer-based as well as the state-feedback controls via the modal decomposition approach. We employ either trigonometric or polynomial dynamic extension. For observer-based control we consider a noisy boundary measurement. First, we show the well-posedness of strong solutions to the closed-loop systems. Then by suggesting a direct Lyapunov method and employing Itô’s formula, we provide mean-square L2 exponential stability analysis of the full-order closed-loop system, leading to linear matrix inequality (LMI) conditions for finding the observer dimension and as large as possible noise intensity bound for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity and large enough observer dimension (for observer-based control). We further show that in the case of state-feedback and linear noise, the system is always stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system with deterministic measurement. Numerical simulations are carried out to illustrate the efficiency of our method. For both state-feedback and observer-based controls, the trigonometric extension always allows for a larger noise than the polynomial one in the example.

A direct sampling method for simultaneously recovering electromagnetic inhomogeneous inclusions of different nature

In this work, we propose a novel direct sampling method (DSM) to recover the support of two different types of electromagnetic inhomogeneous inclusions simultaneously, with only one or two sets of noisy boundary measurement data. The DSM leverages upon an important mutually almost orthogonality property between the fundamental solutions of the forward problem and some proper families of probing functions. Two proper families of probing functions that possess desired properties are proposed to reconstruct the support of inhomogeneous inclusions accurately in the direction that is parallel to or vertical to the measurement surface separately. For the two families of probing functions, the mutually almost orthogonality property is carefully verified through both theoretical justifications and numerical experiments. The novel DSM is fast to compute, effective under limited measurement data, and very stable and robust against random noise of reasonable size. All these features are verified through extensive numerical experiments.

The Method of Fundamental Solutions for the 3D Laplace Inverse Geometric Problem on an Annular Domain

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples.

Artificial neural network approximations of Cauchy inverse problem for linear PDEs

A novel artificial neural network method is proposed for solving Cauchy inverse problems. Using multiple-layers network as an approximation we present a non-mesh discretization to solve the problems. The existence and convergence are shown to establish the well-posedness of neural network approximations for the Cauchy inverse problems. Numerical results on 2D to 8D cases show that compared to finite element method, the neural network approach easier extends to high dimensional case. The stability and accuracy of the proposed network approach are investigated by the experiments with noisy boundary and irregular computational domain. Our studies conclude that the neural network method alleviates the influence of noise and it is observed that networks with wider and deeper hidden layers could lead to better approximation.

BSN++: Complementary Boundary Regressor with Scale-Balanced Relation Modeling for Temporal Action Proposal Generation

Generating human action proposals in untrimmed videos is an important yet challenging task with wide applications. Current methods often suffer from the noisy boundary locations and the inferior quality of confidence scores used for proposal retrieving. In this paper, we present BSN++, a new framework which exploits complementary boundary regressor and relation modeling for temporal proposal generation. First, we propose a novel boundary regressor based on the complementary characteristics of both starting and ending boundary classifiers. Specifically, we utilize the U-shaped architecture with nested skip connections to capture rich contexts and introduce bi-directional boundary matching mechanism to improve boundary precision. Second, to account for the proposal-proposal relations ignored in previous methods, we devise a proposal relation block to which includes two self-attention modules from the aspects of position and channel. Furthermore, we find that there inevitably exists data imbalanced problems in the positive/negative proposals and temporal durations, which harm the model performance on tail distributions. To relieve this issue, we introduce the scale-balanced re-sampling strategy. Extensive experiments are conducted on two popular benchmarks: ActivityNet-1.3 and THUMOS14, which demonstrate that BSN++ achieves the state-of-the-art performance. Not surprisingly, the proposed BSN++ ranked 1st place in the CVPR19 - ActivityNet challenge leaderboard on temporal action localization task.

Simultaneous inversion of the potential term and the fractional orders in a multi-term time-fractional diffusion equation

In the present paper, we devote our effort to a nonlinear inverse problem for simultaneously recovering the potential function and the fractional orders in a multi-term time-fractional diffusion equation from the noisy boundary Cauchy data in the one-dimensional case. The uniqueness for the inverse problem is derived based on the analytic continuation, the Laplace transformation and the Gel’fand–Levitan theory. Finally, the Levenberg–Marquardt regularization method with a regularization parameter chosen by a sigmoid-type function is applied for finding a stable approximate solution. Three numerical examples are provided to show the effectiveness of the proposed method.

A modified quasi-reversibility method for inverse source problem of Poisson equation

ABSTRACT In this article, we consider an inverse source problem for Poisson equation in a strip domain. That is to determine source term in the Poisson equation from a noisy boundary data. This is an ill-posed problem in the sense of Hadamard, i.e., small changes in the data can cause arbitrarily large changes in the results. Before we give the main results about our proposed problem, we display some useful lemmas at first. Then we propose a modified quasi-reversibility regularization method to deal with the inverse source problem and obtain a convergence rate by using an a priori regularization parameter choice rule. Numerical examples are provided to show the effectiveness of the proposed method.

NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling (“residual”) points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at Reτ∼1,000, but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems. We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. We also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings.

Inverse Source Problem for a Multiterm Time-Fractional Diffusion Equation with Nonhomogeneous Boundary Condition

This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.

BOUNDARY REGULARIZED BUILDING FOOTPRINT EXTRACTION FROM SATELLITE IMAGES USING DEEP NEURAL NETWORKS

Abstract. In recent years, an ever-increasing number of remote satellites are orbiting the Earth which streams vast amount of visual data to support a wide range of civil, public and military applications. One of the key information obtained from satellite imagery is to produce and update spatial maps of built environment due to its wide coverage with high resolution data. However, reconstructing spatial maps from satellite imagery is not a trivial vision task as it requires reconstructing a scene or object with high-level representation such as primitives. For the last decade, significant advancement in object detection and representation using visual data has been achieved, but the primitive-based object representation still remains as a challenging vision task. Thus, a high-quality spatial map is mainly produced through complex labour-intensive processes. In this paper, we propose a novel deep neural network, which enables to jointly detect building instance and regularize noisy building boundary shapes from a single satellite imagery. The proposed deep learning method consists of a two-stage object detection network to produce region of interest (RoI) features and a building boundary extraction network using graph models to learn geometric information of the polygon shapes. Extensive experiments show that our model can accomplish multi-tasks of object localization, recognition, semantic labelling and geometric shape extraction simultaneously. In terms of building extraction accuracy, computation efficiency and boundary regularization performance, our model outperforms the state-of-the-art baseline models.

Estimating the dynamics of systems with noisy boundaries

In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution.This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary.In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory.We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

Singular boundary method for 2D and 3D heat source reconstruction

This paper presents the singular boundary method (SBM) in conjunction with the regularization method to recover the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. To avoid the internal measurements, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis. As a result, the source inverse problems can be transformed to a direct fourth-order governing equation, which can be tackled by the SBM. The SBM utilizes the fundamental solutions to approximate the solutions and desingularizes the source singularities of fundamental solutions with origin intensity factors (OIFs), which are derived for both 2D and 3D cases. The regularization method is introduced to deal with noisy boundary. The feasibility and accuracy of the proposed method is validated by both 2D and 3D cases.

Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations

This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the “global” boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.