Backward Problem Research Articles

A Boundary-Type Numerical Procedure to Solve Nonlinear Nonhomogeneous Backward-in-Time Heat Conduction Equations

In this paper, an explicit boundary-type numerical procedure, including a constraint-type fictitious time integration method (FTIM) and a two-point boundary solution of the Lie-group shooting method (LGSM), is constructed to tackle nonlinear nonhomogeneous backward heat conduction problems (BHCPs). Conventional methods cannot effectively overcome numerical instability to solve inverse problems that lack initial conditions and take a long time to calculate, even using different variable transformations and regularization techniques. Therefore, an explicit-type numerical procedure is developed from the FTIM and the LGSM to avoid numerical instability and numerical iterations. First, a two-point boundary solution of the LGSM is introduced into the numerical algorithm. Then, the maximum and minimum values of the initial guess value can be determined linearly from the boundary conditions at the initial and final times. Finally, an explicit-type boundary-type numerical procedure, including a boundary value solution and constraint-type FTIM, can directly avoid the numerical iterative problems of BHCPs. Several nonlinear examples are tested. Based on the numerical results shown, this boundary-type numerical procedure using a two-point solution can directly obtain an approximated solution and can achieve stable convergence to boundary conditions, even if numerical iterations occur. Furthermore, the numerical efficiency and accuracy are better than in the previous literature, even with an increased computational time span without the regularization technique.

MIOpt: optimization framework for backward problems on the basis of the concept of materials integration

MIOpt: optimization framework for backward problems on the basis of the concept of materials integration

The quasi-reversibility regularization method for backward problems of the time-fractional diffusion-wave equation

This paper is devoted to two backward problems in a time-fractional diffusion-wave equation, with the aid of extra measurement data at a final time. Since these two problems are ill-posed, a quasi-reversibility method is obtained by employing eigenfunction expansion, and the existence, uniqueness and regularity of the regularized solutions are also proven. We can get order-optimal error estimates from the a-priori parameter choice rule. Finally, several numerical examples of both one- and two-dimension show that the proposed regularization method is effective and stable.

Homotopy analysis method for discrete quasi-reversibility mollification method of nonhomogeneous backward heat conduction problem

Abstract In this article, the inverse time problem is investigated. Regarding the ill-posed linear problem, utilize the quasi-reversibility method first. This problem has been regularized and after that provides an iterative regularizing strategy for noisy input data that are based on homotopy. For the regularizing solution, the error analysis is proved when we employ noisy measurement data as our initial guess. Finally, numerical implementations are presented.

A dimension-augmented physics-informed neural network (DaPINN) with high level accuracy and efficiency

Physics-informed neural networks (PINNs) have been widely applied in different fields due to their effectiveness in solving partial differential equations (PDEs). However, the accuracy and efficiency of PINNs need to be considerably improved for scientific and commercial purposes. To address this issue, we systematically propose a novel dimension-augmented physics-informed neural network (DaPINN), which simultaneously and significantly improves the accuracy and efficiency of the base PINN. In the DaPINN model, we manipulate the dimensionality of the network input by inserting additional sample features and then incorporate the expanded dimensionality into the loss function. Moreover, we verify the effectiveness of power series augmentation, Fourier series augmentation and replica augmentation in both forward and backward problems. In most experiments, the error of DaPINN is 1 ∼2 orders of magnitude lower than that of the base PINN. The results show that the DaPINN outperforms the original PINN in terms of both accuracy and efficiency with a reduced dependence on the number of sample points. We also discuss the computational complexity of the DaPINN, its network size implications, other implementations of the DaPINN and the compatibility of DaPINN's methods with residual-based adaptive refinement (RAR), self-adaptive physics-informed neural networks (SA-PINNs) and gradient-enhanced physics-informed neural networks (gPINNs).

A variational technique of mollification applied to backward heat conduction problems

A variational technique of mollification applied to backward heat conduction problems

A new regularization for time-fractional backward heat conduction problem

Abstract It is well known that the backward heat conduction problem of recovering the temperature u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} at a time t ≥ 0 {t\geq 0} from the knowledge of the temperature at a later time, namely g := u ⁢ ( ⋅ , τ ) {g:=u(\,\cdot\,,\tau)} for τ > t {\tau>t} , is ill-posed, in the sense that small error in g can lead to large deviation in u ⁢ ( ⋅ , t ) {u(\,\cdot\,,t)} . However, in the case of a time fractional backward heat conduction problem (TFBHCP), the above problem is well-posed for t > 0 {t>0} and ill-posed for t = 0 {t=0} . We use this observation to obtain stable approximate solutions as solutions for t ∈ ( 0 , τ ] {t\in(0,\tau]} with t as regularization parameter for approximating the solution at t = 0 {t=0} , and derive error estimates under suitable source conditions. We shall also provide some numerical examples to illustrate the approximation properties of the regularized solutions.

Tracking Contaminant Transport Backwards with an Operator-Splitting Method

Recovering the past movement of a contaminant plume from measurements of its current values is a challenging problem in hydrology. Moreover, modeling the movement of a contaminant plume backwards is an ill-posed problem due to the unstable and non-unique nature of the resulting solution. Therefore, standard numerical methods become unstable, making it impossible to simulate existing contaminant transport models with reversed time. This paper presents two major contributions to solve the backward problem. Firstly, a stable and consistent numerical method based on an operator-splitting concept which is effective in tracking back the contaminant movement, and secondly, an optimal condition for the choice of mesh width that enables the error during computer simulation to stay within a reasonable bound. The numerical method was validated by introducing errors of varied strengths at the starting point and reconstructing the contaminant profiles backwards at any given time.

A numerical method for a backward problem of a linear stochastic Kuramoto-Sivashinsky equation

This paper concerns a backward problem for a linear stochastic Kuramoto–Sivashinsky equation, which aims to reconstruct the initial value from the mean measurement at the terminal time. By transforming the backward problem into a regularized optimization one, a regularization method together with its numerical implementation in a finite dimensional space is proposed for solving the backward problem. Finally, we show effectiveness of the proposed reconstruction method by several numerical examples.

A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition u(x,T)=βu(x,0)+φ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β=0, then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If β=1, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter β, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β≥1, or β<0, then the problem is well-posed; if β∈(0,1), then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.

Stochastic estimation of Green’s functions with application to diffusion and advection-diffusion-reaction problems

A stochastic method is described for estimating Green’s functions (GF’s), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF’s, within any geometry, the technique solves the central challenge in obtaining Green’s function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF’s that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green’s function, G(x,t|x′,t′), and the backward Kolmogorov problem governing the transition density, p(x,t|x′,t′), of the stochastic process used in Green’s function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green’s function estimation can be used, and for subsequently estimating the transition density and associated Green’s function. Second, a naive estimator for the transition density is proposed and tested. Third, Green’s function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF’s, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF’s.

The backward problem in plasma-assisted combustion: Experiments of nanosecond pulsed discharges driven by flames

Plasma technologies are a promising way of addressing a number of challenges in combustion, ranging from stabilization and flame-holding for hypersonic vehicles, to enabling low-emissions propulsion and power. Whereas most works in the literature have centered on the forward problem: the impact of the plasma on different combustion metrics, including static and dynamic flame stabilization; fewer studies have been devoted to the backward problem: or the significance of having a strongly inhomogeneous environment, often unsteady in time, on the discharge characteristics. In this perspectives article, the backward problem is addressed in a hierarchical manner. First, the implications of the discharge proceeding in an inhomogeneous background gas, with gradients in density and composition, and nonuniform flows, are discussed. Next, these foundation experiments are used as elementary building blocks to explain experiments of nanosecond pulsed discharges in dynamic combustion environments, including a transient laminar flame and a combustor undergoing dynamics. Finally, the strong coupling between the forward and backward problem is illustrated with an imaging study, using transparent electrodes, of a mesoscale premixed flame and a nanosecond pulsed dielectric barrier discharge. The experiments demonstrate the importance of considering the pulse-to-pulse evolution due to changes in the background environment when using plasma assistance in unsteady combustion processes, including active control of combustion dynamics.

Constructing Muslim Identity in a Secular State: The Strategic Role of Two Singapore Islamic Organizations

In Singapore, Islamic identity matters mainly because Muslims and Malays have special constitutional status. However, state policies seem to contradict the status while the community is still dealing with the problem of backwardness in educational and economic development. This article examines the profile and strategic role of two Islamic organizations, MUIS (Majlis Ugama Islam Singapura, the Islamic Religious Council of Singapore) and Pergas (Persatuan Ulama dan Guru-Guru Agama Islam Singapura, Singapore Islamic Scholars, and Religious Teachers Association), in accommodating the expression and reconstructing Singaporean Muslim identity. Through intensive library research and using an interdisciplinary approach from social constructivist and fiqh of minorities’ perspectives, this article found that both have played a very important role in the expression and construction of Singaporean Muslim identity. There have been dynamic contestation and cooperation between the state policies towards the Muslim community. However, the global effect of Islamist extremism and terrorism has united them in terms of religious thought, attitude, and the formulation of the ideal Singaporean Muslim identity. The changing process of their stances toward the government’s policies was effective due to the function of fiqh of minorities in contextualizing Islamic teachings in the context of Singapore as a secular state.

Assessing wellbeing: Profiling and socioeconomic status of Kenyir Lake side community, Malaysia

The Kenyir Lake side community offers numerous advantages to the local community. However, the problems of backwardness and poverty have been identified as the government's main challenges in its efforts to develop the community and maximise the advantages. Therefore, this study was conducted to identify the Kenyir Lake community's profile and assess its well-being. The study was conducted in three sub-districts near Tasik Kenyir, namely Kuala Berang, Hulu Telemong, and Jenagor, with a total of 510 respondents, who are heads of households (HOH). This study was carried out using a quantitative approach using questionnaire with a simple random sampling method. The findings of this study offered demographic profiling and discovered nine indicators of well-being, namely 1) Life Achievement, 2) Health Level, 3) Family Relationship, 4) Community Relationship, 5) Spiritual Level, 6) Safety Level and Social Problems, 7) Income or Finance, 8) Basic Facilities, and 9) Communication Technology. The study found that most respondents are satisfied with their lives now compared to 10 years ago. This study will help many parties to develop the Kenyir Lake Side Community, starting from the local authorities to the highest administration in the country.

An iterative generalized quasi-boundary value regularization method for the backward problem of time fractional diffusion-wave equation in a cylinder

In this paper, we consider the backward problem for a time fractional diffusion-wave equation in a cylinder. The ill-posedness and a conditional stability of the inverse problem are proved. Based on the generalized quasi-boundary value regularization method, we propose an iterative generalized quasi-boundary value regularization method to deal with the inverse problem, and this iterative method has a higher convergence rate. The convergence rates of the regularized solution under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method.

Galerkin Method for a Backward Problem of Time-Space Fractional Symmetric Diffusion Equation

We investigate a backward problem of the time-space fractional symmetric diffusion equation with a source term, wherein the negative Laplace operator −Δ contained in the main equation belongs to the category of uniformly symmetric elliptic operators. The problem is ill-posed because the solution does not depend continuously on the measured data. In this paper, the existence and uniqueness of the solution and the conditional stability for the inverse problem are given and proven. Based on the least squares technique, we construct a Galerkin regularization method to overcome the ill-posedness of the considered problem. Under a priori and a posteriori selection rules for the regularization parameter, the Hölder-type convergence results of optimal order for the proposed method are derived. Meanwhile, we verify the regularized effect of our method by carrying out some numerical experiments where the initial value function is a smooth function or a non-smooth one. Numerical results show that this method works well in dealing with the backward problem of the time-space fractional parabolic equation.

The backward problem for an inhomogeneous time-fractional diffusion-wave equation in an axis-symmetric cylinder

The backward problem for an inhomogeneous time-fractional diffusion-wave equation in an axis-symmetric cylinder

Conditionally positive definite unilateral weighted shifts

In a recent paper [15], Hilbert space operators T with the property that each sequence of the form {‖Tnh‖2}n=0∞ is conditionally positive definite in a semigroup sense were introduced. In the present paper, this line of research is continued in depth in the case of unilateral weighted shifts. The conditional positive definiteness of unilateral weighted shifts is characterized in terms of formal moment sequences. The description of the representing triplet, the main object canonically associated with such operators, is provided. The backward extension problem for conditionally positive definite unilateral weighted shifts is solved, revealing a new feature that does not appear in the case of other operator classes. Finally, the flatness problem in this context is discussed with emphases on unexpected differences from the analogous problem for subnormal unilateral weighted shifts.

A variational framework for second-order backward discrete boundary value problems

A variational framework for second-order backward discrete boundary value problems

Advancements in Numerical Methods for Forward and Inverse Problems in Functional near Infra-Red Spectroscopy: A Review

In the field of biomedical image reconstruction, functional near infra-red spectroscopy (fNIRs) is a promising technology that uses near infra-red light for non-invasive imaging and reconstruction. Reconstructing an image requires both forward and backward problem-solving in order to figure out what the image’s optical properties are from the boundary data that has been measured. Researchers are using a variety of numerical methods to solve both the forward and backward problems in depth. This study will show the latest improvements in numerical methods for solving forward and backward problems in fNIRs. The physical interpretation of the forward problem is described, followed by the explanation of the state-of-the-art numerical methods and the description of the toolboxes. A more in-depth discussion of the numerical solution approaches for the inverse problem for fNIRs is also provided.