Backward Problem Research Articles

A backward problem for stochastic Kuramoto-Sivashinsky equation: Conditional stability and numerical solution

In this paper, we consider a backward problem in time for a linear stochastic Kuramoto-Sivashinsky equation. Firstly, we present two Carleman estimates incorporating weight functions independent of the variable x for the stochastic Kuramoto-Sivashinsky equation. Subsequently, we employ these two Carleman estimates to establish conditional stability for the backward problem in two distinct scenarios: when 0<t0<T and when t0=0. Lastly, we transform the backward problem in time into the minimization of a regularized Tikhonov functional. This functional is solved by the conjugate gradient algorithm based on the gradient formula tailored for the regularized functional. Numerical examples related to the recovery of continuous and discontinuous initial values illustrate the effectiveness of the conjugate gradient algorithm.

Fault tolerant motion planning for a quadrotor subject to complete rotor failure

Assuring safety of a quadrotor subject to rotor failure has been heavily investigated at the control level in view of fault tolerant control (FTC) approach. Yet, the existing FTCs are often concerned with tracking the reference motion even when that reference may not be safely trackable due to the physical constraints of the quadrotor. This paper tackles the faulty quadrotor safety at the planner level, proposing a fault tolerant motion planner. Starting from the formal backward reachability problem formulation, the proposed motion planner generates the time trajectory of the coupled rotational and translational motions that safely guide the faulty quadrotor. The generated trajectory is theoretically guaranteed to be tracked by the embedded FTC without violating the physical constraints. Further, the trajectory is prescribed as an analytical closed-form expression and thus suitable for real-time emergency maneuvers. The effectiveness of the proposed motion planner is numerically validated in conjunction with the different FTC techniques and compared to the existing planning method. The simulation results clearly signify that the proposed planner can successfully complement the fault tolerance of quadrotor. The supplements including code implementations are available on GitHub repository: https://github.com/HMCL-UNIST/Fault-tolerant-motion-planner.

Nesterov acceleration‐based iterative method for backward problem of distributed‐order time‐fractional diffusion equation

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

Regularized kernel function methods for the backward heat conduction problem

In this letter, we are interested in the regularized numerical techniques for backward heat conduction problems, which aim at the detection of the previous status of physical field from its present information. It is well known that backward heat conduction problems are ill-posed. The small perturbation in the final status may result in large change in the previous status of physical field. It is difficult to get the stable numerical solutions of backward heat conduction problems. Based on the reproducing kernel theory, the letter aims at finding a new regularized kernel function methods for backward heat conduction problems.

SEDCPT: A secure and efficient Dynamic Searchable Encryption scheme with cluster padding assisted by TEE

Dynamic Searchable Symmetric Encryption (DSSE), which enables users to search and update encrypted data on an untrusted server without decryption, is a proven method when dealing with availability issues for outsourced data. However, the existing DSSE schemes suffer from forward and backward privacy problems. Although forward and backward privacy DSSE schemes can prevent these attacks, they still suffer from other challenges, such as count attacks, high communication overhead, and low computing efficiency. To solve the above problems simultaneously, we propose a secure and efficient dynamic searchable encryption scheme, which integrates a clustering algorithm, padding strategy, and trusted execution environments (TEE). The purpose of the scheme is to prevent count attacks while ensuring forward and backward privacy security. Our scheme also reduces the computing overhead and storage cost of the client by using TEE (e.g. Intel Software Guard Extension, Intel SGX) while maximizing the protection of client privacy. Furthermore, the scheme provides a secure hardware isolation environment for the cluster padding and search/update process to prevent malicious attacks from external software or super administrators. Finally, we provide corresponding security proofs and experimental evaluation which demonstrate both the security and efficiency of our scheme, respectively.

Finite elements for Matérn-type random fields: Uncertainty in computational mechanics and design optimization

This work highlights an approach for incorporating realistic uncertainties into scientific computing workflows based on finite elements, focusing on prevalent applications in computational mechanics and design optimization. We leverage Matérn-type Gaussian random fields (GRFs) generated using the SPDE method to model aleatoric uncertainties, including environmental influences, variating material properties, and geometric ambiguities. Our focus lies on delivering practical GRF realizations that accurately capture imperfections and variations and understanding how they impact the predictions of computational models as well as the shape and topology of optimized designs. We describe a numerical algorithm based on solving a generalized SPDE to sample GRFs on arbitrary meshed domains. The algorithm leverages established techniques and integrates seamlessly with the open-source finite element library MFEM and associated scientific computing workflows, like those found in industrial and national laboratory settings. Our solver scales efficiently for large-scale problems and supports various domain types, including surfaces and embedded manifolds. We showcase its versatility through biomechanics and topology optimization applications, emphasizing the potential to influence these domains. The flexibility and efficiency of SPDE-based GRF generation empowers us to run large-scale optimization problems on 2D and 3D domains, including finding optimized designs on embedded surfaces, and to generate design features and topologies beyond the reach of conventional techniques. Moreover, these capabilities allow us to model and quantify geometric uncertainties on reconstructed submanifolds, such as the interpolated surfaces of cerebral aneurysms provided by postprocessing CT scans. In addition to offering benefits in these specific domains, the proposed techniques transcend specific applications and generalize to arbitrary forward and backward problems in uncertainty quantification involving finite elements.

A regularization strategy to a backward problem for general nonlinear parabolic equations in Sobolev spaces

This article is devoted to solving a backward problem for general nonlinear parabolic equations in Sobolev spaces. The problem is hardly solved by computation since it is severely ill‐posed in Hadamard's sense. Using the Fourier transform and the truncation of high‐frequency components, we construct a regularized solution for the problem from the data given inexactly. As usual for ill‐posed problems, the rate of convergence can be obtained under additional smoothness assumptions on the regularity of the exact solution, the so‐called source conditions. By using a source condition in a generalized form, we prove a convergence estimate in a Sobolev space . Corresponding to different levels of the source condition, the convergence rates are improved gradually. We apply our theory to the backward problem with a convection–diffusion operator and a cubic nonlinearity. Finally, the numerical results are presented to test the validity of the proposed regularization method.

Modified error-in-constitutive-relation (MECR) framework for the characterization of linear viscoelastic solids

We develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described within the framework of standard generalized materials. To this end, we formulate the viscoelastic behavior in terms of the (Helmholtz) free energy potential and a dissipation potential. Assuming the availability of full-field interior kinematic data, the constitutive mismatch between the kinematic quantities (strains and internal thermodynamic variables) and their “stress” counterparts (Cauchy stress tensor and that of thermodynamic tensions), commonly referred to as the ECR functional, is established with the aid of Legendre–Fenchel gap functionals linking the thermodynamic potentials to their energetic conjugates. We then proceed by introducing the modified ECR (MECR) functional as a linear combination between its ECR parent and the kinematic data misfit, computed for a trial set of constitutive parameters. The affiliated stationarity conditions then yield two coupled evolution problems, namely (i) the forward evolution problem for the (trial) displacement field driven by the constitutive mismatch, and (ii) the backward evolution problem for the adjoint field driven by the data mismatch. This allows us to establish compact expressions for the MECR functional and its gradient with respect to the viscoelastic constitutive parameters. For generality, the formulation is established assuming both time-domain (i.e. transient) and frequency-domain data. We illustrate the developments in a two-dimensional setting by pursuing the multi-frequency MECR reconstruction of (i) piecewise-homogeneous standard linear solid, and (b) smoothly-varying Jeffreys viscoelastic material.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

Deep Neural Networks with Spacetime RBF for Solving Forward and Inverse Problems in the Diffusion Process

Numerical method of the exact control for the elastic string problem with moving boundary

The objective of this article is to investigate, analytically and numerically, the exact control problem associated with a mathematical model that describes the vertical vibrations of elastic strings, with moving contours. To determine the approximate numerical solution, following the description of the HUM (Hilbert Uniqueness Method), the finite element method associated with the Newmark method was used for the adjunct problem and the backward problem. To calculate the optimal control, the conjugated gradient method is used. The properties of a numerical problem must be consistent with the theoretical mathematical one. However, in order to obtain the numerical control, the grid is of fundamental importance, so we used the Bi-Grid technique. In this way, we developed an algorithm to solve this problem. Numerical simulations were performed for different initial data and moving contours, with and without the Bi-Grid algorithm and the numerical results confirmed the consistency between the theoretical and numerical results.

Unraveling forward and backward source problems for a nonlocal integrodifferential equation: A journey through operational calculus for Dzherbashian‐Nersesian operator

AbstractThis article primarily aims at introducing a novel operational calculus of Mikusiński's type for the Dzherbashian‐Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems of a differential equation that features Dzherbashian‐Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann‐Liouville integral. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed.

On multi-type Cannings models and multi-type exchangeable coalescents

A multi-type neutral Cannings population model with migration and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type coalescent sharing the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions of transition probabilities into parts concerning reproduction and migration respectively. The following section deals with a different but closely related multi-type Cannings model with mutation and fixed total population size but stochastically varying subpopulation sizes. The latter model is analyzed forward and backward in time with an emphasis on its behavior as the total population size tends to infinity. Forward in time, multi-type limiting branching processes arise for large population size. Its backward structure and related open problems are briefly discussed.

A Symmetry of Boundary Functions Method for Solving the Backward Time-Fractional Diffusion Problems

In the paper, we develop three new methods for estimating unknown initial temperature in a backward time-fractional diffusion problem, which is transformed to a space-dependent inverse source problem for a new variable in the first method. Then, the initial temperature can be recovered by solving a second-order boundary value problem. The boundary functions and a unique zero element constitute a group symmetry. We derive energetic boundary functions in the symmetry group as the bases to retrieve the source term as an unknown function of space and time. In the second method, the solution bases are energetic boundary functions, and then by collocating the governing equation we obtain the expansion coefficients for retrieving the entire solution and initial temperature. For the first two methods, boundary fluxes are over-specified to retrieve the initial condition. In the third method, we give two boundary conditions and a final time temperature to construct the bases in another symmetry group; the governing equation is collocated to a linear system to obtain the whole solution (initial temperature involved). These three methods are assessed and compared by numerical experiments.

The quasi-reversibility regularization method for backward problem of the multi-term time-space fractional diffusion equation

In this paper, we investigate a backward problem of 1D multi-term time–space fractional diffusion equation, which is ill-posed. With the help of quasi-reversibility regularization method, we give the regularization solution, based on some properties of Fourier transform and Mittag-Leffler functions. Besides, we establish a-priori and a-posteriori convergence estimates, respectively. Finally, we illustrate two numerical examples to verify the effectiveness and feasibility of our proposed method.

On a Backward Problem for the Rayleigh–Stokes Equation with a Fractional Derivative

The Rayleigh–Stokes equation with a fractional derivative is widely used in many fields. In this paper, we consider the inverse initial value problem of the Rayleigh–Stokes equation. Since the problem is ill-posed, we adopt the Tikhonov regularization method to solve this problem. In addition, this paper not only analyzes the ill-posedness of the problem but also gives the conditional stability estimate. Finally, the convergence estimates are proved under two regularization parameter selection rules.

Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm

We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.

The backward problem for time-fractional evolution equations

In this paper, we consider the backward problem for fractional in time evolution equations ∂ t α u ( t ) = Au ( t ) with the Caputo derivative of order 0 < α ≤ 1 , where A is a self-adjoint and bounded above operator on a Hilbert space H. First, we extend the logarithmic convexity technique to the fractional framework by analyzing the properties of the Mittag–Leffler functions. Then we prove conditional stability estimates of Hölder type for initial conditions under a weaker norm of the final data. Finally, we give several applications to show the applicability of our abstract results.

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization method

Dynamic gradient reactivation for backward compatible person re-identification

We study the backward compatible problem for person re-identification (Re-ID), which aims to constrain the features of an updated new model to be comparable with the existing features from the old model in galleries. Most of the existing works adopt distillation-based methods, which focus on pushing new features to imitate the distribution of the old ones. However, the distillation-based methods are intrinsically sub-optimal since it forces the new feature space to imitate the inferior old feature space. To address this issue, we propose the Ranking-based Backward Compatible Learning (RBCL), which directly optimizes the ranking metric between new features and old features. Different from previous methods, RBCL only pushes the new features to find best-ranking positions in the old feature space instead of strictly alignment, and is in line with the ultimate goal of backward retrieval. However, the sharp sigmoid function used to make the ranking metric differentiable also incurs the gradient vanish issue, therefore stems the ranking refinement during the later period of training. To address this issue, we propose the Dynamic Gradient Reactivation (DGR), which can reactivate the suppressed gradients by adding dynamically computed constants during the forward step. To further help target the best-ranking positions, we include the Neighbor Context Agents (NCAs) to approximate the entire old feature space during training. Unlike previous works that mainly test on the in-domain settings, we make the early attempt to introduce the cross-domain settings (including both supervised and unsupervised) for the backward compatible person Re-ID task, which are more challenging yet meaningful. The experimental results on all five settings show that the proposed RBCL outperforms previous state-of-the-art methods by large margins.