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We consider the nonlinear inverse problem for a second order hyperbolic equation with unknown coefficient depending on the time. We establish the existence and uniqueness of regular solutions which are used for constructing a soltuion to the inverse problem under consideration.
Abstract We consider the solution of nonlinear inverse problems where the forward problem is a discretization of a partial differential equation. Such problems are notoriously difficult to solve in practice and require minimizing a combination of a data-fit term and a regularization term. The main computational bottleneck of typical algorithms is the direct estimation of the data misfit. Therefore, likelihood-free approaches have become appealing alternatives. Nonetheless, difficulties in generalization and limitations in accuracy have hindered their broader utility and applicability. In this work, we use a paired autoencoder framework as a likelihood-free estimator (LFE) for inverse problems. We show that the use of such an architecture allows us to construct a solution efficiently and to overcome some known open problems when using LFEs. In particular, our framework can assess the quality of the solution and improve on it if needed. We demonstrate the viability of our approach using examples from full waveform inversion and inverse electromagnetic imaging.
A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion.
Inverse problem of optimal regulator for class of systems with integral type performance indices
The uncertainty in spatially heterogeneous Manning’s n fields is quantified using a novel formulation and numerical solution of stochastic inverse problems for physics-based models. The uncertainty is quantified in terms of a probability measure and the physics-based model considered here is the state-of-the-art ADCIRC model although the presented methodology applies to other hydrodynamic models. An accessible overview of the formulation and solution of the stochastic inverse problem in a mathematically rigorous framework based on measure theory is presented in this paper. Technical details that arise in practice by applying the framework to determine the Manning’s n parameter field in a shallow water equation model used for coastal hydrodynamics are presented and an efficient computational algorithm and open source software package are developed. A new notion of “condition” for the stochastic inverse problem is defined and analyzed as it relates to the computation of probabilities. Finally, this notion of condition is investigated to determine effective output quantities of interest of maximum water elevations to use for the inverse problem for the Manning’s n parameter and the effect on model predictions is analyzed.
We consider the problem of the ambiguity of the inverse problem of polarimetry for a homogeneous anisotropic medium characterised by orthogonal eigenpolarisations generated by non-commutativity of Jones and Mueller matrices of elementary types of anisotropy. The differences between two variants of the orthogonality conditions for the eigenpolarisations of an arbitrary anisotropic medium, the existence of which is due to this ambiguity, are investigated. The results obtained can be used to analyse the adequacy of the solution of the inverse problem of this class and in the problems of the synthesis of polarisation elements with given characteristics. (paper)
In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.
One of the prime challenges in the Bayesian solution of inverse problems governed by partial differential equations (PDEs) is the computational cost of repeatedly evaluating numerical PDE models, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. This paper presents a data-driven projection-based model reduction technique to reduce this computational cost. The proposed technique has two distinctive features. To begin, the model reduction strategy is tailored to inverse problems: the snapshots used to construct the reduced-order model are computed adaptively from the posterior distribution. Posterior exploration and model reduction are thus pursued simultaneously. Second, to avoid repeated evaluations of the full-scale numerical model as in a standard MCMC method, we couple the full-scale model and the reduced-order model together in the MCMC algorithm. This maintains accurate inference while reducing its overall computational cost. In numerical experiments considering steady-state flow in a porous medium, the data-driven reduced-order model achieves better accuracy than a reduced-order model constructed using the classical approach. It also improves posterior sampling efficiency by several orders of magnitude compared with a standard MCMC method.
Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.
Two prime bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. However, incomplete or noisy data, the state variation and parameter dependence of the forward model, and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting—both in the parameter space of the inverse problem and in the state space of the forward model. To this end, we show how to jointly construct low-dimensional subspaces of the parameter space and the state space in order to accelerate the Bayesian solution of the inverse problem. As a byproduct of state dimension reduction, we also show how to identify low-dimensional subspaces of the data in problems with high-dimensional observations. These subspaces enable approximation of the posterior as a product of two factors: (i) a projection of the posterior onto a low-dimensional parameter subspace, wherein the original likelihood is replaced by an approximation involving a reduced model; and (ii) the marginal prior distribution on the high-dimensional complement of the parameter subspace. We introduce and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations. The resulting posterior approximations can rapidly be characterized using standard sampling techniques, e.g., Markov chain Monte Carlo. Two numerical examples demonstrate the accuracy and efficiency of our approach: inversion of an integral equation in atmospheric remote sensing, where the data dimension is very high; and the inference of a heterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters.
We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.
It is shown that inverse problems of steady-state creep bending of plates in both the geometrically linear and nonlinear formulations can be represented in a variational formulation. Steady-state values of the obtained functionals corresponding to the solutions of the problems of inelastic deformation and springback are determined by applying a finite element procedure to the functionals. Optimal laws of creep deformation are formulated using the criterion of minimizing damage in the functionals of the inverse problems. The formulated problems are reduced to the problems solved by the finite element method using MSC.Marc software. Currently, forming of light metals poses tremendous challenges due to their low ductility at room temperature and their unusual deformation characteristics at hot-cold work: strong asymmetry between tensile and compressive behavior, and a very pronounced anisotropy. We used the constitutive models of steady-state creep of initially transverse isotropy structural materials the kind of the stress state has influence. The paper gives basics of the developed computer-aided system of design, modeling, and electronic simulation targeting the processes of manufacture of wing integral panels. The modeling results can be used to calculate the die tooling, determine the panel processibility, and control panel rejection in the course of forming.
We model transient diffusion in heterogeneous materials using a novel physics-informed neural networks framework (PINNs) termed Adaptive interface physics-informed neural networks or AdaI-PINNs (Roy et al. arXiv preprint arXiv:2406.04626, 2024). AdaI-PINNs utilize different activation functions with trainable slopes tailored to each material region within the computational domain, allowing for a fully automated and adaptive PINNs approach to model interface problems with strongly and weakly discontinuous solutions. To enhance its performance in highly heterogeneous transient diffusion systems, we prescribe a suite of robust practices, including appropriate non-dimensionalization of equations, a biased sampling method, Glorot initialization, and the hard enforcement of boundary and initial conditions. Here we evaluate the efficacy of the proposed method on several benchmark forward and inverse problems. Comparative studies on one-dimensional and two-dimensional benchmark problems reveal that the modified AdaI-PINNs outperform its unmodified counterpart, achieving root-mean-square errors that are at least two orders of magnitude better in forward problems. For inverse problems, the maximum errors in the approximated diffusion coefficients by modified AdaI-PINNs are four orders of magnitude better than those of the unmodified version. Additionally, modified AdaI-PINNs demonstrate improved stability in problems with large material mismatches.
Inverse problems in radiative transfer applied to remote measurements of IR radiation from planetary atmospheres, noting temperature and water vapor inversions
The relationships of a variety of general computational methods (and variances) for treating illposed problems such as geophysical inverse problems are considered. Differences in approach and interpretation based on varying assumptions as to, e.g., the nature of measurement uncertainties are discussed along with the factors to be considered in selecting an approach. The reliability of the results of such computation is addressed.
The usual inversion procedures for determining the oceanic circulation take the density field as given. This does not explicitly account for the mismatch between the data (which are typically a single realization of the density field) and the circulation being estimated (which may be a time mean or some other smoothed quantity), but explicit consideration of the uncertainty of the density data renders the problem nonlinear, and new estimation methods have to be used. As an example, a nonlinear inversion is successfully applied here to the simple analytical model of Stommel and Veronis (1981). It is also shown that their linear procedure, if properly used, yields a much better result than they obtained.
The mass sensing superiority of a micro/nanomechanical resonator sensor over conventional mass spectrometry has been, or at least, is being firmly established. Because the sensing mechanism of a mechanical resonator sensor is the shifts of resonant frequencies, how to link the shifts of resonant frequencies with the material properties of an analyte formulates an inverse problem. Besides the analyte/adsorbate mass, many other factors such as position and axial force can also cause the shifts of resonant frequencies. The in-situ measurement of the adsorbate position and axial force is extremely difficult if not impossible, especially when an adsorbate is as small as a molecule or an atom. Extra instruments are also required. In this study, an inverse problem of using three resonant frequencies to determine the mass, position and axial force is formulated and solved. The accuracy of the inverse problem solving method is demonstrated and how the method can be used in the real application of a nanomechanical resonator is also discussed. Solving the inverse problem is helpful to the development and application of mechanical resonator sensor on two things: reducing extra experimental equipments and achieving better mass sensing by considering more factors.
Inverse problem of radiative transfer as applied to remote measurements of infrared radiation from planetary atmospheres