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A method for estimating an unknown domain in elliptic boundary value problems is considered. The problem is formulated as an inverse problem of integral equations of the second kind. A computational method is developed using a splice collocation scheme. The results can be applied to the inverse problem of impedance computed tomography (ICT) for image reconstruction.
Abstract Of great relevance to climate engineering is the systematic relationship between the radiative forcing to the climate system and the response of the system, a relationship often represented by the linear response function (LRF) of the system. However, estimating the LRF often becomes an ill-posed inverse problem due to high-dimensionality and nonunique relationships between the forcing and response. Recent advances in machine learning make it possible to address the ill-posed inverse problem through regularization and sparse system fitting. Here, we develop a convolutional neural network (CNN) for regularized inversion. The CNN is trained using the surface temperature responses from a set of Green’s function perturbation experiments as imagery input data together with data sample densification. The resulting CNN model can infer the forcing pattern responsible for the temperature response from out-of-sample forcing scenarios. This promising proof of concept suggests a possible strategy for estimating the optimal forcing to negate certain undesirable effects of climate change. The limited success of this effort underscores the challenges of solving an inverse problem for a climate system with inherent nonlinearity. Significance Statement Predicting the climate response for a given climate forcing is a direct problem, while inferring the forcing for a given desired climate response is often an inverse, ill-posed, problem, posing a new challenge to the climate community. This study makes the first attempt to infer the radiative forcing for a given target pattern of global surface temperature response using a deep learning approach. The resulting deeply trained convolutional neural network inversion model shows promise in capturing the forcing pattern corresponding to a given surface temperature response, with a significant implication on the design of an optimal solar radiation management strategy for curbing global warming. This study also highlights the technical challenges that future research should prioritize in seeking feasible solutions to the inverse climate problem.
We develop a new methodology for extracting Compton form factors (CFFs) from deeply virtual exclusive reactions such as the unpolarized DVCS cross section using a specialized inverse problem solver, a variational autoencoder inverse mapper (VAIM). The VAIM-CFF framework not only allows us access to a fitted solution set possibly containing multiple solutions in the extraction of all 8 CFFs from a single cross section measurement, but also accesses the lost information contained in the forward mapping from CFFs to cross section. We investigate various assumptions and their effects on the predicted CFFs such as cross section organization, number of extracted CFFs, use of uncertainty quantification technique, and inclusion of prior physics information. We then use dimensionality reduction techniques such as principal component analysis to visualize the missing physics information tracked in the latent space of the VAIM framework. Through re-framing the extraction of CFFs as an inverse problem, we gain access to fundamental properties of the problem not comprehensible in standard fitting methodologies: exploring the limits of the information encoded in deeply virtual exclusive experiments.
One of the main purposes of geodesy is to determine the gravity field of the Earth in the space outside its physical surface. This purpose can be pursued without any particular knowledge of the internal density even if the exact shape of the physical surface of the Earth is not known, though this seems to entangle the two domains, as it was in the old Stoke's theory before the appearance of Molodensky's approach. Nevertheless, even when large, dense and homogeneous data sets are available, it was always recognized that subtracting from the gravity field the effect of the outer layer of the masses (topographic effect) yields a much smoother field. This is obviously more important when a sparse data set is bad so that any smoothing of the gravity field helps in interpolating between the data without raising the modeling error, this approach is generally followed because it has become very cheap in terms of computing time since the appearance of spectral techniques. The mathematical description of the Inverse Gravimetric Problem (IGP) is dominated mainly by two principles, which in loose terms can be formulated as follows: the knowledge of the external gravity field determines mainly the lateral variations of the density; and the deeper the density anomaly giving rise to a gravity anomaly, the more improperly posed is the problem of recovering the former from the latter. The statistical relation between rho and n (and its inverse) is also investigated in its general form, proving that degree cross-covariances have to be introduced to describe the behavior of rho. The problem of the simultaneous estimate of a spherical anomalous potential and of the external, topographic masses is addressed criticizing the choice of the mixed collection approach.
The ability to solve inverse problems – inferring unknown parameters, structures, or states of a system from observed data – is essential for advancing scientific discovery and innovation capabilities for the DOE mission. Basic research needs and challenges are particularly acute in emerging areas such as the interactive, data-driven, modeling and simulation of digital twins; decision support for experiments at DOE scientific user facilities; and for other complex systems and workflows. Inverse problems are at the heart of understanding and controlling complex systems due to factors such as observational data with varying modalities and fidelities, inherent uncertainties in physical measurements and numerical models, and the computational demands of rapid and high-fidelity simulations. The convergence of recent scientific computing trends – scientific machine learning, artificial intelligence, and computing advances such as exascale computing – is creating unprecedented opportunities. These advancements offer the potential to revolutionize how we approach inverse problems to extract actionable insights with the required level of accuracy and computational efficiency. This workshop and the Call for Position Papers are vital steps in bringing together experts to collectively explore and identify the new computational and mathematical directions needed in inverse methods for complex systems under uncertainty.
It is known from Lighthill's exact solution of the incompressible inverse problem that in the inverse design problem, the surface pressure distribution and the free stream speed cannot both be prescribed independently. This implies the existence of a constraint on the prescribed pressure distribution. The same constraint exists at compressible speeds. Presented here is an inverse design method for transonic airfoils. In this method, the target pressure distribution contains a free parameter that is adjusted during the computation to satisfy the regularity condition. Some design results are presented in order to demonstrate the capabilities of the method.
The increasing availability of precipitation observations from space, e.g., from the Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) Mission, has fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can handle large data sets in computationally efficient ways while optimally reproducing desired properties of the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism for downscaling satellite precipitation observations, which explicitly allows for the preservation of some key geometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to high-intensity regions embedded within lower-intensity areas), coherent spatial structures (due to regions of slowly varying rainfall),and thicker-than-Gaussian tails of precipitation gradients and intensities. Specifically, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational approach (variational downscaling) where the regularization term is selected to impose the desired smoothness in the solution while allowing for some steep gradients(called 1-norm or total variation regularization). We demonstrate the duality between this geometrically inspired solution and its Bayesian statistical interpretation, which is equivalent to assuming a Laplace prior distribution for the precipitation intensities in the derivative (wavelet) space. When the observation operator is not known, we discuss the effect of its misspecification and explore a previously proposed dictionary-based sparse inverse downscaling methodology to indirectly learn the observation operator from a database of coincidental high- and low-resolution observations. The proposed method and ideas are illustrated in case studies featuring the downscaling of a hurricane precipitation field.
The main results of the grant were (1) finishing the manuscript of a proof of completeness of the Poincare modes in an incompressible nonviscous fluid corotating with a rigid ellipsoidal boundary, (2) partial completion of a manuscript describing a definition of helicity that resolved questions in the literature about calculating the helicities of vector fields with complicated topologies, and (3) the beginning of a reexamination of the inverse problem of inferring properties of the geomagnetic field B just outside the core-mantle boundary (CMB) from measurements of elements of B at and above the earth's surface. This last work has led to a simple general formalism for linear and nonlinear inverse problems that appears to include all the inversion schemes so far considered for the uniqueness problem in geomagnetic inversion. The technique suggests some new methods for error estimation that form part of this report.
Eigenvalue problem of inverse scattering in quantum mechanics considered in terms of orthonormal and biorthonormal wave functions
The use of simulated and controlled acoustic emission signals forms the basis of a powerful tool for the detailed study of various deformation and wave interaction processes in materials. The results of experiments and signal analyses of acoustic emission resulting from point sources such as various types of indentation-produced cracks in brittle materials and the growth of fatigue cracks in 7075-T6 aluminum panels are discussed. Recent work dealing with the modeling and subsequent signal processing of an extended source of emission in a material is reviewed. Results of the forward problem and the inverse problem are presented with the example of a source distributed through the interior of a specimen.
A fully Sinc-Galerkin method for recovering the spatially varying stiffness parameter in fourth-order time-dependence problems with fixed and cantilever boundary conditions is presented. The forward problems are discretized with a sinc basis in both the spatial and temporal domains. This yields an approximation solution which converges exponentially and is valid on the infinite time interval. When the forward methods are applied to parameter recovery problems, the resulting inverse problems are ill-posed. Tikhonov regularization is applied and the resulting minimization problems are solved via a quasi-Newton/trust region algorithm. The L-curve method is used to determine an appropriate value of the regularization parameter. Numerical results which highlight the method are given for problems with both fixed and cantilever boundary conditions.
A spherical harmonic model of the magnetic field of Jupiter is obtained from vector magnetic field observations acquired by the Juno spacecraft during its first nine polar orbits about the planet. Observations acquired during eight of these orbits provide the first truly global coverage of Jupiter's magnetic field with a coarse longitudinal separation of ~45 deg between perijoves. The magnetic field is represented with a degree 20 spherical harmonic model for the planetary ("internal") field, combined with a simple model of the magnetodisc for the field ("external") due to distributed magnetospheric currents. Partial solution of the underdetermined inverse problem using generalized inverse techniques yields a model ("Juno Reference Model through Perijove 9") of the planetary magnetic field with spherical harmonic coefficients well determined through degree and order 10, providing the first detailed view of a planetary dynamo beyond Earth.
Detailed knowledge of the local environments of spin defects in semiconductors, such as nitrogenvacancy (NV) centers in diamond or divacancies in silicon carbide, is crucial for optimizing control and entanglement protocols in quantum sensing and information applications. However, at present a direct experimental characterization of individual defect environments is not scalable, as conventional spin-bath measurements are time consuming and difficult to automate. Achieving high-throughput characterization requires short experiments to probe the spin bath. However, with fewer and noisier measurements, the inverse problem of recovering spin-bath properties from measured data becomes ill posed, with multiple spin baths having a high likelihood of yielding the same data. In this work, we present a set of computational tools to resolve the ill-posed inverse problem of recovering the atomic positions and hyperfine couplings of random nuclei surrounding spin defects from sparse, noisy experimental coherence data, which can be obtained in hours. Here, we use a trans-dimensional Bayesian approach that incorporates ab initio data to yield full posterior distributions over nuclear spin environments, enabling robust recovery from limited data. We also provide practical tools and guidelines to determine the limits of detectability for hyperfine couplings under specific dynamical decoupling sequences and sampling conditions. In addition, we demonstrate how the tools developed here, in combination with ab initio simulations of spin baths, can guide the design of efficient experimental protocols for application-specific high-throughput screening. To showcase the utility of our approach, we apply it to design fast dynamical decoupling experiments to characterize the spin baths often individual NV centers in diamond. While the primary focus is on accelerating spin-bath characterization of spin defects, this Bayesian approach also lays the foundation for digital-twin studies of spin defects, where a virtual model of the spin-defect system evolves in real time with ongoing experimental measurements. Together, the set of tools we designed and applied paves the way for scalable deployment of spin defects in semiconductors for quantum sensing and information applications.
To solve a geophysical inverse problem means applying measurements to determine the parameters of the selected model. The inverse problem is formulated as the Bayesian inference. The Gaussian probability density functions are applied in the Bayes's equation. The CHAMP satellite gravity data are determined at the altitude of 400 kilometer altitude over the South part of the Pannonian basin. The model of interpretation is the right vertical cylinder. The parameters of the model are obtained from the minimum problem solved by the Simplex method.
Rotational spectroscopy is the most accurate method for determining structures of molecules in the gas phase. It is often assumed that a rotational spectrum is a unique “fingerprint” of a molecule. The availability of large molecular databases and the development of artificial intelligence methods for spectroscopy make the testing of this assumption timely. In this paper, we pose the determination of molecular structures from rotational spectra as an inverse problem. Within this framework, we adopt a funnel-based approach to search for molecular twins, which are two or more molecules, which have similar rotational spectra but distinctly different molecular structures. Here we demonstrate that there are twins within standard levels of computational accuracy by generating rotational constants for many molecules from several large molecular databases, indicating that the inverse problem is ill-posed. However, some twins can be distinguished by increasing the accuracy of the theoretical methods or by performing additional experiments.
The inverse problem in empirical geomagnetic modeling is investigated, with critical examination of recently published studies. Particular attention is given to the use of Bayesian inference (BI) to select the damping parameter lambda in the uniqueness portion of the inverse problem. The mathematical bases of BI and stochastic inversion are explored, with consideration of bound-softening problems and resolution in linear Gaussian BI. The problem of estimating the radial magnetic field B(r) at the earth core-mantle boundary from surface and satellite measurements is then analyzed in detail, with specific attention to the selection of lambda in the studies of Gubbins (1983) and Gubbins and Bloxham (1985). It is argued that the selection method is inappropriate and leads to lambda values much larger than those that would result if a reasonable bound on the heat flow at the CMB were assumed.
The inverse problem of recovering an electrical conductivity of the form Gamma(x) = 1 + (k-1)(sub Chi(D)) (Chi(D) is the characteristic function of D) on a region omega is a subset of 2-dimensional Euclid space from boundary data is considered, where D is a subset of omega and k is some positive constant. A linearization of the forward problem is formed and used in a least squares output method for approximately solving the inverse problem. Convergence results are proved and some numerical results presented.
Abstract This paper presents a statistical forward model for a Compton imaging system, called Compton imager. This system, under development at the University of Illinois Urbana Champaign, is a variant of Compton cameras with a single type of sensors which can simultaneously act as scatterers and absorbers. This imager is convenient for imaging situations requiring a wide field of view. The proposed statistical forward model is then used to solve the inverse problem of estimating the location and energy of point-like sources from observed data. This inverse problem is formulated and solved in a Bayesian framework by using a Metropolis within Gibbs algorithm for the estimation of the location, and an expectation-maximization algorithm for the estimation of the energy. This approach leads to more accurate estimation when compared with the deterministic standard back-projection approach, with the additional benefit of uncertainty quantification in the low photon imaging setting.