Eulogy: Kyriacos D. Papailiou (1939–2021)

A major task of geophysics is to make quantitative statements about the interior of the earth. For this reason, inverse problems are an important area of geophysical research and industrial application. Figure 1 shows how many texts present inverse problems. The earth model is an element of a mathematical space that contains all allowable parameterizations of the earth’s properties (or at least those properties relevant to a given experiment); this space is referred to as model space . The physics of the problem determines which data d correspond to a given model m . The problem of computing the model response (synthetic “data”) given a model is called the forward problem . The corresponding data reside in a mathematical space that is called data space . In many applications, one records the data, and the goal is to find the corresponding model. The task is called the inverse problem , as shown in Figure 1. FIG. 1. The conventional view of inverse problems: find the model that predicts the measurements. Unfortunately, Figure 1 is wrong. There is a simple reason for this. In general the model that one seeks is a continuous function of the space variables with infinitely many degrees of freedom. For example, the 3-D velocity structure in the earth has infinitely many degrees of freedom. On the other hand, the data space is always of finite dimension because any real experiment can only result in a finite number of measurements. A simple count of variables shows that the mapping from the data to a model cannot be unique; or equivalently, there must be elements of the model space that have no influence on the data. This lack of uniqueness is apparent even for problems involving idealized, noise-free measurements. The problem only becomes worse when the uncertainties of real …

Analysis of tissue and arterial blood temperatures in the resting human forearm. 1948.

Previous article Next article Determination of an Unknown Heat Source from Overspecified Boundary DataJ. R. CannonJ. R. Cannonhttps://doi.org/10.1137/0705024PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. R. Cannon, Determination of an unknown coefficient in a parabolic differential equation, Duke Math. J., 30 (1963), 313–323 10.1215/S0012-7094-63-03033-3 MR0157121 (28:358) 0117.06901 CrossrefISIGoogle Scholar[2] J. R. Cannon, Determination of certain parameters in heat conduction problems, J. Math. Anal. Appl., 8 (1964), 188–201 10.1016/0022-247X(64)90061-7 MR0160047 (28:3261) 0131.32104 CrossrefGoogle Scholar[3] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla \cdot k(u)\nabla u=0$ from overspecified boundary data, J. Math. Anal. Appl., 18 (1967), 112–114 10.1016/0022-247X(67)90185-0 MR0209634 (35:531) 0151.15901 CrossrefISIGoogle Scholar[4] J. R. Cannon and , D. L. Filmer, The determination of unknown parameters in analytic systems of ordinary differential equations, SIAM J. Appl. Math., 15 (1967), 799–809 10.1137/0115069 MR0218632 (36:1716) 0251.34002 LinkISIGoogle Scholar[5] J. R. Cannon, , Jim Douglas, Jr. and , B. Frank Jones, Jr., Determination of the diffusivity of an isotropic medium, Internat. J. Engrg. Sci., 1 (1963), 453–455 10.1016/0020-7225(63)90002-8 MR0160045 (28:3259) CrossrefGoogle Scholar[6] J. R. Cannon and , B. Frank Jones, Jr., Determination of the diffusivity of an anisotropic medium, Internat. J. Engrg. Sci., 1 (1963), 457–460 10.1016/0020-7225(63)90003-X MR0160046 (28:3260) CrossrefGoogle Scholar[7] J. R. Cannon and , J. H. Halton, The irrotational solution of an elliptic differential equation with an unknown coefficient, Proc. Cambridge Philos. Soc., 59 (1963), 680–682 MR0149064 (26:6560) 0117.07101 CrossrefISIGoogle Scholar[8] Jim Douglas, Jr. and , B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. II. Numerical approximation, J. Math. Mech., 11 (1962), 919–926 MR0153988 (27:3949) 0112.32603 ISIGoogle Scholar[9] B. Frank Jones, Jr., The determination of a coefficient in a parabolic differential equation. I. Existence and uniqueness, J. Math. Mech., 11 (1962), 907–918 MR0153987 (27:3948) 0112.32602 ISIGoogle Scholar[10] B. Frank Jones, Jr., Various methods for finding unknown coefficients in parabolic differential equations, Comm. Pure Appl. Math., 16 (1963), 33–44 MR0152760 (27:2735) 0119.08302 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Identifying a space-dependent source term in distributed order time-fractional diffusion equationsMathematical Control and Related Fields, Vol. 0, No. 0 | 1 Jan 2022 Cross Ref Identification of stationary source in the anomalous diffusion equationInverse Problems in Science and Engineering, Vol. 29, No. 13 | 21 November 2021 Cross Ref A modified quasi-reversibility method for inverse source problem of Poisson equationInverse Problems in Science and Engineering, Vol. 29, No. 12 | 22 March 2021 Cross Ref Inverse modeling of contaminant transport for pollution source identification in surface and groundwaters: a reviewGroundwater for Sustainable Development, Vol. 15 | 1 Nov 2021 Cross Ref Convergence Analysis of a Crank–Nicolson Galerkin Method for an Inverse Source Problem for Parabolic Equations with Boundary ObservationsApplied 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Stationary caseInverse Problems in Science and Engineering, Vol. 15, No. 8 | 18 December 2007 Cross Ref The boundary-element method for the determination of a heat source dependent on one variableJournal of Engineering Mathematics, Vol. 54, No. 4 | 3 January 2006 Cross Ref Inverse source problem for a transmission problem for a parabolic equationJournal of Inverse and Ill-posed Problems, Vol. 14, No. 1 | 1 Jan 2006 Cross Ref Identification of a point source in a linear advection–dispersion–reaction equation: application to a pollution source problemInverse Problems, Vol. 21, No. 3 | 9 May 2005 Cross Ref Conditional stability in determining a heat sourceJournal of Inverse and Ill-posed Problems, Vol. 12, No. 3 | 1 Jun 2004 Cross Ref Lipschitz stability in inverse parabolic problems by the Carleman estimateInverse Problems, Vol. 14, No. 5 | 1 January 1999 Cross Ref Determination of the unknown sources in the heat-conduction equationComputational Mathematics and Modeling, Vol. 8, No. 4 | 1 Oct 1997 Cross Ref An estimation method for point sources of multidimensional diffusion equationApplied Mathematical Modelling, Vol. 21, No. 2 | 1 Feb 1997 Cross Ref A reliable estimation method for locations of point sources for an n-dimensional Poisson equationApplied Mathematical Modelling, Vol. 20, No. 11 | 1 Nov 1996 Cross Ref Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary dataInverse Problems, Vol. 10, No. 6 | 1 January 1999 Cross Ref Conditional Stability in Determination of Densities of Heat Sources in a Bounded DomainControl and Estimation of Distributed Parameter Systems: Nonlinear Phenomena | 1 Jan 1994 Cross Ref Conditional stability in determination of force terms of heat equations in a rectangleMathematical and Computer Modelling, Vol. 18, No. 1 | 1 Jul 1993 Cross Ref Inverse problems and Carleman estimatesInverse Problems, Vol. 8, No. 4 | 1 January 1999 Cross Ref Determination of an unknown function in a parabolic equation with an overspecified conditionMathematical Methods in the Applied Sciences, Vol. 13, No. 5 | 1 Nov 1990 Cross Ref Some stability estimates for a heat source in terms of overspecified data in the 3-D heat equationJournal of Mathematical Analysis and Applications, Vol. 147, No. 2 | 1 Apr 1990 Cross Ref Parameter identification in hyperbolic and parabolic partial differential equations of cylindrical geometry from overspecified boundary dataInternational Journal of Engineering Science, Vol. 28, No. 10 | 1 Jan 1990 Cross Ref Identification of certain physical parameters in hyperbolic boundary value problems from overspecified boundary dataInternational Journal of Engineering Science, Vol. 26, No. 8 | 1 Jan 1988 Cross Ref An Inverse Problem for a Nonlinear Elliptic Differential EquationMichael Pilant and William RundellSIAM Journal on Mathematical Analysis, Vol. 18, No. 6 | 1 August 2006AbstractPDF (793 KB)An inverse problem for the heat equationInverse Problems, Vol. 2, No. 4 | 1 January 1999 Cross Ref A Note on an Inverse Problem Related to the 3-D Heat EquationInverse Problems | 1 Jan 1986 Cross Ref Determination of a Source Term in a Linear Parabolic Differential Equation with Mixed Boundary ConditionsInverse Problems | 1 Jan 1986 Cross Ref Nonparametric algorithm for input signals identification in static distributed-parameter systemsIEEE Transactions on Automatic Control, Vol. 29, No. 7 | 1 Jul 1984 Cross Ref Parameter determination in parabolic partial differential equations from overspecified boundary dataInternational Journal of Engineering Science, Vol. 20, No. 6 | 1 Jan 1982 Cross Ref Determination of an unknown non-homogeneous term in a linear partial differential equation .from overspecified boundary dataApplicable Analysis, Vol. 10, No. 3 | 2 May 2007 Cross Ref Distributed parameter system indentification A survey†International Journal of Control, Vol. 26, No. 4 | 16 May 2007 Cross Ref Determination of a source term in a linear parabolic partial differential equationZeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 27, No. 3 | 1 May 1976 Cross Ref Some general remarks on improperly posed problems for partial differential equationsSymposium on Non-Well-Posed Problems and Logarithmic Convexity | 22 August 2006 Cross Ref Estimation of parameters in partial differential equations from noisy experimental dataChemical Engineering Science, Vol. 26, No. 6 | 1 Jun 1971 Cross Ref Determination of an unknown forcing function in a hyperbolic equation from overspecified dataAnnali di Matematica Pura ed Applicata, Vol. 85, No. 1 | 1 Dec 1970 Cross Ref The character of non-uniqueness in the conductivity modelling problem for the earthPure and Applied Geophysics PAGEOPH, Vol. 80, No. 1 | 1 Jan 1970 Cross Ref Volume 5, Issue 2| 1968SIAM Journal on Numerical Analysis199-459 History Submitted:29 September 1967Published online:03 August 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705024Article page range:pp. 275-286ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Fixed-Point Algorithms for Inverse Problems in Science and Engineering presents some ofthe most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.

We have developed a multisource full-waveform inversion algorithm using a dynamic phase encoding strategy with dual-randomization—both the position and polarity of simultaneous sources are randomized and changed every iteration. The dynamic dual-randomization is used to promote the destructive interference of crosstalk noise resulting from blending a large number of common shot gathers into a supergather. We compare our multisource algorithm with various algorithms in a numerical experiment using the 3D SEG/EAGE overthrust model and show that our algorithm provides a higher-quality velocity tomogram than the other methods that use only monorandomization. This suggests that increasing the degree of randomness in phase encoding should improve the quality of the inversion result.

Biosecurity and Bioterrorism: Biodefense Strategy, Practice, and ScienceVol. 2, No. 1 Synopsis of January 22-23, 2004, MeetingPublished Online:5 Jul 2004https://doi.org/10.1089/153871304322964327AboutSectionsPDF/EPUB ToolsPermissionsDownload CitationsTrack CitationsAdd to favorites Back To Publication ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetailsCited bySource characterization of atmospheric releases using stochastic search and regularized gradient optimization28 June 2011 | Inverse Problems in Science and Engineering, Vol. 19, No. 8 Volume 2Issue 1Jan 2004 To cite this article:Synopsis of January 22-23, 2004, Meeting.Biosecurity and Bioterrorism: Biodefense Strategy, Practice, and Science.Jan 2004.41-45.http://doi.org/10.1089/153871304322964327Published in Volume: 2 Issue 1: July 5, 2004PDF download

In this paper we prove some new converse and saturation results for Tikhonov regularization of linear ill-posed problems Tx=y, where T is a linear operator between two Hilbert spaces.

Introduction With only a few minor exceptions, electrical geophysical methods can be divided into two categories: (1) galvanic source methods, directly coupling or injecting electrical current into the ground via electrodes, and (2) inductive source methods, inducing eddy currents into the ground via time-varying magnetic fields using coils not in direct contact with the ground. Ground-penetrating radar (GPR) can be considered an extreme high-frequency, dielectric-properties-sensitive example of the latter. Self-potential (SP) effects, due to electrochemical mechanisms in the ground, and telluric current methods (where only ambient voltages, induced by natural EM sources such as the oscillating magnetosphere of the earth, are measured) can arguably be included in the former, but are both passive methods and are beyond the scope of this chapter.

This study presents a methodology to treat performance-based seismic design (PBSD) as an inverse engineering problem, where design parameters are directly derived to achieve specific performance objectives (POs). By implementing explainable machine learning (ML) models, this methodology directly maps design variables and performance metrics, thereby tackling the computational inefficiencies associated with performance-based design. The resultant ML model is integrated as an evaluation function into a genetic optimization algorithm to solve the inverse problem. The developed methodology is then applied to two different inventories of steel and concrete moment frames in Los Angeles and Charleston to obtain sectional properties of frame members that minimize expected annualized seismic loss in terms of repair costs. The results show high accuracy of the surrogate models (e.g. R2 > 90%) across a diverse set of building types, geometries, seismic design and site hazard, where the optimization algorithm could identify the optimum values of members' properties for a fixed set of geometric variables, consistent with engineering principles.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

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.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

We introduce stochastic asymptotical regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We demonstrate that SAR can quantify the uncertainty in error estimates for inverse problems. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an order-optimal regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

In Bayesian inverse problems, it is common to consider several hyperparameters that define the prior and the noise model that must be estimated from the data. In particular, we are interested in linear inverse problems with additive Gaussian noise and Gaussian priors defined using Matérn covariance models. In this case, we estimate the hyperparameters using the maximum a posteriori (MAP) estimate of the marginalized posterior distribution. However, this is a computationally intensive task since it involves computing log determinants. To address this challenge, we consider a stochastic average approximation (SAA) of the objective function and use the preconditioned Lanczos method to compute efficient approximations of the function and gradient evaluations. We propose a new preconditioner that can be updated cheaply for new values of the hyperparameters and an approach to compute approximations of the gradient evaluations, by reutilizing information from the function evaluations. We demonstrate the performance of our approach on static and dynamic seismic tomography problems.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

Spline-based methods have been applied to inverse problems in science and engineering.Those studies have shown that if proper spline bases can be chosen, problem complexity can be significantly reduced while increasing estimation accuracy and robustness against the disturbances.We proposed non-parametric regression spline based approach for the solution of inverse ECG problem and assessed its robustness against measurement noise, variation of the heart size from its true size, and their combination.Our model defines the spline functions in terms of spatial coordinate variables based on the given epicardial surface geometry.The results demonstrated that, proposed method performed better than the Tikhonov regularization and can be feasible alternative for the inverse ECG problem.

It has been hypothesized that during a motion task the central nervous system controls the skeletal muscles partitioning them into synergetic groups, hence effectively reducing the dimensionality of the control problem. The identification of muscle groups that are co-activated remains an open problem: its solution could have important implications in the design of training or rehabilitation protocols. In this article, we combine Bayesian inverse problem techniques and data science algorithms to identify muscle synergies in human motion from the motion tracker time series of positions of fiducial markers on the body during the task. The inverse problem of estimating the muscle activation patterns from the motion tracking data is cast in the Bayesian framework, and the posterior distribution of muscle activations is explored using Myobolica, a Gibbs-sampler-based Markov chain Monte Carlo sampler. A low-rank approximation of the muscle activation patterns is then obtained via a sparsity promoting Bayesian non-negative matrix factorization of the sample mean, where the sparse coefficient vectors correspond to groups of muscles that show co-activation over the sample.This article is part of the theme issue 'Frontiers of applied inverse problems in science and engineering'.

Chapter 1 - Forward and Inverse Problems in Science and Engineering