Forward and inverse problems of optical engineering when a two-dimensional illuminance distribution is being formed

In the last 25 years, mathematical and numerical modeling of cardiovascular problems has experienced a spectacular progress. Moving progressively from the stage of idealized models supporting traditional in vitro and in vivo approaches – basically with proofs of concept – Cardiovascular Mathematics has been transforming into a unique tool for patient-specific analysis. This has been made possible by the strong integration with imaging devices. Nowadays, the work pipeline starting from a medical image in the Digital Imaging and Communications in Medicine (DICOM) format to retrieve the patient-specific geometry of an artery or the left ventricle to be used as computational domain in a numerical simulation is well established. This has initiated a paradigm shift in the interactions between mathematicians, bioengineers, and medical doctors. The latter are progressively getting used to in silico analysis as an important component of their research [3]. The introduction of numerical procedures as a part of an established clinical routine and more in general of a consolidated support to the decision-making process of physicians still requires some steps both in terms of infrastructures (to bring computational tools to the operating room or the bedside) and methods. In particular, the quality of the numerical results needs to be assessed and certified. The reliability of simulations calls for an accurate quantification and possibly reduction of uncertainty. In this scenario and in view of terrific advancements of measuring techniques, an important research line – quite established in other fields – is data assimilation [4,6,9], that is, the integration of numerical simulations and measurements. We may say that numerical models provide a background knowledge (based on physical principles and constitutive laws, not patient-specific), whereas measures give a foreground (individual) information; accuracy of in silico procedures relies on the correct integration of these two levels of knowledge of the problem [9,10]. As a matter of fact, numerical models depend on parameters and boundary/initial conditions. Their knowledge is in practice usually incomplete, inaccurate, and noisy. In some cases, the sensitivity itself of the final results to these parameters is unclear and needs to be quantified. Such an uncertainty may be significantly reduced by the availability of patient-specific measures; on the other hand, the quality of measures can be strongly enhanced by comparison with mathematical models (after all, noise is in general not educated, it does not know PDEs . . . ) [4,5,7,8,9,11]. We think therefore that, in the future, a crucial step for the clinical application of Cardiovascular Mathematics will be the passage from numerical simulation to data assimilation. To be concrete, borrowing a motto by Michel Fortin supporting mesh adaptivity in finite element simulations, stating that a ‘grid should not be considered a data, but an unknown of the problem’, we think that many parameters currently tuned on the basis of nonpatient specific criteria (even in patient-specific geometries) should be considered part of the unknowns to be found by assimilation procedures.

BackgroundThe aim of electroencephalogram (EEG) source localization is to find the brain areas responsible for EEG waves of interest. It consists of solving forward and inverse problems. The forward problem is solved by starting from a given electrical source and calculating the potentials at the electrodes. These evaluations are necessary to solve the inverse problem which is defined as finding brain sources which are responsible for the measured potentials at the EEG electrodes.MethodsWhile other reviews give an extensive summary of the both forward and inverse problem, this review article focuses on different aspects of solving the forward problem and it is intended for newcomers in this research field.ResultsIt starts with focusing on the generators of the EEG: the post-synaptic potentials in the apical dendrites of pyramidal neurons. These cells generate an extracellular current which can be modeled by Poisson's differential equation, and Neumann and Dirichlet boundary conditions. The compartments in which these currents flow can be anisotropic (e.g. skull and white matter). In a three-shell spherical head model an analytical expression exists to solve the forward problem. During the last two decades researchers have tried to solve Poisson's equation in a realistically shaped head model obtained from 3D medical images, which requires numerical methods. The following methods are compared with each other: the boundary element method (BEM), the finite element method (FEM) and the finite difference method (FDM). In the last two methods anisotropic conducting compartments can conveniently be introduced. Then the focus will be set on the use of reciprocity in EEG source localization. It is introduced to speed up the forward calculations which are here performed for each electrode position rather than for each dipole position. Solving Poisson's equation utilizing FEM and FDM corresponds to solving a large sparse linear system. Iterative methods are required to solve these sparse linear systems. The following iterative methods are discussed: successive over-relaxation, conjugate gradients method and algebraic multigrid method.ConclusionSolving the forward problem has been well documented in the past decades. In the past simplified spherical head models are used, whereas nowadays a combination of imaging modalities are used to accurately describe the geometry of the head model. Efforts have been done on realistically describing the shape of the head model, as well as the heterogenity of the tissue types and realistically determining the conductivity. However, the determination and validation of the in vivo conductivity values is still an important topic in this field. In addition, more studies have to be done on the influence of all the parameters of the head model and of the numerical techniques on the solution of the forward problem.

Inverse problem solution is an integral part of data interpretation for well testing in petroleum reservoirs. In case of two-phase well tests with water injection, forward problem is based on the multiphase flow model in porous media and solved numerically. The inverse problem is based on a misfit or likelihood objective function. Adjoint methods have proved robust and efficient for gradient calculation of the objective function in this type of problems. However, if time-lapse electrical resistivity measurements during the well test are included in the objective function, both the forward and inverse problems become multiphysical, and straightforward application of the adjoint method is problematic. In this paper we present a novel adjoint algorithm for the inverse problems considered. It takes into account the structure of cross dependencies between flow and electrical equations and variables, as well as specifics of the equations (mixed parabolic-hyperbolic for flow and elliptic for electricity), numerical discretizations and grids, and measurements in the inverse problem. Decomposition is proposed for the adjoint problem which makes possible step-wise solution of the electric adjoint equations, like in the forward problem, after which a cross-term is computed and added to the right-hand side of the flow adjoint equations at this timestep. The overall procedure provides accurate gradient calculation for the multiphysical objective function while preserving robustness and efficiency of the adjoint methods. Example cases of the adjoint gradient calculation are presented and compared to straightforward difference-based gradient calculation in terms of accuracy and efficiency.

Researchers have investigated the dynamics of oscillation of a single-link assistive manipulator and carried out vibration analysis of the same due to its great application in real life. Thus, studying the dynamics of flexible assistive robots and developing the model is indeed an interesting research topic. It is possible that the exact solution to the equations associated with such a single-link manipulator does not always exist. We used a numerical method, viz., Galerkin’s method to solve the governing differential equations of the problem, and the corresponding frequency of oscillation was calculated. This is known as the forward problem. On the other hand, the inverse problem is to compute values of the unknowns when the frequency is known by experiment, but the values of the remaining parameters are not known. With this backdrop, the purpose of this chapter is to investigate forward and inverse problems related to single-link robotic manipulators using Euler’s beam theory to define the system dynamics. Firstly, the frequencies of the first four modes of vibration have been calculated. Furthermore, the frequency is assumed to be known (experimental outcome), and the inverse case is studied using Young’s modulus and density of the material of manufacturing of the manipulator as unknown parameters. Besides, the system of single-link manipulators has been studied in uncertain environments using fuzzy theory for forward and inverse problems.

P 211. Efficient method for computing the TMS-induced electric field in spherically symmetric head models, the triangle Construction

Many materials (e.g., rubber or biologic tissues) are "nearly" incompressible and often assumed to be incompressible in their constitutive equations. This assumption hinders realistic analyses of wave motion including acoustoelasticity. In this study, this constraint is relaxed and the reflected waves from nearly incompressible, hyper-elastic materials are examined. Specifically, reflection coefficients are considered from the interface of water and uni-axially prestretched rubber. Both forward and inverse problems are experimentally and analytically studied with the incident wave perpendicular to the interface. In the forward problem, the wave reflection coefficient at the interface is evaluated with strain energy functions for nearly incompressible materials in order to compute applied strain. For the general inverse problem, mathematical relations are derived that identify both uni-axial strains and normalized material constants from reflected wave data. The validity of this method of analysis is demonstrated via an experiment with stretched rubber. Results demonstrate that applied strains and normalized material coefficients can be simultaneously determined from the reflected wave data alone if they are collected at several different (but unknown) levels of strain. This study therefore indicates that acoustoelasticity, with an appropriate constitutive formulation, can determine strain and material properties in hyper-elastic, nearly incompressible materials.

Inverse Theory and Applications in Geophysics

International Journal of Computational Engineering ScienceVol. 04, No. 01, pp. 19-65 (2003) No AccessFULLY NONLINEAR MODELING AND ANALYSIS OF PRECISION MEMBRANESP. FRANK PAI and LEYLAND G. YOUNGP. FRANK PAIDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author and LEYLAND G. YOUNGDepartment of Mechanical and Aerospace Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA Search for more papers by this author https://doi.org/10.1142/S1465876303000788Cited by:4 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractHigh precision membranes are used in many current space applications. This paper presents a fully nonlinear membrane theory with forward and inverse analyses of high precision membrane structures. The fully nonlinear membrane theory is derived from Jaumann strains and stresses, exact coordinate transformations, the concept of local relative displacements, and orthogonal virtual rotations. In this theory, energy and Newtonian formulations are fully correlated, and every structural term can be interpreted in terms of vectors. Fully nonlinear ordinary differential equations (ODEs) governing the large static deformations of known axisymmetric membranes under known axisymmetric loading (i.e., forward problems) are presented as first-order ODEs, and a method for obtaining numerically exact solutions using the multiple shooting procedure is shown. A method for obtaining the undeformed geometry of any axisymmetric membrane with a known inflated geometry and a known internal pressure (i.e., inverse problems) is also derived. Numerical results from forward analysis are verified using results in the literature, and results from inverse analysis are verified using known exact solutions and solutions from the forward analysis. Results show that the membrane theory and the proposed numerical methods for solving nonlinear forward and inverse membrane problems are accurate.Keywords:Nonlinear membrane theorynumerically exact analysis References M. Salama , M. Lou and H. Fang , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2000 ) . Google Scholar M. S. Smith , R. S. Schallenkamp and L. G. Seely , AIAA International Balloon Technology Conference . Google Scholar D. Gorinevsky , T. Hyde and C. Cabuz , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 2001 ) . Google Scholar V. Firt , Statics, Formfinding and Dynamics of Air-Supported Membrane Structures ( Martinus Nijhoff Publishers , Boston , 1983 ) . Google Scholar A. L. Palisoc, Inflatable Reflector Development Program, Task 3 Report, L'Garde Technical Report, LTR-94-AP-008, 1994 . Google Scholar S. V. Damle et al. , AIAA International Balloon Technology Conference . Google ScholarP. F. Pai, A. N. Palazotto and J. M. Greer Jr., Comput. Struct. 66, 823 (1998), DOI: 10.1016/S0045-7949(98)00004-2. Crossref, Google Scholar H. Furuya and Y. Kawasaki , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference . Google Scholar G. Greschik , M. M. Mikulas and A. Palisoc , Proceedings of the AIAA Structures, Structural Dynamics and Materials Conference ( 1998 ) . Google ScholarJ. Ruze, Proceedings of the IEEE 54, 633 (1966), DOI: 10.1109/PROC.1966.4784. Crossref, Google Scholar A. Palisoc et al. , Fifth SPIE Conference on Space Telescopes and Instruments . Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 6, 459 (1994). Crossref, Google ScholarP. F. Pai and A. N. Palazotto, J. Eng. Mech. 121, 568 (1995), DOI: 10.1061/(ASCE)0733-9399(1995)121:4(568). Crossref, Google ScholarP. F. Pai and A. H. Nayfeh, Nonlinear Dynamics 2, 445 (1991), DOI: 10.1007/BF00045438. Crossref, Google Scholar K. Washizu , Variational Methods in Elasticity & Plasticity , 3rd edn. ( Pergamon Press Inc , New York , 1982 ) . Google ScholarP. F. Pai and A. N. Palazotto, Int. J. Solids Struct. 32, 3047 (1995), DOI: 10.1016/0020-7683(94)00273-Y. Crossref, Google Scholar H. Golden , T. W. Strganac and R. A. Schapery , AIAA International Balloon Technology Conference ( 1997 ) . Google Scholar J. Stoer and R. Bulirsch , Introduction to Numerical Analysis ( Springer-Verlag , New York , 1980 ) . Crossref, Google Scholar FiguresReferencesRelatedDetailsCited By 4Analytical Investigation of Dynamics of Inflatable Parabolic Membrane ReflectorFushou Liu and Dongping Jin1 Jan 2015 | Journal of Spacecraft and Rockets, Vol. 52, No. 1A point collocation method for geometrically nonlinear membranesKyle F. Kolsti and Donald L. Kunz1 Jan 2013 | International Journal of Solids and Structures, Vol. 50, No. 2Numerical and experimental dynamic characteristics of thin-film membranesLeyland G. Young, Suresh Ramanathan, Jiazhu Hu and P. Frank Pai1 May 2005 | International Journal of Solids and Structures, Vol. 42, No. 9-10Modeling, Analysis and Testing of some Deployable/Inflatable StructuresPerngjin Pai and Leyland Young26 June 2012 Recommended Vol. 04, No. 01 Metrics History Received 26 August 2002 Accepted 4 November 2002 KeywordsNonlinear membrane theorynumerically exact analysisPDF download

Geophysical Inverse Theory and Regularization Problems

We compute the solution to the photon diffusion equation using a state-space system. The state-space system under consideration is obtained by applying a FEM technique to the photon diffusion equation. Computing the photon density in the medium is commonly called in computational tomography, the forward problem. We investigate and present some preliminary results on the inverse problem. The inverse problem entails the identification of anomalies (benign or malignant) inside the medium. We investigate the inverse problem using an optimization setup. Specifically, we use a baseline homogeneous experiment that represents the healthy tissue to compare with the actual data. We then form an error signal between the heterogeneous and homogeneous case, which we try to minimize over the absorption coefficient of the true medium. We minimize it using the two-norm of the error signal.

'Acoustic probes have shown promise to be quite effective in determining the solid content in solid-liquid suspensions. However, the presence of small amounts of gas in the waste slurries stored in tanks across the DOE complex prevents straightforward application for characterization of these slurries. The proposed research will develop an acoustic probe for monitoring particle size and volume fraction in slurries in the absence and the presence of gas bubbles. Theoretical Analysis Accomplished: Attenuation of sound waves depends on the size distribution of the solids and the volume fraction of solids. These can in principle be calculated from attenuation measured over a range of frequencies. However, small amounts of bubbles distort the measured attenuation. A typical result from theoretical analysis for the attenuation of solid- gas-liquid systems is given in Figure 1. The total attenuation of a sound wave v(o) equals the sum of contributions by a large number of ''bins'' of particle sizes. This notion yields the following equation for the (hitherto) unknown number density of solid particles as a function of particle radius N(a): j k(o,a)N(a)da = v(o), where the kernel k(o,a) is obtained from analysis. If N(a) is given, the above equation is used to calculate the attenuation v(o). This is referred to as solving the ''forward problem''. Solving for N(a) with v(o) given is the ''inverse problem''. A complication that one faces when trying to solve the inverse problem is that the stated problem is mathematically ill-posed, i.e., small fluctuations in v(o) cause large fluctuations in the result for the number density. Therefore the problem needs to be ''regularized'', i.e., the stated problem needs to be changed slightly such as to make it well-posed. This has been done by others for gas-liquid systems in the past. This approach is currently being applied in the present project to solid-liquid systems. As is shown in Figure 2, it successfully recovers the number density that has been used in the forward problem to generate attenuation data. Having this solution technique giving reliable results for the inverse problems of both gas-liquid and solid-liquid systems, the authors shall apply this method in the near future to solid-gas-liquid systems.'

This paper is about Endo Vascular Elastography (EVE), a new technique for the diagnostic of arterial disease which gives information about plaque mechanical properties. Given this, EVE can provide useful information that could guide therapeutic decisions. Two important problems are considered: a) the Forward Problem (FP) of predicting the strain field of a tissue with a known elasticity and subjected to a known surface traction applied as a boundary condition, and b) the Inverse Problem (IP) of reconstructing the elasticity distribution from the measured displacement field, the boundary conditions and the governing equations. In the FP, arterial elastic properties are disclosed in the tissue displacement induced by a small intraluminal pressure change; the pre- and post- compression ultrasound signals are used to estimate radial (axial) tissue displacement and the corresponding strain. The approach we adopted to solve the IP consists in minimizing the least squares error between observed and predicted displacement fields. It uses an iterative procedure where at each iteration a linear inversion scheme based on a perturbation method is implemented. The paper illustrates the FP and IP implementation we have realized.

Recent work on agent-based models of wealth distribution has yielded nonlinear, non-local Fokker-Planck equations whose steady-state solutions describe empirical wealth distributions with remarkable accuracy using only a few free parameters. Because these equations are often used to solve the 'inverse problem' of determining the free parameters given empirical wealth data, there is much impetus to find fast and accurate methods of solving the 'forward problem' of finding the steady state corresponding to given parameters. In this work, we derive and calibrate a lattice Boltzmann equation for this purpose. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

In this work, physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse problem, where continuous velocity and pressure fields are inferred from scattered-time data on the interface position. We employ a volume of fluid approach, i.e. the auxiliary variable here is the volume fraction of the fluids within each phase. For the forward problem, we solve the two-phase Couette and Poiseuille flow. For the inverse problem, three classical test cases for two-phase modeling are investigated: (i) drop in a shear flow, (ii) oscillating drop and (iii) rising bubble. Data of the interface position over time is generated by numerical simulation. An effective way to distribute spatial training points to fit the interface, i.e. the volume fraction field, and the residual points is proposed. Furthermore, we show that appropriate weighting of losses associated with the residual of the partial differential equations is crucial for successful training. The benefit of using adaptive activation functions is evaluated for both the forward and inverse problem.

Inversion of geophysical data relies on knowledge about how to solve the forward problem, that is, computing data from a given set of model parameters. In many applications of inverse problems, the solution to the forward problem is assumed to be known perfectly, without any error. In reality, solving the forward model (forward-modeling process) will almost always be prone to errors, which we referred to as modeling errors. For a specific forward problem, computation of crosshole tomographic first-arrival traveltimes, we evaluated how the modeling error, given several different approximate forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model. We discovered some general tools by which the modeling error can be quantified and cast into a consistent formulation as an additive Gaussian observation error. We tested a method for generating a sample of the modeling error due to using a simple and approximate forward model, as opposed to a more complex and correct forward model. Then, a probabilistic model of the modeling error was inferred in the form of a correlated Gaussian probability distribution. The key to the method was the ability to generate many realizations from a statistical description of the source of the modeling error, which in this case is the a priori model. The methodology was tested for two synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent with the actual model. Further, using an approximate forward modeling may lead to a dramatic decrease in the computational demands for solving inverse problems.