Browsing by Subject "Inverse problem"
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Item A Systems Biology Approach to Develop Models of Signal Transduction Pathways(2011-10-21) Huang, ZuyiMathematical models of signal transduction pathways are characterized by a large number of proteins and uncertain parameters, yet only a limited amount of quantitative data is available. The dissertation addresses this problem using two different approaches: the first approach deals with a model simplification procedure for signaling pathways that reduces the model size but retains the physical interpretation of the remaining states, while the second approach deals with creating rich data sets by computing transcription factor profiles from fluorescent images of green-fluorescent-protein (GFP) reporter cells. For the first approach a model simplification procedure for signaling pathway models is presented. The technique makes use of sensitivity and observability analysis to select the retained proteins for the simplified model. The presented technique is applied to an IL-6 signaling pathway model. It is found that the model size can be significantly reduced and the simplified model is able to adequately predict the dynamics of key proteins of the signaling pathway. An approach for quantitatively determining transcription factor profiles from GFP reporter data is developed as the second major contribution of this work. The procedure analyzes fluorescent images to determine fluorescence intensity profiles using principal component analysis and K-means clustering, and then computes the transcription factor concentration from the fluorescence intensity profiles by solving an inverse problem involving a model describing transcription, translation, and activation of green fluorescent proteins. Activation profiles of the transcription factors NF-?B, nuclear STAT3, and C/EBP? are obtained using the presented approach. The data for NF-?B is used to develop a model for TNF-? signal transduction while the data for nuclear STAT3 and C/EBP? is used to verify the simplified IL-6 model. Finally, an approach is developed to compute the distribution of transcription factor profiles among a population of cells. This approach consists of an algorithm for identifying individual fluorescent cells from fluorescent images, and an algorithm to compute the distribution of transcription factor profiles from the fluorescence intensity distribution by solving an inverse problem. The technique is applied to experimental data to derive the distribution of NF-?B concentrations from fluorescent images of a NF-?B GFP reporter system.Item The inverse medium problem for Timoshenko beams and frames : damage detection and profile reconstruction in the time-domain(2009-12) Karve, Pranav Madhav; Kallivokas, Loukas F.; Manuel, LanceWe discuss a systematic methodology that leads to the reconstruction of the material profile of either single, or assemblies of one-dimensional flexural components endowed with Timoshenko-theory assumptions. The probed structures are subjected to user-specified transient excitations: we use the complete waveforms, recorded directly in the time-domain at only a few measurement stations, to drive the profile reconstruction using a partial-differential-equation-constrained optimization approach. We discuss the solution of the ensuing state, adjoint, and control problems, and the alleviation of profile multiplicity by means of either Tikhonov or Total Variation regularization. We report on numerical experiments using synthetic data that show satisfactory reconstruction of a variety of profiles, including smoothly and sharply varying profiles, as well as profiles exhibiting localized discontinuities. The method is well suited for imaging structures for condition assessment purposes, and can handle either diffusive or localized damage without need for a reference undamaged state.Item A mixed unsplit-field PML-based scheme for full waveform inversion in the time-domain using scalar waves(2010-05) Kang, Jun Won, 1975-; Kallivokas, Loukas F.; Stokoe, Kenneth H.; Tonon, Fulvio; Ghattas, Omar; Gonzalez, OscarWe discuss a full-waveform based material profile reconstruction in two-dimensional heterogeneous semi-infinite domains. In particular, we try to image the spatial variation of shear moduli/wave velocities, directly in the time-domain, from scant surficial measurements of the domain's response to prescribed dynamic excitation. In addition, in one-dimensional media, we try to image the spatial variability of elastic and attenuation properties simultaneously. To deal with the semi-infinite extent of the physical domains, we introduce truncation boundaries, and adopt perfectly-matched-layers (PMLs) as the boundary wave absorbers. Within this framework we develop a new mixed displacement-stress (or stress memory) finite element formulation based on unsplit-field PMLs for transient scalar wave simulations in heterogeneous semi-infinite domains. We use, as is typically done, complex-coordinate stretching transformations in the frequency-domain, and recover the governing PDEs in the time-domain through the inverse Fourier transform. Upon spatial discretization, the resulting equations lead to a mixed semi-discrete form, where both displacements and stresses (or stress histories/memories) are treated as independent unknowns. We propose approximant pairs, which numerically, are shown to be stable. The resulting mixed finite element scheme is relatively simple and straightforward to implement, when compared against split-field PML techniques. It also bypasses the need for complicated time integration schemes that arise when recent displacement-based formulations are used. We report numerical results for 1D and 2D scalar wave propagation in semi-infinite domains truncated by PMLs. We also conduct parametric studies and report on the effect the various PML parameter choices have on the simulation error. To tackle the inversion, we adopt a PDE-constrained optimization approach, that formally leads to a classic KKT (Karush-Kuhn-Tucker) system comprising an initial-value state, a final-value adjoint, and a time-invariant control problem. We iteratively update the velocity profile by solving the KKT system via a reduced space approach. To narrow the feasibility space and alleviate the inherent solution multiplicity of the inverse problem, Tikhonov and Total Variation (TV) regularization schemes are used, endowed with a regularization factor continuation algorithm. We use a source frequency continuation scheme to make successive iterates remain within the basin of attraction of the global minimum. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on both one- and two-dimensional examples, including the Marmousi benchmark problem, that lead efficiently to the reconstruction of heterogeneous profiles involving both horizontal and inclined layers, as well as of inclusions within layered systems.Item Recovery of the logical gravity field by spherical regularization wavelets approximation and its numerical implementation(2009-05) Shuler, Harrey Jeong; Tapley, Byron D.As an alternative to spherical harmonics in modeling the gravity field of the Earth, we built a multiresolution gravity model by employing spherical regularization wavelets in solving the inverse problem, i.e. downward propagation of the gravity signal to the Earth.s surface. Scale discrete Tikhonov spherical regularization scaling function and wavelet packets were used to decompose and reconstruct the signal. We recovered the local gravity anomaly using only localized gravity measurements at the observing satellite.s altitude of 300 km. When the upward continued gravity anomaly to the satellite altitude with a resolution 0.5° was used as simulated measurement inputs, our model could recover the local surface gravity anomaly at a spatial resolution of 1° with an RMS error between 1 and 10 mGal, depending on the topography of the gravity field. Our study of the effect of varying the data volume and altering the maximum degree of Legendre polynomials on the accuracy of the recovered gravity solution suggests that the short wavelength signals and the regions with high magnitude gravity gradients respond more strongly to such changes. When tested with simulated SGG measurements, i.e. the second order radial derivative of the gravity anomaly, at an altitude of 300 km with a 0.7° spatial resolution as input data, our model could obtain the gravity anomaly with an RMS error of 1 ~ 7 mGal at a surface resolution of 0.7° (< 80 km). The study of the impact of measurement noise on the recovered gravity anomaly implies that the solutions from SGG measurements are less susceptible to measurement errors than those recovered from the upward continued gravity anomaly, indicating that the SGG type mission such as GOCE would be an ideal choice for implementing our model. Our simulation results demonstrate the model.s potential in determining the local gravity field at a finer scale than could be achieved through spherical harmonics, i.e. less than 100 km, with excellent performance in edge detection.Item Regularizing Inverse Problems(2014-06-26) Wang, FangAn inverse problem reconstructs the unknown internal parameters of a subject based on collected data derived synthetically or from real measurements. Inverse problems often lack the well-posedness defined by J. Hadamard; in other words, solutions of inverse problems, namely the reconstructions of the parameters, may not exist, may not be unique or may be unstable. Regularization is a technique that deals with such situations. The well-known Tikhonov regularization method translates the original inverse problem to optimization problems of minimizing the norm of the data misfit plus a weighted regularization functional that incorporates the a priori information we may have about the original problem. The choices of the regularization functional r(q) include ?q??(2@L^(2 ) )??q??(2@H^(1) ), |q|BV and |q|TV. However, each of these has its limitations. In this work, we develop a novel H^(s) seminorm regularization method and present numerical results for model problems. This method relies on the evaluation of the seminorms of an intermediary Hilbert space, namely H^(s) space, that stays between L^(2) and H^(1). The H^(s) seminorm regularization is designed to minimize the undesirable aspects of the existing L^(2) and H^(1) regularization functionals. The H^(s) seminorm regularization also allows discontinuities and stabilizes the perturbations. We study the H^(s) seminorm regularization method both theoretically and numerically. We consider the theoretical analysis of this new regularization method based on a model problem. We show that a stable solution can be achieved with some conditions. In addition, we prove the convergence and guarantee a convergence rate provided additional conditions for the model problem when the considered domain is 1D. Numerically, we produce an approximated discretization of the H^(s) seminorm regularization that can be applied to 1D, 2D or 3D examples. We also provide reconstructions of both continuous and discontinuous parameters from synthetic data and a comparison of these solutions to the ones based on existing L^(2) and H^(1) regularization methods. Furthermore, we also apply the H^(s) seminorm regularization method to a fluorescence optical tomography problem. In summary, we study and implement the H^(s) seminorm regularization method for inverse problems, which can provide a stable solution to the model problem. The numerical results indicate the robustness of the new method and suggests that the H^(s) seminorm regularization method produces the closest approximation of the exact solution than the L^(2) norm and H^(1) seminorm regularization methods for the model problem.Item Some statistical methods for directly and indirecly observed functional data(Texas Tech University, 2008-08) Pang, Johnny; Ruymgaart, Frits; Wang, Alex; Paige, RobertIn this dissertation, we will be concerned with the statistical inference regarding linear models with functional data. For the sake of generality these functional data will be considered as sample elements in an abstract infinite dimensional Hilbert space. In the special instance of the one-sample problem, both directly and indirectly observed functions will be included. It should be stressed that the linear model mentioned above each sample elements itself is a function, so that we have more information than in cases where the data consist of a number of sampled function values. In Chapter 1, we will review some useful properties and formulas of arbitrary random variables and Gaussian random variables in Hilbert spaces. It should be noted that a Gaussian measure will be employed as a dominating measure because there doesn't exist a shift invariant (i.e. Lebesgue) measure on an infinite dimensional Hilbert space. In Chapter 2, linear model in Hilbert space will be considered. We will borrow the notation from the univariate linear model and use matrices to arrive at a convenient notation for linear models in Hilbert spaces. We will show that our estimator of the function parameter has approximately a Gaussian distribution for large sample size. In Chapter 3, three special cases of the main model introduced in Chapter $2$ will be considered. First, the simplest version of the one-sample problem in Hilbert spaces will be introduced together with an application of neighborhood hypotheses. Second, the indirect-one-sample problem in Hilbert spaces will be considered. We will exploit the spectral-cut-off type regularized inverse and consider the MISE of the estimator as a means to investigate its quality. In fact, we will prove that the estimator is rate-optimal. Finally, multi-sample problem will be briefly considered along the same lines as the direct one-sample problem.