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Register a new userAlternatives for Generating a Reduced Basis to Solve the Hyperspectral Diffuse Optical Tomography Model
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The Reduced Basis Method (RBM) is a model reduction technique used to solve parametric PDEs that relies upon a basis set of solutions to the PDE at specific parameter values. To generate this reduced basis, the set of a small number of parameter values must be strategically chosen. We apply a Metropolis algorithm and a gradient algorithm to find the set of parameters and compare them to the standard greedy algorithm most commonly used in the RBM. We test our methods by using the RBM to solve a simplified version of the governing partial differential equation for hyperspectral diffuse optical tomography (hyDOT). The governing equation for hyDOT is an elliptic PDE parameterized by the wavelength of the laser source. For this one-dimensional problem, we find that both the Metropolis and gradient algorithms are potentially superior alternatives to the greedy algorithm in that they generate a reduced basis which produces solutions with a smaller relative error with respect to solutions found using the finite element method and in less time.
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Optical coherence tomography (OCT) is a widely used imaging technique in the micrometer regime, which gained accelerating interest in medical imaging %and material testing in the last twenty years. In up-to-date OCT literature [5,6] certain simplifying assumptions are made for the reconstructions, but for many applications a more realistic description of the OCT imaging process is of interest. In mathematical models, for example, the incident angle of light onto the sample is usually neglected or %although having a huge impact on the laser power inside the sample is usually neglected or a plane wave description for the light-sample interaction in OCT is used, which ignores almost completely the occurring effects within an OCT measurement process. In this article, we make a first step to a quantitative model by considering the measured intensity as a combination of back-scattered Gaussian beams affected by the system. In contrast to the standard plane wave simplification, the presented model includes system relevant parameters such as the position of the focus and the spot size of the incident laser beam, which allow a precise prediction of the OCT data and therefore ultimately serves as a forward model. The accuracy of the proposed model - after calibration of all necessary system parameters - is illustrated by simulations and validated by a comparison with experimental data obtained from a 1300nm swept-source OCT system.
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the physical and velocity/angular variables and the fact that the problem is multiscale in nature. Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test the angular-space reduced order model for the linear radiative transfer equation, the first such effort based on the celebrated reduced basis method (RBM). Our method is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an asymptotic preserving upwind discontinuous Galerkin method for the physical space, and an efficient synthetic accelerated source iteration for the resulting linear system. Addressing the challenge of the parameter values (or angular directions) being coupled through an integration operator, the first novel ingredient of our method is an iterative procedure where the macroscopic density is constructed from the RBM snapshots, treated explicitly and allowing a transport sweep, and then updated afterwards. A greedy algorithm can then proceed to adaptively select the representative samples in the angular space and form a surrogate solution space. The second novelty is a least-squares density reconstruction strategy, at each of the relevant physical locations, enabling the robust and accurate integration over an arbitrarily unstructured set of angular samples toward the macroscopic density. Numerical experiments indicate that our method is highly effective for computational cost reduction in a variety of regimes.
In numerical simulations of many charged systems at the micro/nano scale, a common theme is the repeated solution of the Poisson-Boltzmann equation. This task proves challenging, if not entirely infeasible, largely due to the nonlinearity of the equation and the high dimensionality of the physical and parametric domains with the latter emulating the system configuration. In this paper, we for the first time adapt a mathematically rigorous and computationally efficient model order reduction paradigm, the so-called reduced basis method (RBM), to mitigate this challenge. We adopt a finite difference method as the mandatory underlying scheme to produce the {em truth approximations} of the RBM upon which the fast algorithm is built and its performance is measured against. Numerical tests presented in this paper demonstrate the high efficiency and accuracy of the fast algorithm, the reliability of its error estimation, as well as its capability in effectively capturing the boundary layer.
We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. Second, we propose an emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the emph{a posteriori} error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.
Based on the ACV approach to transplanckian energies, the reduced-action model for the gravitational S-matrix predicts a critical impact parameter b_c ~ R = 2 G sqrt{s} such that S-matrix unitarity is satisfied in the perturbative region b > b_c, while it is exponentially suppressed with respect to s in the region b < b_c that we think corresponds to gravitational collapse. Here we definitely confirm this statement by a detailed analysis of both the critical region b ~ b_c and of further possible contributions due to quantum transitions for b < b_c. We point out, however, that the subcritical unitarity suppression is basically due to the boundary condition which insures that the solutions of the model be ultraviolet-safe. As an alternative, relaxing such condition leads to solutions which carry short-distance singularities presumably regularized by the string. We suggest that through such solutions - depending on the detailed dynamics at the string scale - the lost probability may be recovered.
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