Computational Applied Mathematics and Operations Research

Permanent URI for this community

https://hdl.handle.net/1911/8311

Browse

Browsing Computational Applied Mathematics and Operations Research by Author "Acosta, Sebastian"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Inverse Source Problems for Time-Dependent Radiative Transport
    (2014-05) Acosta, Sebastian
    In the first part of this thesis, I develop a time reversal method for the radiative transport equation to solve two problems: an inverse problem for the recovery of an initial condition from boundary measurements, and the exact boundary controllability of the transport field with finite steering time. Absorbing and scattering effects, modeled by coefficients with low regularity, are incorporated in the formulation of these problems. This time reversal approach leads to a convergent iterative procedure to reconstruct the initial condition provided that the scattering coefficient is sufficiently small. Then, using duality arguments, I show that the solvability of the inverse problem leads to exact controllability of the transport field. The solution approach to both of these problems may have applications in areas such as optical imaging and optimization of radiation delivery. The second portion of the work is dedicated to the simultaneous recovery of a source term f(x) and an isotropic initial condition u0(x), using the single measurement induced by these data. This result is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. More precisely, based on exact boundary controllability, I derive a system of equations for the unknown terms f and u0. The system is shown to be of Fredholm type if s satisfies a certain positivity condition. This condition requires that the radiation visits the region over which f is to be recovered. I show that for generic terms and weakly absorbing media, the inverse problem is well-posed.
  • Thumbnail Image
    Item
    Numerical method of characteristics for one-dimensional blood flow
    (Elsevier, 2015) Acosta, Sebastian; Puelz, Charles; Riviére, Béatrice; Penny, Daniel J.; Rusin, Craig G.
    Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant.