3 research outputs found
Computing Reduced Order Models via Inner-Outer Krylov Recycling in Diffuse Optical Tomography
In nonlinear imaging problems whose forward model is described by a partial
differential equation (PDE), the main computational bottleneck in solving the
inverse problem is the need to solve many large-scale discretized PDEs at each
step of the optimization process. In the context of absorption imaging in
diffuse optical tomography, one approach to addressing this bottleneck proposed
recently (de Sturler, et al, 2015) reformulates the viewing of the forward
problem as a differential algebraic system, and then employs model order
reduction (MOR). However, the construction of the reduced model requires the
solution of several full order problems (i.e. the full discretized PDE for
multiple right-hand sides) to generate a candidate global basis. This step is
then followed by a rank-revealing factorization of the matrix containing the
candidate basis in order to compress the basis to a size suitable for
constructing the reduced transfer function. The present paper addresses the
costs associated with the global basis approximation in two ways. First, we use
the structure of the matrix to rewrite the full order transfer function, and
corresponding derivatives, such that the full order systems to be solved are
symmetric (positive definite in the zero frequency case). Then we apply MOR to
the new formulation of the problem. Second, we give an approach to computing
the global basis approximation dynamically as the full order systems are
solved. In this phase, only the incrementally new, relevant information is
added to the existing global basis, and redundant information is not computed.
This new approach is achieved by an inner-outer Krylov recycling approach which
has potential use in other applications as well. We show the value of the new
approach to approximate global basis computation on two DOT absorption image
reconstruction problems
A Multiscale Method for Model Order Reduction in PDE Parameter Estimation
Estimating parameters of Partial Differential Equations (PDEs) is of interest
in a number of applications such as geophysical and medical imaging. Parameter
estimation is commonly phrased as a PDE-constrained optimization problem that
can be solved iteratively using gradient-based optimization. A computational
bottleneck in such approaches is that the underlying PDEs needs to be solved
numerous times before the model is reconstructed with sufficient accuracy. One
way to reduce this computational burden is by using Model Order Reduction (MOR)
techniques such as the Multiscale Finite Volume Method (MSFV).
In this paper, we apply MSFV for solving high-dimensional parameter
estimation problems. Given a finite volume discretization of the PDE on a fine
mesh, the MSFV method reduces the problem size by computing a
parameter-dependent projection onto a nested coarse mesh. A novelty in our work
is the integration of MSFV into a PDE-constrained optimization framework, which
updates the reduced space in each iteration. We also present a computationally
tractable way of differentiating the MOR solution that acknowledges the change
of basis. As we demonstrate in our numerical experiments, our method leads to
computational savings particularly for large-scale parameter estimation
problems and can benefit from parallelization.Comment: 22 pages, 4 figures, 3 table
Randomization for the Efficient Computation of Parametric Reduced Order Models for Inversion
Nonlinear parametric inverse problems appear in many applications. Here, we
focus on diffuse optical tomography (DOT) in medical imaging to recover unknown
images of interest, such as cancerous tissue in a given medium, using a
mathematical (forward) model. The forward model in DOT is a
diffusion-absorption model for the photon flux. The main bottleneck in these
problems is the repeated evaluation of the large-scale forward model. For DOT,
this corresponds to solving large linear systems for each source and frequency
at each optimization step. Moreover, Newton-type methods, often the method of
choice, require additional linear solves with the adjoint to compute derivative
information. Emerging technology allows for large numbers of sources and
detectors, making these problems prohibitively expensive. Reduced order models
(ROM) have been used to drastically reduce the system size in each optimization
step, while solving the inverse problem accurately. However, for large numbers
of sources and detectors, just the construction of the candidate basis for the
ROM projection space incurs a substantial cost, as matching the full parameter
gradient matrix in interpolatory model reduction requires large linear solves
for all sources and frequencies and all detectors and frequencies for each
parameter interpolation point. As this candidate basis numerically has low
rank, this construction is followed by a rank-revealing factorization that
typically reduces the number of vectors in the candidate basis substantially.
We propose to use randomization to approximate this basis with a drastically
reduced number of large linear solves. We also provide a detailed analysis for
the low-rank structure of the candidate basis for our problem of interest. Even
though we focus on the DOT problem, the ideas presented are relevant to many
other large scale inverse problems and optimization problems.Comment: 18 pages, 5 figure