Research
Research interests
Regularization theory
Nonlinear, dynamic inverse problems
Parameter identification and model discovery for partial differential equations (PDEs)
Data assimilation: online parameter identification
Machine learning for PDEs and inverse problems
Applications: magnetic particle imaging, elastography, helioseismology
Passive imaging with correlation data
Research topics
MPI is a novel tracer based imaging technique invented in 2005, employing a temporally-spatially varying magnetic field with field-free point to excite the nanoparticals inside a body and detect their concentration. MPI explores potential clinical applications like locate cloggings in blood vessels and tumors. In the work, we use the Landau-Lifshitz-Gilbert (LLG) equation to model for the evolution of the magnetization of a magnetic material, and study forward problem and inverse problem of identifying relaxation parameters.
Figure. Magnetization (right) of a particle in response to a dynamic external field (left); middle: initial magnetization vector [GIP]
Learning hidden physics from empirical data [3]
We investigate the problem discover nonlinearity in parameter-dependent PDEs, caused by e.g. inexact or simplified modeling, or by the effect of undiscovered physical laws. The unknown nonlinearity is represented via a neural network of a state, which is also not directly accessible. The learning-informed PDE model has three unknown quantities: physical parameter, state and model nonlinearity. We propose an all-at-once approach to simultaneously recover these unknowns.
From neural-networks to discretization of inverse problems [4]
In inverse problems, the representation of an unknown quantity via neural networks can be realized as a discretization/approximation scheme. Optimization with respect to the
NN approximation error (universal approximation theorem) reveals a need to choose the network size in dependence on the data error to combat overfitting. We study the interplay between regularization (Tikhonov and Landweber) and NNs discretization to show reconstruction convergence when the networks size increas to infinity.
Solar passive imaging with inertial waves
Passive imaging is the field of imaging a medium from observation of waves generated
by ambient noise source, e.g. in helioseismology, where solar oscillations are excited by
near-surface random turbulence. We study the forward and inverse problem of reconstructing viscosity and differential rotation on the Sun from cross-covariance observations of these inertial waves. Correlation-based passive imaging is vastly complex, in part due to exacerbated nonlinearity, higher dimensionality, for which special strategies are required.
Figure. Simultaneous reconstruction of viscosity and rotation.
Figure. Resulting covariance images.
New approach to nonlinear inverse problems: ALL-AT-ONCE [a]
For parameter identification problems, the common approach is to construct the reduced forward operator (RED) via the parameter-to-state map. We propose the new all-at-once approach (AAO) which overcomes many analytical challenges and is more computationally efficient than RED since it bypasses solving exactly PDEs. AAO usually takes more iteration to converge, however, for each iteration the cost is much less than RED (see Table).
Figure. RED (left) vs. AAO (right) reconstruction for source inversion in the semi-linear Allen-Cahn equation. Top: source, middle: state, bottom: state error.
New approach to nonlinear inverse problems: BI-LEVEL [b]
Recently, we develop another new approach called bi-level (BiLev) utilizing the best features of both AAO and RED, allowing us to, respectively, utilize well-posedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs exactly. BiLev hierarchy consists the upper-level for parameter reconstruction with embedding a lower-level state approximation. AAO and BiLev are well-suited for inverse problems with nonlinear PDEs.
Figure. BiLev lower-level state approximation for the quasi-linear LLG describing how random magnetization evolves in an external field pointing upwards.
Figure. Number of iterations for parameter identification using Bi-Lev:
upper iterations (x-axis), corresponding number of lower-iterations (y-axis).
Data assimilation [da]
A challenging aspect of time-dependent inverse problems in dynamical systems and model predictive control, is online/on-the-fly parameter identification. This occurs when data acquisition and recovery of unknown parameters must be done simultaneously during systems operation, as new data continuously becomes available. The required techniques differ significantly from conventional parameter identification methods. Our contribution [da] is one of the first extending the scope of problems to nonlinear parameter laws, employing model reference adaptive systems.
Figure. Illustration data assimilation process.
The tangential cone condition [c]
Convergence proofs of iterative regularization methods for solving nonlinear ill-posed inverse problems such as the Landweber iteration or the iteratively regularized Gauss-Newton method require structural assumptions on the nonlinear forward operator 𝐹 such as the tangential cone condition introduced in 1995 by Scherzer. Although powerful, the verification of this condition can be extremely challenging. In this work, we present a series of time-dependent benchmark inverse problems for which we can establish this important condition.
Figure. Illustration of the tangential cone condition describing nonlinear degree of the forward map 𝐹.
