Research

Research topics

Magnetic Partical Imaging [1] [2]

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]

Traction force microscopy [1]

Traction force microscopy is a method widely used in biophysics and cell biology to determine forces that biological cells apply to their environment. In the experiment, the cells adhere to a soft elastic substrate, which is then deformed in response to cellular traction forces (Fig 1). 

The inverse problem consists in reconstructi the traction stress applied by the cell from microscopy measurements of the substrate deformations. In this work, we consider a linear model, in which 3D forces are applied at a 2D interface, and a nonlinear pure 2D mode. Numerical experiments are carried out on real data (Fig 2).

Inverse acoustic source [1]

We illustrate the connection between adaptive mesh refinement for finite element discretized PDEs and the recently developed bi-level regularization algorithm. By adaptive mesh refinement according to data noise, regularization effect and convergence are immediate consequences. We moreover demonstrate its numerical advantages to the classical Landweber algorithm in terms of time and reconstruction quality for the aeroacoustic setting. This opens up exciting usage within the field of optimal experimental design (OED)

Learning hidden physics from empirical data [1]

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 [2]

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. 

poster

The extended adjoint state and nonlinearity in correlation-based passive imaging [1]

We investigate physics-based passive imaging problems, wherein one infers an unknown medium using ambient noise and correlation of the noise signal. Correlation-based passive imaging is vastly complex, in part due to exacerbated nonlinearity and higher dimensionality. We develop a general backpropagation framework via the so-called extended adjoint state, suitable for any linear PDE and reduces by half the number of required PDE solves. In addition, we analyze the nonlinearity of the correlated model, revealing a surprising tangential cone condition-like structure.

Figure. Left: Noise distributed in medium. Mid: Noise at sensors. Right: Correlation between sensors.

Figure. Extended adjoint state computation. Left: via J-parallel PDE solves, outer product and summation. Right: via N- parallel PDE solves, collect and arrange in column-wise.

Solar passive imaging with inertial waves [2]

We study the modeling, forward and inverse problem to infer viscosity and differential rotation on the Sun from newly discovered inertial waves. Since solar oscillations are excited by near-surface random turbulence, we investigate the inverse problem in the framework of passive-imaging using cross-covariance data.

Figure. Reconstruction of differential rotation.

Figure. Resulting covariance images.

Novel regularization strategies: 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).

Novel regularization strategies: SEQUENTIAL BI-LEVEL [c]

This study significantly improves upon the bi-level algorithm by sequential initialization (Seq-BiLev), yielding accelerated convergence and demonstrable multi-scale effect, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Interestingly, the lower-level trajectory of Seq-BiLev shows a connection to the Incremental Load Method (ILM). We illustrates its universality through several reaction-diffusion applications, in which the nonlinear reaction law needs to be determined.

Figure. Reconstruction of the reaction law in Lane-Emde equation 

Figure. Lower-level trajectory in Fisher equation

Data assimilation in dynamical systems [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 of 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.