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. MPI explores potential clinical applications like locate cloggings in blood vessels and tumors. We use the Landau-Lifshitz-Gilbert (LLG) equation to model for the evolution of the magnetization of a magnetic material, and study 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]

Active forces in actomyosin systems [1]

Knowledge on forces is key to understanding dynamic processes in active biological systems that are able to self-organize. Fig: Fluorescence image of actin-myosin networks encapsulated in water-in-oil droplets. Actin is fluorescently labelled.

Using Stokes equation as model as optical microscopy image as data, we identify forces generated by a filamentmotor network of F-actin and myosin – actomyosin – and exerted on the surrounding fluid, therefore causing a fluid flow

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 reconstruction the traction stress applied by the cell from microscopy measurements of the substrate deformations. We construct a linear 3D traction force and a 2D nonlinear 2D body with real data (Fig 2).

Optimal sensor placement for heat source inversion [1]

Within the field of optimal experimental design, sensor placement refers to the act of finding the optimal locations of data collecting sensors, with the aim to minimize uncertainty in reconstruction of an unknown parameter from finite data. We investigate sensor placement for reconstructing a heat source given final time measurement via the redundant-dominant 𝑝-continuation algorithm to minimizing the trace of posterior covariance, obtaining binary A-optimal sensor design.

Figure. Left: Source domain and grid of candidate sensor locations. Right: Spatial-dependent diffusion parameter 

Figure. Left: Optimal experimental for 36 sensor, plotted together with its pointwise variance field. Right: Theoretical lower bound for A-optimality, proposed A-optimal designs vs random designs.

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]

In passive imaging, one infers an unknown medium using ambient noise source and correlation of the noise signal. Correlation-based 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 elliptic 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]

We develop, for the first time, a hierarchical optimization-regularization called bi-level (BiLev) allowing, to respectively, utilize well-posedness of the solution  map in the classical reduced framework, and to bypass solving any nonlinear PDE exactly. BiLev hierarchy consists the upper-level for parameter reconstruction embedding a multi-precision lower-level for PDE state approximation. Our bi-level algorithms are well-suited to solve large scale inverse problems in nonlinear evolution PDEs.

Figure. BiLev lower-level state approximation for the Landau–Lifshitz–Gilbert equation 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 lower-level warm-starting (Seq-BiLev) for acceleration effect. We moreover demonstrable multi-scale effect in the lower-level, while retaining regularizing effect and allows for the usage of inexact PDE solvers. Interestingly, the lower-level trajectory of Seq-BiLev exposes a connection to the Incremental Load Method used in nonlinear elasticity. We illustrates the universality through several reaction-diffusion applications with hidden reaction discovery.

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

Figure. Lower-level trajectory in Fisher equation

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)

The tangential cone condition [c]

Convergence proofs of iterative regularization methods for solving nonlinear ill-posed inverse problems require the tangential cone condition (TCC) to converges. Although powerful, the verification of this condition can be extremely challenging. In this work, we present, for the first time, a general framework for establishing TCC for nonlinear evolution problems, and demonstrate this for  a series of  benchmark problems.

Figure. Illustration of the tangential cone condition describing nonlinear degree of the forward map 𝐹.