Third-Party Funds
Locally adaptive methods for free discontinuity problems
Funder: The Crafoord Foundation
Periode: Oct. 2022 - Sept. 2024
PI: Andreas Langer In many modern applications, like medical image reconstruction, the solution of interest is described by a discontinuous function. Unfortunately, it is a priori not known where the discontinuities are located and their location has to be determined numerically. In fact, computing discontinuities wrongly or inaccurately could be either fatal (as in obstacle and tumor detection) or scientifically costly. In order to preserve edges and discontinuities in image processing, the minimization of a functional consisting of a the total variation term is common and widely used. Typical applications, where such an optimization problem is solved are for example: image denoising, image deblurring, and reconstruction of partial Fourier-data (magnetic resonance imaging).
In this project we solve the constituted problems by constructing methods which automatically adjust to the discontinuities of the underlying (observed) data. The adaptive adjustment of the derived methods occur on two sides, on the regularization side, where the regularization is locally adjusted, and on the discretization side, where an optimal discretization is learned. This leads to methods, which adaptively (in an iterative manner) reduce the reconstruction error, allowing to obtain the discontinuities more accurately and hence yield qualitatively better reconstructions than state-of-the-art methods, while keeping the computational complexity of the underlying problem on a manageable size. Additionally to these attributes the derived methods will possess further desirable properties, as robustness with respect to the given data and they allow for different discretization techniques.
Periode: Oct. 2022 - Sept. 2024
PI: Andreas Langer In many modern applications, like medical image reconstruction, the solution of interest is described by a discontinuous function. Unfortunately, it is a priori not known where the discontinuities are located and their location has to be determined numerically. In fact, computing discontinuities wrongly or inaccurately could be either fatal (as in obstacle and tumor detection) or scientifically costly. In order to preserve edges and discontinuities in image processing, the minimization of a functional consisting of a the total variation term is common and widely used. Typical applications, where such an optimization problem is solved are for example: image denoising, image deblurring, and reconstruction of partial Fourier-data (magnetic resonance imaging).
In this project we solve the constituted problems by constructing methods which automatically adjust to the discontinuities of the underlying (observed) data. The adaptive adjustment of the derived methods occur on two sides, on the regularization side, where the regularization is locally adjusted, and on the discretization side, where an optimal discretization is learned. This leads to methods, which adaptively (in an iterative manner) reduce the reconstruction error, allowing to obtain the discontinuities more accurately and hence yield qualitatively better reconstructions than state-of-the-art methods, while keeping the computational complexity of the underlying problem on a manageable size. Additionally to these attributes the derived methods will possess further desirable properties, as robustness with respect to the given data and they allow for different discretization techniques.
EXC 2075, PN5-2A: Data-driven optimisation algorithms for local dynamic model adaptivity
Funder: German Research Foundation (Deutsche Forschungsgemeinschaft, DFG)
Periode: Dec. 2019 – May 2023
PI: Andreas Langer We develop an optimization framework that will on-the-fly identify parameters in an adaptive computational model for gas storage. These parameters might be physical unknown values of interest such as the permeability, or indicate which type of physical model is locally used (multi-modelling). Thereby the choice of the local model is based on a model hierarchy. Our optimization framework formulates a problem, where the objective function contains local criteria and the fulfillment of the model equation may appear as constraint.
Periode: Dec. 2019 – May 2023
PI: Andreas Langer We develop an optimization framework that will on-the-fly identify parameters in an adaptive computational model for gas storage. These parameters might be physical unknown values of interest such as the permeability, or indicate which type of physical model is locally used (multi-modelling). Thereby the choice of the local model is based on a model hierarchy. Our optimization framework formulates a problem, where the objective function contains local criteria and the fulfillment of the model equation may appear as constraint.
Automatic detection of moving objects in high-resolution image sequences using novel decomposition methods
Funder: Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden-Württemberg and the University of Stuttgart
Periode: April 2018 – March 2021
PI: Andreas Langer The determination of motion fields plays a fundamental role in the field of self-driving cars. There it is of highest interest to detect automatically threats by moving objects so that a vehicle can react to them independently (artificial intelligence). Thereby the environment is recorded in detail as a sequence of images. The apparent movement (called optical flow), i.e. the change in color intensities in the pixels, can then be determined from this sequence of images. Since high-resolution image sequences lead to large amounts of data, known methods can only calculate the optical flow from such image sequences very slowly. In order to be able to carry out these calculations in real time, as is necessary for autonomous driving, new efficient solvers are required. In this project, we will construct such efficient methods for calculating the optical flow using domain decomposition methods (DDMs). It has been shown several times that DDMs are one of the most successful methods to design efficient numerical solvers for large problems. The reason for this is that they allow the computational effort to be decomposed and a sequence of smaller problems to be solved.
To keep the computational effort as low as possible, we also propose a finite element discretisation of the problem. This allows us to perform an adaptive discretization of the problem, which further reduces the computational effort. In order to perform an adaptive refinement, we will derive a posteriori error estimates.
Periode: April 2018 – March 2021
PI: Andreas Langer The determination of motion fields plays a fundamental role in the field of self-driving cars. There it is of highest interest to detect automatically threats by moving objects so that a vehicle can react to them independently (artificial intelligence). Thereby the environment is recorded in detail as a sequence of images. The apparent movement (called optical flow), i.e. the change in color intensities in the pixels, can then be determined from this sequence of images. Since high-resolution image sequences lead to large amounts of data, known methods can only calculate the optical flow from such image sequences very slowly. In order to be able to carry out these calculations in real time, as is necessary for autonomous driving, new efficient solvers are required. In this project, we will construct such efficient methods for calculating the optical flow using domain decomposition methods (DDMs). It has been shown several times that DDMs are one of the most successful methods to design efficient numerical solvers for large problems. The reason for this is that they allow the computational effort to be decomposed and a sequence of smaller problems to be solved.
To keep the computational effort as low as possible, we also propose a finite element discretisation of the problem. This allows us to perform an adaptive discretization of the problem, which further reduces the computational effort. In order to perform an adaptive refinement, we will derive a posteriori error estimates.