Multi-Query Analysis

The majority of research at our institute in the area of multi-query analysis focuses on uncertainty quantification, global sensitivity analysis and (shape) optimization.

Contacts at IMCS

Uncertainty Quantification

Uncertainty quantification (UQ) assesses, manages, and reduces uncertainties in models and simulations. It involves determining the various sources of uncertainty affecting model outputs by propagating input uncertainties through the model. This is often achieved by sampling fromthe model output distribution by repeatedly evaluating the model with varying inputs. However, in computationally expensive models, such as those in biomechanics, this process can become costly due to the need for many simulations.

At IMCS, we develop advanced UQ algorithms and technologies, including surrogate modeling, implemented in our open-source code, QUEENS, to support large-scale computational models. Understanding the uncertainty in model outputs helps make more informed decisions, which is crucial in high-risk fields like surgical planning and critical infrastructure management.

Key Publications

Wirthl, B., Brandstaeter, S., Nitzler, J., Schrefler, B.A., Wall, W.A. (2023): Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems, International Journal for Numerical Methods in Biomedical Engineering, 39, e3675, DOI (Open Access)
Brandstaeter, S., Fuchs, S.L., Biehler, J., Aydin, R.C., Wall, W.A., Cyron, C.J. (2021): Global Sensitivity Analysis of a Homogenized Constrained Mixture Model of Arterial Growth and Remodeling, Journal of Elasticity, 145, 191–221, DOI (Open Access)

Current Project

Global Sensitivity Analysis

Parallel coordinates plot for global sensitvity analysis Global sensitivity analysis (GSA) assesses the contribution of input parameters to the uncertainty in a model’s output. Unlike local sensitivity analysis, GSA considers the entire range of input variations. It evaluates how changes in all input parameters affect the model’s output simultaneously and over their entire range. This approach provides a comprehensive understanding of the relative importance of each parameter and helps identify which inputs most significantly influence the model’s behavior.
We develop and enhance the latest GSA algorithms and apply them to fields where they are currently underused but would be highly beneficial, or even essential, including numerical methods development and biomedical engineering applications.

Key Publications

Wirthl, B., Brandstaeter, S., Nitzler, J., Schrefler, B.A., Wall, W.A. (2023): Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems, International Journal for Numerical Methods in Biomedical Engineering, 39, e3675, DOI (Open Access)
Brandstaeter, S., Fuchs, S.L., Biehler, J., Aydin, R.C., Wall, W.A., Cyron, C.J. (2021): Global Sensitivity Analysis of a Homogenized Constrained Mixture Model of Arterial Growth and Remodeling, Journal of Elasticity, 145, 191–221, DOI (Open Access)

Current Projects

Optimization

Optimization is a ubiquitous task in engineering sciences and a very illustrative example for a multi-query scenario: Many of the relevant optimization problems that engineers encounter in their everyday life are constrained by partial-differential equations (PDEs), e.g., the conservation laws of the real-world application to be optimized. In order to find a solution to these problems, the forward model that solves the governing PDE system needs to be evaluated multiple times to compute the objective function (and its gradients). To reduce the computational costs associated with the repeated evaluation of the costly forward model, optimization tasks are often solved by first constructing a surrogate model that can approximate the full-order model at much lower computational costs to reduce the overall computation time.

At IMCS, research in shape optimization is conducted to optimize the arrangement of fiber reinforcements in fiber-reinforced concrete parts and ultimately optimize resource usage (see also our page with application highlights).

Key Publications

Kremheller, J., Brandstaeter, S., Schrefler, B.A., Wall, W.A. (2021): Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model. International Journal for Numerical Methods in Biomedical Engineering, 37. DOI (Open Access)

Current Projects

Inverse Problems

An inverse problem involves determining unknown system states from observed data, typically using mathematical models that link the parameters to the observations. Inverse problems aim to infer the unknown model inputs that generate a known output, essentially reversing the process of a forward problem, where outputs are computed from known inputs. These problems are often ill-posed, meaning they may have multiple solutions, no solution, or solutions highly sensitive to small changes in the data. Therefore, they require advanced solution algorithms to ensure stable and meaningful results. Inverse problems appear in various fields, including civil engineering and biomechanics. One important class of inverse problems is parameter identification, which estimates parameter values based on indirect or noisy measurements. At IMCS, we explore numerical techniques for identifying patient-specific parameters in biomechanics models. We strongly emphasise developing algorithms for Bayesian inverse problems, a specialized type of inverse problem rooted in Bayesian probability. In this framework, Bayesian inference is used to update our belief about the parameters based on prior knowledge and the likelihood of observing the data given the model. This approach integrates both the uncertainty in the parameters (represented by a prior distribution) and the uncertainty in the data (described by a likelihood function) to compute a posterior distribution reflecting the updated belief after incorporating the data.

Key Publications

Sahin, T., von Danwitz, M., Popp, A. (2024): Solving forward and inverse problems of contact mechanics using physics-informed neural networks, Advanced Modeling and Simulation in Engineering Sciences, 11:11, DOI (Open Access) . arXiv
Willmann, H., Nitzler, J., Brandstäter, S., Wall, W.A. (2022): Bayesian calibration of coupled computational mechanics models under uncertainty based on interface deformation. Advanced Modeling and Simulation in Engineering Sciences 9, 24. DOI (Open Access)
Kremheller, J., Brandstaeter, S., Schrefler, B.A., Wall, W.A. (2021): Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model. International Journal for Numerical Methods in Biomedical Engineering, 37. DOI (Open Access)

Current Projects

Simulation Analytics

In applied computational engineering and developing novel numerical methods, analyzing the properties of simulation models—such as convergence behavior, scaling properties, and parameter influences—is a crucial task. At IMCS, we believe that developing and implementing advanced simulation analysis techniques and maintaining a common open-source software framework to facilitate these tasks can significantly enhance the sustainability, repeatability, and accessibility of research. To this end, we focus on addressing the challenges associated with simulation analytics, particularly the need for repeated simulation runs. These can be difficult to manage and quickly become computationally expensive, especially for complex, large-scale models.

Key Publications

Firmbach, M., Steinbrecher, I., Popp, A., Mayr, M. (2024): An approximate block factorization preconditioner for mixed-dimensional beam-solid interaction, Computer Methods in Applied Mechanics and Engineering, 431:117256, DOI (Open Access)
Wiesner, T.A., Mayr, M., Popp, A., Gee, M.W., Wall, W.A. (2021): Algebraic multigrid methods for saddle point systems arising from mortar contact formulations, International Journal for Numerical Methods in Engineering, 122:3749-3779, DOI (Open Access) , arXiv

Current Projects