arXiv subjects

Simon W. Funke

Publications and source records attributed to Simon W. Funke.

14 recordsLinked to original sources

arXiv2021

Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks.

math.NA↗

arXiv2021

A scalable solver for a stochastic, hybrid cellular automaton model of personalized breast cancer therapy

Mathematical modeling and simulation is a promising approach to personalized cancer medicine. Yet, the complexity, heterogeneity and multi-scale nature of cancer pose significant computational challenges. Coupling discrete cell-based models with continuous models using hybrid cellular automata is a powerful approach for mimicking biological complexity and describing the dynamical exchange of information across different scales. However, when clinically relevant cancer portions are taken into account, such models become computationally very expensive. While efficient parallelization techniques for continuous models exist, their coupling with discrete models, particularly cellular automata, necessitates more elaborate solutions. Building upon FEniCS, a popular and powerful scientific computing platform for solving partial differential equations, we developed parallel algorithms to link stochastic cellular automata with differential equations ( https://bitbucket.org/HTasken/cansim ). The algorithms minimize the communication between processes that share cellular automata neighborhood values while also allowing for reproducibility during stochastic updates. We demonstrated the potential of our solution on a complex hybrid cellular automaton model of breast cancer treated with combination chemotherapy. On a single-core processor, we obtained nearly linear scaling with an increasing problem size, whereas weak parallel scaling showed moderate growth in solving time relative to increase in problem size. Finally we applied the algorithm to a problem that is 500 times larger than previous work, allowing us to run personalized therapy simulations based on heterogeneous cell density and tumor perfusion conditions estimated from magnetic resonance imaging data on an unprecedented scale.

q-bio.QM↗

arXiv2020

Automatic shape derivatives for transient PDEs in FEniCS and Firedrake

In industry, shape optimization problems are of utter importance when designing structures such as aircraft, automobiles and turbines. For many of these applications, the structure changes over time, with a prescribed or non-prescribed movement. Therefore, it is important to capture these features in simulations when optimizing the design of the structure. Using gradient based algorithms, deriving the shape derivative manually can become very complex and error prone, especially in the case of time-dependent non-linear partial differential equations. To ease this burden, we present a high-level algorithmic differentiation tool that automatically computes first and second order shape derivatives for partial differential equations posed in the finite element frameworks FEniCS and Firedrake. The first order shape derivatives are computed using the adjoint method, while the second order shape derivatives are computed using a combination of the tangent linear method and the adjoint method. The adjoint and tangent linear equations are symbolically derived for any sequence of variational forms. As a consequence our methodology works for a wide range of PDE problems and is discretely consistent. We illustrate the generality of our framework by presenting several examples, spanning the range of linear, non-linear and time-dependent PDEs for both stationary and transient domains.

math.OC↗

arXiv2019

A multimesh finite element method for the Navier-Stokes equations based on projection methods

The multimesh finite element method is a technique for solving partial differential equations on multiple non-matching meshes by enforcing interface conditions using Nitsche's method. Since the non-matching meshes can result in arbitrarily cut cells, additional stabilization terms are needed to obtain a stable variational formulation. In this contribution we extend the multimesh finite element method to the Navier-Stokes equations based on the incremental pressure correction scheme. For each step in the pressure correction scheme, we derive a multimesh finite element formulation with suitable stabilization terms. The overall scheme yields expected spatial and temporal convergence rates on the Taylor-Green problem, and demonstrates good agreement for the drag and lift coefficients on the Turek-Schafer benchmark (DFG benchmark 2D-3). Finally, we illustrate the capabilities of the proposed scheme by optimizing the layout of obstacles in a channel.

math.NA↗

arXiv2019

Can diffusion alone explain brain-wide distribution of a CSF tracer within 24 hours?

The recently proposed glymphatic system suggests that bulk flow is important for clearing waste from the brain, and as such may underlie the development of e.g. Alzheimer's disease. The glymphatic hypothesis is still controversial and several biomechanical modeling studies at the micro-level have at least partially dismissed the system and its assumptions. In contrast, at the macro-level, there are many experimental findings in support of bulk flow. Here, we will investigate to what extent the CSF tracer distributions seen in novel magnetic resonance imaging (MRI) investigations over hours and days are suggestive of bulk flow or diffusion. In order to include the complex geometry of the brain, the heterogeneous CSF flow around the brain, and the transport over the time-scale of days, we employed the methods of partial differential constrained optimization to identify the apparent diffusion coefficient (ADC) that would correspond best to the MRI findings. We found that the computed ADC in grey and white matter was respectively 23\% and 82\% larger than the ADC estimated with DTI, which suggests that diffusion may not be the only mechanism governing transport.

math.NA↗

arXiv2018

Shape Optimization using the Finite Element Method on Multiple Meshes

An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other typically lacks robustness or is computationally expensive. This paper proposes a different approach, in which the computational domain is represented by multiple, independent meshes. A Nitsche based finite element method is used to weakly enforce continuity over the non-matching mesh interfaces. The optimization is preformed using an iterative gradient method, in which the shape-sensitivities are obtained by employing the Hadamard formulas and the adjoint approach. An optimize-then-discretize approach is chosen due to its independence of the FEM framework. Since the individual meshes may be moved freely, re-meshing or mesh deformations can be entirely avoided in cases where the geometry changes consists of rigid motions or scaling. By this free movement, we obtain robust and computational cheap mesh adaptation for optimization problems even for large domain changes. For general geometry changes, the method can be combined with mesh-deformation or re-meshing techniques to reduce the amount of deformation required. We demonstrate the capabilities of the method on several examples, including the optimal placement of heat emitting wires in a cable to minimize the chance of overheating, the drag minimization in Stokes flow, and the orientation of 25 objects in a Stokes flow.

math.NA↗

arXiv2018

High-level python abstractions for optimal checkpointing in inversion problems

Inversion and PDE-constrained optimization problems often rely on solving the adjoint problem to calculate the gradient of the objec- tive function. This requires storing large amounts of intermediate data, setting a limit to the largest problem that might be solved with a given amount of memory available. Checkpointing is an approach that can reduce the amount of memory required by redoing parts of the computation instead of storing intermediate results. The Revolve checkpointing algorithm o ers an optimal schedule that trades computational cost for smaller memory footprints. Integrat- ing Revolve into a modern python HPC code and combining it with code generation is not straightforward. We present an API that makes checkpointing accessible from a DSL-based code generation environment along with some initial performance gures with a focus on seismic applications.

cs.MS↗

arXiv2017

Automated adjoints of coupled PDE-ODE systems

Mathematical models that couple partial differential equations (PDEs) and spatially distributed ordinary differential equations (ODEs) arise in biology, medicine, chemistry and many other fields. In this paper we discuss an extension to the FEniCS finite element software for expressing and efficiently solving such coupled systems. Given an ODE described using an augmentation of the Unified Form Language (UFL) and a discretisation described by an arbitrary Butcher tableau, efficient code is automatically generated for the parallel solution of the ODE. The high-level description of the solution algorithm also facilitates the automatic derivation of the adjoint and tangent linearization of coupled PDE-ODE solvers. We demonstrate the capabilities of the approach on examples from cardiac electrophysiology and mitochondrial swelling.

math.NA↗

arXiv2016

Design optimisation and resource assessment for tidal-stream renewable energy farms using a new continuous turbine approach

This paper presents a new approach for optimising the design of tidal stream turbine farms. In this approach, the turbine farm is represented by a turbine density function that specifies the number of turbines per unit area and an associated continuous locally-enhanced bottom friction field. The farm design question is formulated as a mathematical optimisation problem constrained by the shallow water equations and solved with efficient, gradient-based optimisation methods. The resulting method is accurate, computationally efficient, allows complex installation constraints, and supports different goal quantities such as to maximise power or profit. The outputs of the optimisation are the optimal number of turbines, their location within the farm, the overall farm profit, the farm's power extraction, and the installation cost. We demonstrate the capabilities of the method on a validated numerical model of the Pentland Firth, Scotland. We optimise the design of four tidal farms simultaneously, as well as individually, and study how farms in close proximity may impact upon one another.

cs.CE↗

arXiv2016

An iteration count estimate for a mesh-dependent steepest descent method based on finite elements and Riesz inner product representation

Existing implementations of gradient-based optimisation methods typically assume that the problem is posed in Euclidean space. When solving optimality problems on function spaces, the functional derivative is then inaccurately represented with respect to $\ell^2$ instead of the inner product induced by the function space. This error manifests as a mesh dependence in the number of iterations required to solve the optimisation problem. In this paper, an analytic estimate is derived for this iteration count in the case of a simple and generic discretised optimisation problem. The system analysed is the steepest descent method applied to a finite element problem. The estimate is based on Kantorovich's inequality and on an upper bound for the condition number of Galerkin mass matrices. Computer simulations validate the iteration number estimate. Similar numerical results are found for a more complex optimisation problem constrained by a partial differential equation. Representing the functional derivative with respect to the inner product induced by the continuous control space leads to mesh independent convergence.

math.OC↗

arXiv2016

On the validity of tidal turbine array configurations obtained from steady-state adjoint optimisation

Extracting the optimal amount of power from an array of tidal turbines requires an intricate understanding of tidal dynamics and the effects of turbine placement on the local and regional scale flow. Numerical models have contributed significantly towards this understanding, and more recently, adjoint-based modelling has been employed to optimise the positioning of the turbines in an array in an automated way and improve on simple, regular man-made configurations. Adjoint-based optimisation of high-resolution and ideally 3D transient models is generally a very computationally expensive problem. As a result, existing work on the adjoint optimisation of tidal turbine placement has been mostly limited to steady-state simulations in which very high, non-physical values of the background viscosity are required to ensure that a steady-state solution exists. However, such compromises may affect the reliability of the modelled turbines, their wakes and interactions, and thus bring into question the validity of the computed optimal turbine positions. This work considers a suite of idealised simulations of flow past tidal turbine arrays in a 2D channel. It compares four regular array configurations, detailed by Divett et al. (2013), with the configuration found through adjoint optimisation in a steady-state, high-viscosity setup. The optimised configuration produces considerably more power. The same configurations are then used to produce a suite of transient simulations that do not use constant high-viscosity, and instead use large eddy simulation (LES) to parameterise the resulting turbulent structures. It is shown that the LES simulations produce less power than that predicted by the constant high-viscosity runs. Nevertheless, they still follow the same trends in the power curve throughout time, with optimised layouts continuing to perform significantly better than simplified configurations.

physics.flu-dyn↗

arXiv2014

Deflation techniques for finding distinct solutions of nonlinear partial differential equations

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and fluid mechanics that permit distinct solutions.

math.NA↗

arXiv2014

Hybrid global-local optimisation algorithms for the layout design of tidal turbine arrays

Tidal stream power generation represents a promising source of renewable energy. In order to extract an economically useful amount of power, tens to hundreds of tidal turbines need to be placed within an array. The layout of these turbines can have a significant impact on the power extracted and hence on the viability of the site. Funke et al. formulated the question of the best turbine layout as an optimisation problem constrained by the shallow water equations and solved it using a local, gradient-based optimisation algorithm. Given the local nature of this approach, the question arises of how optimal the layouts actually are. This becomes particularly important for scenarios with complex bathymetry and layout constraints, both of which typically introduce locally optimal layouts. Optimisation algorithms which find the global optima generally require orders of magnitude more iterations than local optimisation algorithms and are thus infeasible in combination with an expensive flow model. This paper presents an analytical wake model to act as an efficient proxy to the shallow water model. Based upon this, a hybrid global-local two-stage optimisation approach is presented in which turbine layouts are first optimised with the analytical wake model via a global optimisation algorithm, and further optimised with the shallow water model via a local gradient-based optimisation algorithm. This procedure is applied to a number of idealised cases and a more realistic case with complex bathymetry in the Pentland Firth, Scotland. It is shown that in cases where bathymetry is considered, the two-stage optimisation procedure is able to improve the power extracted from the array by as much as 25% compared to local optimisation for idealised scenarios and by as much as 12% for the more realistic Pentland Firth scenario whilst in many cases reducing the overall computation time by approximately 35%.

math.OC↗

arXiv2013

A framework for the automation of generalised stability theory

The traditional approach to investigating the stability of a physical system is to linearise the equations about a steady base solution, and to examine the eigenvalues of the linearised operator. Over the past several decades, it has been recognised that this approach only determines the asymptotic stability of the system, and neglects the possibility of transient perturbation growth arising due to the nonnormality of the system. This observation motivated the development of a more powerful generalised stability theory (GST), which focusses instead on the singular value decomposition of the linearised propagator of the system. While GST has had significant successes in understanding the stability of phenomena in geophysical fluid dynamics, its more widespread applicability has been hampered by the fact that computing the SVD requires both the tangent linear operator and its adjoint: deriving the tangent linear and adjoint models is usually a considerable challenge, and manually embedding them inside an eigensolver is laborious. In this paper, we present a framework for the automation of generalised stability theory, which overcomes these difficulties. Given a compact high-level symbolic representation of a finite element discretisation implemented in the FEniCS system, efficient C++ code is automatically generated to assemble the forward, tangent linear and adjoint models; these models are then used to calculate the optimally growing perturbations to the forward model, and their growth rates. By automating the stability computations, we hope to make these powerful tools a more routine part of computational analysis. The efficiency and generality of the framework is demonstrated with applications drawn from geophysical fluid dynamics, phase separation and quantum mechanics.

cs.MS↗