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Following axon pathfinding, growth cones transition from stochastic filopodial exploration to the formation of a limited number of synapses. How the interplay of filopodia and synapse assembly ensures robust connectivity in the brain has remained a challenging problem. Here, we developed a new 4D analysis method for filopodial dynamics and a data-driven computational model of synapse formation for R7 photoreceptor axons in developing Drosophila brains. Our live data support a 'serial synapse formation' model, where at any time point only a single 'synaptogenic' filopodium suppresses the synaptic competence of other filopodia through competition for synaptic seeding factors. Loss of the synaptic seeding factors Syd-1 and Liprin-α leads to a loss of this suppression, filopodial destabilization and reduced synapse formation, which is sufficient to cause the destabilization of entire axon terminals. Our model provides a filopodial 'winner-takes-all' mechanism that ensures the formation of an appropriate number of synapses.
Optimization of clinical radiofrequency hyperthermia by use of MR-thermography in a hybrid system
(2010)
The paper presents a new affine invariant theory on asymptotic mesh independence of Newton's method in nonlinear PDEs. Compared to earlier attempts, the new approach is both much simpler and more natural from the algorithmic point of view. The theory is exemplified at collocation methods for ODE boundary value problems and at finite element methods for elliptic PDE problems.
A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown.
Spectral Deferred Correction methods for adaptive electro-mechanical coupling in cardiac simulation
(2014)
We investigate spectral deferred correction (SDC) methods for time stepping
and their interplay with spatio-temporal adaptivity, applied to the solution
of the cardiac electro-mechanical coupling model. This model consists
of the Monodomain equations, a reaction-diffusion system modeling the cardiac
bioelectrical activity, coupled with a quasi-static mechanical model describing
the contraction and relaxation of the cardiac muscle. The numerical
approximation of the cardiac electro-mechanical coupling is a challenging
multiphysics problem, because it exhibits very different spatial and temporal
scales. Therefore, spatio-temporal adaptivity is a promising approach
to reduce the computational complexity. SDC methods are simple iterative
methods for solving collocation systems. We exploit their flexibility for combining
them in various ways with spatio-temporal adaptivity. The accuracy
and computational complexity of the resulting methods are studied on some
numerical examples.
Spectral Deferred Correction methods for adaptive electro-mechanical coupling in cardiac simulation
(2017)
We investigate spectral deferred correction (SDC) methods for time stepping
and their interplay with spatio-temporal adaptivity, applied to the solution
of the cardiac electro-mechanical coupling model. This model consists
of the Monodomain equations, a reaction-diffusion system modeling the cardiac
bioelectrical activity, coupled with a quasi-static mechanical model describing
the contraction and relaxation of the cardiac muscle. The numerical
approximation of the cardiac electro-mechanical coupling is a challenging
multiphysics problem, because it exhibits very different spatial and temporal
scales. Therefore, spatio-temporal adaptivity is a promising approach
to reduce the computational complexity. SDC methods are simple iterative
methods for solving collocation systems. We exploit their flexibility for combining
them in various ways with spatio-temporal adaptivity. The accuracy
and computational complexity of the resulting methods are studied on some
numerical examples.
Pulse thermography of concrete structures is used in civil engineering for detecting voids, honeycombing and delamination. The physical situation is readily modeled by Fourier's law. Despite the simplicity of the PDE structure, quantitatively realistic numerical 3D simulation faces two major obstacles. First, the short heating pulse induces a thin boundary layer at the heated surface which encapsulates all information and therefore has to be resolved faithfully. Even with adaptive mesh refinement techniques, obtaining useful accuracies requires an unsatisfactorily fine discretization. Second, bulk material parameters and boundary conditions are barely known exactly. We address both issues by a semi-analytic reformulation of the heat transport problem and by parameter identification. Numerical results are compared with measurements of test specimens.
Pulse thermography of concrete structures is used in civil engineering for detecting voids, honeycombing and delamination. The physical situation is readily modeled by Fourier's law. Despite the simplicity of the PDE structure, quantitatively realistic numerical 3D simulation faces two major obstacles. First, the short heating pulse induces a thin boundary layer at the heated surface which encapsulates all information and therefore has to be resolved faithfully. Even with adaptive mesh refinement techniques, obtaining useful accuracies requires an unsatisfactorily fine discretization. Second, bulk material parameters and boundary conditions are barely known exactly. We address both issues by a semi-analytic reformulation of the heat transport problem and by parameter identification. Numerical results are compared with measurements of test specimens.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient.
In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient.
A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.
In several inital value problems with particularly expensive right hand side evaluation or implicit step computation, there is a trade-off between accuracy and computational effort. We consider inexact spectral deferred correction (SDC) methods for solving such initial value problems. SDC methods are interpreted as fixed point iterations and, due to their corrective iterative nature, allow to exploit the accuracy-work-tradeoff for a reduction of the total computational effort. On one hand we derive error models bounding the total error in terms of the evaluation errors. On the other hand, we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal local tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance. The properties of optimal local tolerances and the predicted efficiency gain compared to simpler heuristics, and a reasonable practical performance, are illustrated on simple numerical examples.
In several inital value problems with particularly expensive right hand side computation, there is a trade-off between accuracy and computational effort in evaluating the right hand sides. We consider inexact spectral deferred correction (SDC) methods for solving such non-stiff initial value problems. SDC methods are interpreted as fixed point iterations and, due to their corrective iterative nature, allow to exploit the accuracy-work-tradeoff for a reduction of the total computational effort. On one hand we derive an error model bounding the total error in terms of the right hand side evaluation errors. On the other hand, we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance.
Estimation of time of death based on a single measurement of body
core temperature is a standard procedure in forensic medicine.
Mechanistic models using simulation of heat transport promise
higher accuracy than established phenomenological models in
particular in nonstandard situations, but involve many not exactly
known physical parameters. Identifying both time of death and
physical parameters from multiple temperature measurements is
one possibility to reduce the uncertainty significantly.
In this paper, we consider the inverse problem in a Bayesian setting
and perform both local and sampling-based uncertainty
quantification, where proper orthogonal decomposition is used as
model reduction for fast solution of the forward model. Based on
the local uncertainty quantification, optimal design of experiments
is performed in order to minimize the uncertainty in the time of
death estimate for a given number of measurements. For reasons
of practicability, temperature acquisition points are selected from
a set of candidates in different spatial and temporal locations.
Applied to a real corpse model, a significant accuracy improvement
is obtained already with a small number of measurements.
Estimation of time of death based on a single measurement of body
core temperature is a standard procedure in forensic medicine.
Mechanistic models using simulation of heat transport promise
higher accuracy than established phenomenological models in
particular in nonstandard situations, but involve many not exactly
known physical parameters. Identifying both time of death and
physical parameters from multiple temperature measurements is
one possibility to reduce the uncertainty significantly.
In this paper, we consider the inverse problem in a Bayesian setting
and perform both local and sampling-based uncertainty
quantification, where proper orthogonal decomposition is used as
model reduction for fast solution of the forward model. Based on
the local uncertainty quantification, optimal design of experiments
is performed in order to minimize the uncertainty in the time of
death estimate for a given number of measurements. For reasons
of practicability, temperature acquisition points are selected from
a set of candidates in different spatial and temporal locations.
Applied to a real corpse model, a significant accuracy improvement
is obtained already with a small number of measurements.
Temperature-based estimation of time of death (ToD) can be per-
formed either with the help of simple phenomenological models of corpse
cooling or with detailed mechanistic (thermodynamic) heat transfer mod-
els. The latter are much more complex, but allow a higher accuracy of
ToD estimation as in principle all relevant cooling mechanisms can be
taken into account.
The potentially higher accuracy depends on the accuracy of tissue and
environmental parameters as well as on the geometric resolution. We in-
vestigate the impact of parameter variations and geometry representation
on the estimated ToD based on a highly detailed 3D corpse model, that
has been segmented and geometrically reconstructed from a computed to-
mography (CT) data set, differentiating various organs and tissue types.
From that we identify the most crucial parameters to measure or estimate,
and obtain a local uncertainty quantifcation for the ToD.
Temperature-based estimation of time of death (ToD) can be per-
formed either with the help of simple phenomenological models of corpse
cooling or with detailed mechanistic (thermodynamic) heat transfer mod-
els. The latter are much more complex, but allow a higher accuracy of
ToD estimation as in principle all relevant cooling mechanisms can be
taken into account.
The potentially higher accuracy depends on the accuracy of tissue and
environmental parameters as well as on the geometric resolution. We in-
vestigate the impact of parameter variations and geometry representation
on the estimated ToD based on a highly detailed 3D corpse model, that
has been segmented and geometrically reconstructed from a computed to-
mography (CT) data set, differentiating various organs and tissue types.
Reasons for the failure of adaptive methods to deliver improved efficiency when integrating monodomain models for myocardiac excitation are discussed. Two closely related techniques for reducing the computational complexity of linearly implicit integrators, deliberate sparsing and splitting, are investigated with respect to their impact on computing time and accuracy.
Reasons for the failure of adaptive methods to deliver improved efficiency when integrating monodomain models for myocardiac excitation are discussed. Two closely related techniques for reducing the computational complexity of linearly implicit integrators, deliberate sparsing and splitting, are investigated with respect to their impact on computing time and accuracy.
The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.
The paper provides a detailed analysis of a short step interior point algorithm applied to linear control constrained optimal control problems. Using an affine invariant local norm and an inexact Newton corrector, the well-known convergence results from finite dimensional linear programming can be extended to the infinite dimensional setting of optimal control. The present work complements a recent paper of Weiser and Deuflhard, where convergence rates have not been derived. The choice of free parameters, i.e. the corrector accuracy and the number of corrector steps, is discussed.
The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples.
Regional hyperthermia is a cancer therapy aiming at heating tumors using phased array applicators. This article provides an overview over current mathematical challenges of delivering individually optimal treatments. The focus is on therapy planning and identification of technical as well as physiological quantities from MR thermometry measurements.
Parabolic reaction-diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction-diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples.
Spectral deferred correction methods for solving stiff ODEs are known to converge rapidly towards the collocation limit solution on equidistant grids, but show a much less favourable contraction on non-equidistant grids such as
Radau-IIa points. We interprete SDC methods as fixed point iterations for the collocation system and propose new DIRK-type sweeps for stiff problems based on purely linear algebraic considerations. Good convergence is recovered also
on non-equidistant grids. The properties of different variants are explored on a couple of numerical examples.
The biodomain model of cardioelectric excitation consists of a reaction‐diffusion equation, an elliptic algebraic constraint, and a set of pointwise ODEs. Fast reaction enforces small time steps, such that for common mesh sizes the reaction‐diffusion equation is easily solved implicitly due to a dominating mass matrix. In contrast, the elliptic constraint does not benefit from small time steps and requires a comparably expensive solution. We propose a delayed residual compensation that improves the solution of the elliptic constraint and thus alleviates the need for long iteration times.
The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples.
Spectral deferred correction methods for solving stiff ODEs are known to converge rapidly towards the collocation limit solution on equidistant grids, but show a much less favourable contraction on non-equidistant grids such as
Radau-IIa points. We interprete SDC methods as fixed point iterations for the collocation system and propose new DIRK-type sweeps for stiff problems based on purely linear algebraic considerations. Good convergence is recovered also
on non-equidistant grids. The properties of different variants are explored on a couple of numerical examples.
Inside Finite Elements
(2016)
All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered as far as it gives insight into the construction of algorithms.Throughout the exercises a complete FE-solver for scalar 2D problems will be implemented in Matlab/Octave.
Adaptation of antenna profiles for control of MR guided hyperthermia (HT) in a hybrid MR-HT system
(2007)
An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solves. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.
Background
Mathematical optimisation models have recently been applied to identify ideal Automatic External Defibrillator (AED) locations that maximise coverage of Out of Hospital Cardiac Arrest (OHCA). However, these fixed location models cannot relocate existing AEDs in a flexible way, and have nearly exclusively been applied to urban regions. We developed a flexible location model for AEDs, compared its performance to existing fixed location and population models, and explored how these perform across urban and rural regions.
Methods
Optimisation techniques were applied to AED deployment and OHCA coverage was assessed. A total of 2802 geolocated OHCAs occurred in Canton Ticino, Switzerland, from January 1st 2005 to December 31st 2015.
Results
There were 719 AEDs in Canton Ticino. 635 (23%) OHCA events occurred within 100m of an AED, with 306 (31%) in urban, and 329 (18%) in rural areas. Median distance from OHCA events to the nearest AED was 224m (168m urban vs. 269m rural). Flexible location models performed better than fixed location and population models, with the cost to deploy 20 new AEDs instead relocating 171 existing AEDs to new locations, improving OHCA coverage to 38%, compared to 26% using fixed models, and 24% with the population based model.
Conclusions
Optimisation models for AEDs placement are superior to population models and should be strongly considered by communities when selecting areas for AED deployment. Compared to other models, flexible location models increase overall OHCA coverage, and decreases the distance to nearby AEDs, even in rural areas, while saving significant financial resources.
Cardiac electrograms are an important tool to study the spread of excitation waves inside the heart, which in turn underlie muscle contraction. Electrograms can be used to analyse the dynamics of these waves, e.g. in fibrotic tissue. In computational models, these analyses can be done with greater detail than during minimally invasive in vivo procedures. Whilst homogenised models have been used to study electrogram genesis, such analyses have not yet been done in cellularly resolved models. Such high resolution may be required to develop a thorough understanding of the mechanisms behind abnormal excitation patterns leading to arrhythmias. In this study, we derived electrograms from an excitation propagation simulation in the Extracellular, Membrane, Intracellular (EMI) model, which represents these three domains explicitly in the mesh. We studied the effects of the microstructural excitation dynamics on electrogram genesis and morphology. We found that electrograms are sensitive to the myocyte alignment and connectivity, which translates into micro-fractionations in the electrograms.
Adaptive Gaussian Process Regression for Efficient Building of Surrogate Models in Inverse Problems
(2023)
In a task where many similar inverse problems must be solved, evaluating costly simulations is impractical. Therefore, replacing the model y with a surrogate model y(s) that can be evaluated quickly leads to a significant speedup. The approximation quality of the surrogate model depends strongly on the number, position, and accuracy of the sample points. With an additional finite computational budget, this leads to a problem of (computer) experimental design. In contrast to the selection of sample points, the trade-off between accuracy and effort has hardly been studied systematically. We therefore propose an adaptive algorithm to find an optimal design in terms of
position and accuracy. Pursuing a sequential design by incrementally appending the computational budget leads to a convex and constrained optimization problem. As a surrogate, we construct a Gaussian process regression model. We measure the global approximation error in terms of its impact on the accuracy of the identified parameter and aim for a uniform absolute tolerance, assuming that y(s) is computed by finite element calculations. A priori error estimates and a coarse estimate of computational effort relate the expected improvement of the surrogate model error to computational effort, resulting in the most efficient combination of sample point and evaluation tolerance. We also allow for improving the accuracy of already existing sample points by continuing previously truncated finite element solution procedures.
The growing discrepancy between CPU computing power and memory bandwidth drives more and more numerical algorithms into a bandwidth-
bound regime. One example is the overlapping Schwarz smoother, a highly effective building block for iterative multigrid solution of elliptic equations with higher order finite elements. Two options of reducing the required
memory bandwidth are sparsity exploiting storage layouts and representing matrix entries with reduced precision in floating point or fixed point
format. We investigate the impact of several options on storage demand and contraction rate, both analytically in the context of subspace correction methods and numerically at an example of solid mechanics. Both perspectives agree on the favourite scheme: fixed point representation of Cholesky factors in nested dissection storage.