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We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $$H^1$$ H 1 and subject to pointwise lower and upper bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau–Yosida-type penalty approach to treat the control constraint and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton differentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.

A multitude of industries, including energy and process engineering, as well as academia are researching and utilizing new fluid substances to further the aim of sustainability. Knowledge of the thermodynamic properties of these substances is a prerequisite, if they are to be utilized to their fullest potential. To date, the way to acquire reliable knowledge of the thermodynamic behavior is through measurements. The ensuing experimental data are then used to develop equations of state, which efficiently embody the gained knowledge of the behavior of the fluid substance, allow for interpolation and, to some extent, extrapolation. However, the acquisition of low-uncertainty experimental data, and thus the development of accurate equations of state, is often time-consuming and expensive. For substances for which suitable force field models exist, molecular modeling and simulation are well-suited to generate thermodynamic data or to augment experimental data, however, at the expense of larger uncertainties. The major goal of this work is to present a new approach for the development of equations of state using (1) symbolic regression, which is a machine learning based model development approach, (2) optimal experimental design, and (3) efficient data acquisition. We demonstrate this approach using the example of density data of an air-like binary mixture ($$0.2094\,\hbox {O}_{2}\,+\,0.7906\,\hbox {N}_{2}$$ 0.2094 O 2 + 0.7906 N 2 ) over the temperature range from $${100}\,{\textrm{K}}$$ 100 K to $${300}\,{\textrm{K}}$$ 300 K at pressures of up to $${8}\,{\textrm{MPa}}$$ 8 MPa , which covers the gaseous, liquid, and supercritical regions. For this purpose, an experimental data set published by von Preetzmann et al. (Int. J. Thermophys. 42, 2021) and molecular simulation data sampled in this work are used. The two data sets are compared in terms of acquisition time, cost, and uncertainty, showing that an optimized combination of experimental and simulation data leads to lower cost while maintaining low uncertainties.

We propose a novel discrete concept for the total generalized variation (TGV), which has originally been derived to reduce the staircasing effect in classical total variation (TV) regularization, in image denoising problems. We describe discrete, second-order TGV for piecewise constant functions on triangular meshes, thus allowing the TGV functional to be applied to more general data structures than pixel images, and in particular in the context of finite element discretizations. Particular attention is given to the description of the kernel of the TGV functional, which, in the continuous setting, consists of linear polynomials. We discuss how to take advantage of this kernel structure using piecewise constant functions on triangular meshes. Numerical experiments include denoising and inpainting problems for images defined on non-standard grids, including data from a 3D scanner.

Background Hemophilia A is an X-linked genetic disorder caused by a mutation in the gene for factor VIII (FVIII) protein that reduces the ability of blood to clot. Clinical drug trials have shown the potential of adeno-associated virus (AAV) gene therapy as a one-time treatment for hemophilia A that can produce sustained high levels of FVIII. However, a gradual decline in protein levels has been observed in patients after 2-4 years. The hypothesis being tested in the Herzog Lab is that an interlinked immune and cellular stress response could be causing the loss of expression. Methods Two groups of Hemophilia A mice were administered AAV therapy, with one group receiving recurrent doses of Rapamycin. Blood samples were taken at weeks 4, 8, 12 and 14. Mice were euthanized at weeks 4, 8, and 14, and their livers were harvested. qPCR was used to measure AAV copy numbers and FVIII mRNA at 4, 8, and 14 weeks. Cryosections of mice livers from weeks 4, 8, and 14 were stained with antibodies for FVIII protein and CD8. Results qPCR showed roughly half as much AAV copy numbers in the rapamycin group at all time points, and little difference in FVIII mRNA between the groups. There was also a large decrease in AAV copy numbers and FVIII mRNA in both groups between 8 and 14 weeks. Immunohistochemistry showed less CD8 and more FVIII signal in mice treated with rapamycin. Discussion Experiments are currently being performed to investigate the decline in AAV copy numbers and mRNA between weeks 8 and 14. The immunohistochemistry data shows a relationship between increased FVIII protein levels and decreased cellular immune response but does not explain the gradual decline in FVIII. Further investigation into FVIII expression following AAV gene therapy could lead to an effective one-time treatment for hemophilia A.

We consider optimization problems with manifold-valued constraints. These generalize classical equality and inequality constraints to a setting in which both the domain and the codomain of the constraint mapping are smooth manifolds. We model the feasible set as the preimage of a submanifold with corners of the codomain. The latter is a subset which corresponds to a convex cone locally in suitable charts. We study first- and second-order optimality conditions for this class of problems. We also show the invariance of the relevant quantities with respect to local representations of the problem.

The identification of the interface of an inclusion in a diffusion process is considered. This task is viewed as a parameter identification problem in which the parameter space bears the structure of a shape manifold. A corresponding optimum experimental design (OED) problem is formulated in which the activation pattern of an array of sensors in space and time serves as experimental condition. The goal is to improve the estimation precision within a certain subspace of the infinite dimensional tangent space of shape variations to the manifold, and to find those shape variations of best and worst identifiability. Numerical results for the OED problem obtained by a simplicial decomposition algorithm are presented.

Decreasing the time required for accurate thermodynamic property measurements is extremely desirable for model development that can respond to the needs of science and industry within a short time frame. Here, we demonstrate the application of optimal experimental design to measurements of thermodynamic properties. The technique is explored using the fitting of a Schilling-type equation from published $$(p, \rho , T)$$ ( p , ρ , T ) -measurements of ethylene glycol and propylene glycol. The analysis shows that a fixed-exponent fit using (p, T)-measurements along the five most informative isotherms produces models of relative density errors comparable to those obtained using the data along all investigated isotherms, i.e., eight or nine. It is also argued that a calculation of optimal isotherms prior to the measurement series can further increase the precision at no additional experimental effort.

In this paper, we introduce a definition of Fenchel conjugate and Fenchel biconjugate on Hadamard manifolds based on the tangent bundle. Our definition overcomes the inconvenience that the conjugate depends on the choice of a certain point on the manifold, as previous definitions required. On the other hand, this new definition still possesses properties known to hold in the Euclidean case. It even yields a broader interpretation of the Fenchel conjugate in the Euclidean case itself. Most prominently, our definition of the Fenchel conjugate provides a Fenchel--Moreau theorem for geodesically convex, proper, lower semicontinuous functions. In addition, this framework allows us to develop a theory of separation of convex sets on Hadamard manifolds, and a strict separation theorem is obtained.

This paper studies a new class of integration schemes for the numerical solution of semiexplicit differential-algebraic equations of differentiation index 2 in Hessenberg form. Our schemes provide the flexibility to choose different discretizations in the differential and algebraic equations. At the same time they are designed to have a property called variational consistency, i.e., the choice of the discretization of the constraint determines the discretization of the Lagrange multiplier. For the case of linear constraints we prove convergence of order $r+1$, both for the state and the multiplier if piecewise polynomials of order $r$ are used. These results are also verified numerically.

Fiber optical strain sensors are used to measure the strain at a particular sensor position inside the fiber. In order to deduce the strain in the surrounding matrix material, one can employ the strain transfer principle. Its application is based on the assumption that the presence of the fiber does not impede the deformation of the matrix material in fiber direction. In fact, the strain transfer principle implies that the strain in fiber direction inside the fiber carries over verbatim to the strain inside the matrix material. For a comparatively soft matrix material, however, this underlying assumption may not be valid. To overcome this drawback, we propose to superimpose the matrix material with a one-dimensional model of the fiber, which takes into account its elastic properties. The finite element solution of this model yields a more accurate prediction of the strain inside the fiber in fiber direction at low computational costs.

The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to as dimensional consistency, suggests a natural combination of the parameters intrinsic to the problem. It turns out that the scaling obtained in this way leads to robustness with respect to problem parameters in many relevant cases. As a consequence, we advertise dimensional consistency based preconditioning as a new and systematic way to designing parameter robust preconditoners for saddle-point systems arising from models for physical phenomena.

This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g., the Fenchel–Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel–Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas–Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm may even converge on manifolds of positive curvature.

We consider large-scale dynamical systems in which both the initial state and some parameters are unknown. These unknown quantities must be estimated from partial state observations over a time window. A data assimilation framework is applied for this purpose. Specifically, we focus on large-scale linear systems with multiplicative parameter-state coupling as they arise in the discretization of parametric linear time-dependent partial differential equations. Another feature of our work is the presence of a quantity of interest different from the unknown parameters, which is to be estimated based on the available data. In this setting, we employ a simplicial decomposition algorithm for an optimal sensor placement and set forth formulae for the efficient evaluation of all required quantities. As a guiding example, we consider a thermo-mechanical PDE system with the temperature constituting the system state and the induced displacement at a certain reference point as the quantity of interest.

Lai and Chan [Computer Vis. Image Underst., 115 (2011), pp. 1647--1661] introduced an analogue of the total variation image reconstruction approach of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), ...

The identification of Lame parameters in linear elasticity is considered. An optimum experimental design problem is formulated, which aims at minimizing the size of an associated confidence ellipso...

In this study we present an extension of a model of an elastic crane transporting a load by means of controlling the crane trolley motion and the crane rotation. In addition to the model considered in Kimmerle et al. (2017), we allow for rotations of the crane and include damping of the trolley and moments of inertia as well. We derive a fully coupled system of ordinary differential equations (ODE), representing the trolley and load (modelled as a pendulum), and partial differential equations (PDE), i.e. the linear elasticity equations for the deformed crane beam. The objective to be minimized is a linear combination of the terminal time, the control effort, the kinetic energy of the load, and penalty terms for the terminal conditions. We show the Frechet-differentiability of the mechanical displacement field with respect to the location of the boundary condition that is moving. This is a crucial point for a further mathematical analysis on the existence of optimal controls and the derivation of necessary optimality conditions. Finally we present first results for the full time-optimal control of the extended model.

We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is d...

A quasistatic, thermoviscoplastic model at small strains with linear kinematic hardening, von Mises yield condition and mixed boundary conditions is considered. The existence of a unique weak solution is proved by means of a fixed-point argument, and by employing maximal parabolic regularity theory. The weak continuity of the solution operator is also shown. As an application, the existence of a global minimizer of a class of optimal control problems is proved.

An analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. The analysis of the total variation of the normal vector field is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is shown that spheres are stationary points when the total variation of the normal is minimized under an area constraint. Shape calculus is used to characterize the relevant derivatives. Since the total variation functional is non-differentiable whenever the boundary contains flat regions, an extension of the split Bregman method to manifold valued functions is proposed.

In this article an optimal sensor placement problem for a thermo-elastic solid body model is considered. Temperature sensors are placed in a near-optimal way so that their measurements allow an acc...

In this article, the well-known conjugate gradient (CG) method for linear systems in Hilbert spaces is extended to a reflexive Banach space setting. In this setting, the Riesz isomorphism has to be replaced by the duality mapping. Due to the nonlinearity of the duality mapping, the short term recursion and conjugacy of search directions cannot be maintained simultaneously. The well-posedness of the proposed iteration and its global convergence are shown under appropriate conditions. Error bounds and stopping criteria are presented as well. The results extend to a limited-memory variant of the algorithm. The behavior of the method is demonstrated by numerical examples.

Summary The investigation of a reaction–diffusion system with new nonlinear reactions kinetics to model the fermentation of wine is presented. The reactive part extends existing ordinary differential models by taking into account oxygen availability and ethanol toxicity. The presence of spatial diffusion and the inclusion of a heat equation allow geometrical and thermal effects in the model. Existence, uniqueness, and regularity of solutions to this system of reaction–diffusion partial differential equations are discussed. A boundary optimal control problem is formulated with the purpose of steering an ideal fermentation process. This optimal control problem is theoretically investigated and numerically solved in the adjoint method framework. The existence of an optimal control is proved, and its solution is characterized by means of the corresponding optimality system. To solve this system, an implicit–explicit splitting approach is considered, and a Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization scheme is implemented. Results of numerical experiments demonstrate the validity of the fermentation model and of the proposed control strategy. Copyright © 2016 John Wiley & Sons, Ltd.

Elliptic optimal control problems with pointwise state gradient constraints are considered. A quadratic penalty approach is employed together with a semismooth Newton iteration. Three different preconditioners are proposed and the ensuing spectral properties of the preconditioned linear Newton saddle-point systems are analyzed dependent on the penalty parameter. A new bound for the smallest positive eigenvalue is proved. Since the analysis is carried out in function space it will ensure mesh independent convergence behavior of suitable Krylov subspace methods such as Minres, also in discretized settings. A path-following strategy with a preconditioned inexact Newton solver is implemented and numerical results are provided.

An optimal control problem of static plasticity with linear kinematic hardening and von Mises yield condition is studied. The problem is treated in its primal formulation, where the state system is a variational inequality of the second kind. First-order necessary optimality conditions are obtained by means of an approximation by a family of control problems with state system regularized by Huber-type smoothing, and a subsequent limit analysis. The equivalence of the optimality conditions with the C-stationarity system for the equivalent dual formulation of the problem is proved. Numerical experiments are presented, which demonstrate the viability of the Huber-type smoothing approach.

An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element computations. The analysis of the functional is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. It is found to agree with the discrete total mean curvature known in discrete differential geometry. A split Bregman iteration is proposed for the solution of discretized shape optimization problems, in which the total variation of the normal appears as a regularizer. Unlike most other priors, such as surface area, the new functional allows for piecewise flat shapes. As two applications, a mesh denoising and a geometric inverse problem of inclusion detection type involving a partial differential equation are considered. Numerical experiments confirm that polyhedral shapes can be identified quite accurately., Comment: arXiv admin note: substantial text overlap with arXiv:1902.07240

The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart–Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV- $$L^2$$ denoising and inpainting, can be implemented in low- and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.

Karush-Kuhn-Tucker (KKT) conditions for equality and inequality constrained optimization problems on smooth manifolds are formulated. Under the Guignard constraint qualification, local minimizers are shown to admit Lagrange multipliers. The linear independence, Mangasarian-Fromovitz, and Abadie constraint qualifications are also formulated, and the chain "LICQ implies MFCQ implies ACQ implies GCQ" is proved. Moreover, classical connections between these constraint qualifications and the set of Lagrange multipliers are established, which parallel the results in Euclidean space. The constrained Riemannian center of mass on the sphere serves as an illustrating numerical example.

Optimal transport problems pose many challenges when considering their numerical treatment. We investigate the solution of a PDE-constrained optimisation problem subject to a particular transport equation arising from the modelling of image metamorphosis. We present the nonlinear optimisation problem, and discuss the discretisation and treatment of the nonlinearity via a Gauss--Newton scheme. We then derive preconditioners that can be used to solve the linear systems at the heart of the (Gauss--)Newton method. With the optical flow in mind, we further propose the reduction of dimensionality by choosing a radial basis function discretisation that uses the centres of superpixels as the collocation points. Again, we derive suitable preconditioners that can be used for this formulation., 28 pages, 5 figures, submitted manuscript

We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub.

Today there exists a huge demand for technologies which enable and facilitate the mass production of fiber reinforced composites. Injection molding of Short Fiber Reinforced Plastics (SFRP) is a quite popular method especially in the automotive industry, providing high stiffness levels on the one hand and complex moldable shapes on the other hand. Due to the high cost of mold production and injection molding machines, nowadays lots of research is done to improve models and to develop software for the simulation of this process. This allows to detect problems with the mold design and optimization of the part performance and quality at an early stage of the development.

Optimal sensor placement problems are considered for a thermoelastic solid body. The main motivation is the real-time capable prediction of the displacement of the tool center point (TCP) in a machine tool by temperature measurements. A reduced order model based on proper orthogonal decomposition is used to describe the temperature field. The quality of the TCP displacement estimation is measured in terms of the associated covariance matrix. Based on this criterion, a sequential placement algorithm is described which stops when a certain prediction quality is reached. Numerical tests are provided.