The inverse problem of supervised reconstruction of depth-variable (time-dependent) parameters in ordinary differential equations is considered, with the typical application of finding weights of a neural ordinary differential equation (NODE) for a residual network with time continuous layers. The differential equation is treated as an abstract and isolated entity, termed a standalone NODE (sNODE), to facilitate for a wide range of applications. The proposed parameter reconstruction is performed by minimizing a cost functional covering a variety of loss functions and penalty terms. Regularization via penalty terms is incorporated to enhance ethical and trustworthy AI formulations. A nonlinear conjugate gradient mini-batch optimization scheme (NCG) is derived for the training having the benefit of including a sensitivity problem. The model (differential equation)-based approach is thus combined with a data-driven learning procedure. Mathematical properties are stated for the differential equation and the cost functional. The adjoint problem needed is derived together with the sensitivity problem. The sensitivity problem itself can estimate changes in the output under perturbation of the trained parameters. To preserve smoothness during the iterations, the Sobolev gradient is calculated and incorporated. Numerical results are included to validate the procedure for a NODE and synthetic datasets and compared with standard gradient approaches. For stability, using the sensitivity problem, a strategy for adversarial attacks is constructed, and it is shown that the given method with Sobolev gradients is more robust than standard approaches for parameter identification.

Evolution of brain tumours backwards in time is studied using well-established brain tumour growth models being semilinear parabolic equations of reaction-diffusion type. To run the models backwards, the tumour cell density data at a fixed (final) time is used, rendering an inverse ill-posed problem. This problem is recast as the minimisation of a cost functional matching the data against the solution at a final time of a forward parabolic model having the initial cell density as a control function. Regularisation is incorporated via penalising terms involving Sobolev norms. Mathematical properties of the semilinear parabolic equations are shown in Sobolev-Bochner spaces including uniqueness of a solution to the inverse problem. Differentiability of the control-to-state map is established rendering a sensitivity problem. The derivative of the cost functional is calculated and the adjoint state is derived via the Lagrange formalism. A non-linear conjugate gradient method (NCG) is presented for the minimisation. Numerical realisation of the minimisation on the BraTS'20 dataset is included using a standard finite difference discretisation of the space and time derivatives, showing that tumour evolution backwards in time can be accomplished and that the initial tumour cell density can be reconstructed. Comparison is done with a non-linear Landweber method.

Rothes method combined with a fundamental sequences method is considered for the numerical solution of the nonstationary (unsteady) homogeneous Stokes problem in two-dimensional doubly connected domains. The Stokes system is reduced, using Rothes method, to a sequence of stationary inhomogeneous problems with a known sequence of fundamental solutions. The stationary problems are discretized by a fundamental sequences method; this means searching for the solution as a linear combination of elements of the fundamental sequence and matching the given boundary conditions in order to find the coefficients in the expansion of the solution. No additional reduction of the inhomogeneous problems is needed, making it an efficient method and different from standard strategies of the method of fundamental solutions. Results of numerical experiments are given, and these confirm the applicability of the proposed approach.

A method of fundamental solutions (MFS) is presented for the ill-posed linear inverse problem consisting of the reconstruction of boundary data on the inner boundary for the hyperbolic system of elastodynamics in planar annular domains from known essential and natural boundary conditions on the outer boundary. This corresponds to the problem of finding elastic wave propagation in a structure from measured data being the displacement and traction on a portion of the boundary of the structure. The time-dependent lateral Cauchy problem is reduced to a sequence of elliptic systems by applying the Laguerre transform. A sequence of fundamental solutions to the elliptic equations are derived. Linear combination of elements of this sequence of fundamental solutions is used to generate an approximation to the elliptic Cauchy problems. By placing source points outside of the solution domain, and collocating on the boundary, linear equations are obtained for finding the coefficients in the MFS approximation. It is outlined that the sequence of fundamental solutions of the elliptic systems constitutes a linearly independent and dense set on the boundary with respect to the L-2-norm. Tikhonov regularization in combination with the L-curve rule is incorporated to generate a stable solution to the obtained systems of linear equations. The proposed MFS approximation for the time-dependent lateral Cauchy problem is supported by numerical results.

We present an overview of recent methods based on semidiscretization (in time) for the inverse ill-posed problems of finding the solutions of evolution equations according to the time-like Cauchy data. Specifically, the values of function and normal derivative are given on a portion of the lateral boundary of a space-time cylinder and the corresponding data should be generated on the remaining lateral part of the cylinder either for the heat equation or for the wave equation. The procedure of semidiscretization in time is based on the application either of the Laguerre transform or of the Rothe method (finite-difference approximation), and has a specific feature that similar sequences of elliptic problems are obtained for the heat and wave equations, and only the values of some parameters are different. The elliptic equations are solved numerically either by the boundary integral approach involving the Nystrom method or by the method of fundamental solutions. The theoretical properties are formulated together with discretization strategies in the space. Systems of linear equations are obtained for finding either the values of densities or the coefficients. The Tikhonov regularization is applied for the stable solution of the linear equations. The presented numerical results show that the proposed strategies give good accuracy in combination with economic computational costs.

En omskrivning av cosinussatsen presenteras. Från arbeten av Stjernhjelm och Pitiscus visas att omskrivningen kan tolkas som en tillämpning av sekantsatsen. Vidare erhålls en formel för att beräkna längderna av de två segment som bildas då en sida i en triangel delas av en höjd, givet triangelns sidor. En sådan explicit formel går tillbaka till Ptolemaios. Hänvisningar med historik kring cosinussatsen och trigonometriska tabeller ges också.

A summary in English is given at the end on page 10.

To enhance understanding when analysing the intersection between a cone and a plane, it is helpful to have a simple 3-dimensional paper model of the intersection. In this note, it is shown how the conic sections appear when the cone is unwrapped (flattened). Equations for the flattened conic intersections together with a printable figure and some LaTeX code for experiments are included

We work with a well-known model of reaction-diffusion type for brain tumour growth and accomplish full 3-dimensional (3d) simulations of the tumour in time on two types of imaging data, the 3d Shepp-Logan head phantom image and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository. The source term is such that we have logistic growth. These simulations are obtained using standard finite difference approximations with novel calculations to increase speed and accuracy. Moreover, biological background to the model, its well-posedness together with a variational formulation are given. The variational formulation enable the feasibility of different derivations and modifications of the model.

We propose a nonlinear Landweber method for the inverse problem of locating the brain tumour source (origin where the tumour formed) based on well-established models of reaction-diffusion type for brain tumour growth. The approach consists of recovering the initial density of the tumour cells starting from a later state, which can be given by a medical image, by running the model backwards. Moreover, full three-dimensional simulations are given of the tumour source localization on two types of data, the three-dimensional Shepp-Logan phantom and an MRI T1-weighted brain scan. These simulations are obtained using standard finite difference discretizations of the space and time derivatives, generating a simple approach that performs well.

The problem of reconstructing the solution to a second-order elliptic equation in a doubly-connected domain from knowledge of the solution and its normal derivative on the outer part of the boundary of the solution domain, that is from Cauchy data, is considered. An iterative method is given to generate a stable numerical approximation to this inverse ill-posed problem. The procedure is physically feasible in that boundary data is updated with data of the same type in the iterations, meaning that Dirichlet values is updated with Dirichlet values from the previous step and Neumann values by Neumann data. Proof of convergence and stability are given by showing that the proposed method is an extension of the Landweber method for an operator equation reformulation of the Cauchy problem. Connection with the alternating method is discussed. Numerical examples are included confirming the feasibility of the suggested approach.