arXiv subjects

Tommi Brander

Publications and source records attributed to Tommi Brander.

12 recordsLinked to original sources

arXiv2024

Inverting the sound speed profile from multi-beam echo sounder data and historical measurements -- a simulation study

The ocean's opacity poses challenges for security, as new technology, e.g. underwater drones, offers new opportunities for illegal activities, such as smuggling and terrorism. A network of unmanned surface vehicles (USV) and autonomous underwater vehicles (AUV) offers a potential underwater surveillance solution, but demands high autonomy and compact hardware. For improved situational awareness and efficient operation, sonar performance models may provide the network with sensor coverage maps, but this requires constantly updated environmental information, in particular the present sound speed profile (SSP). We propose the inversion of SSPs from multibeam echo sounder (MBES) data in an environment with known topography. The method exploits the two-way travel time from the MBES to the bottom, comparing the measurements to modelled travel time for a proposed SSP model. An acoustic raytracer models the travel time for the SSP model. The inversion problem is shown to be non-unique when basing the cost function on the two-way travel time alone. This is resolved by incorporating a Tikhonov-type regularization term for inclusion of a priori knowledge on the SSPs in addition to the travel time in the final cost function. Empirical orthogonal functions (EOFs) are derived from a historical SSP data set, and variance for the EOF coefficients are determined from the same data set. The EOF coefficient distributions are assumed Gaussian and used in the regularization term to limit the search space of the inversion algorithm to physically feasible SSPs. A neural network determines the regularization parameters. The method's validity and sensitivity to errors is assessed using synthetic sonar data for the Norwegian Trench. The method accurately recovers SSPs with average root mean square errors of 0.83 m/s. For comparison, the error obtained using state-of-the-art climatology (WOA) is 2.6 m/s.

eess.SP↗

arXiv2023

Inverse problems for a model of biofilm growth

A bacterial biofilm is an aggregate of micro-organisms growing fixed onto a solid surface, rather than floating freely in a liquid. Biofilms play a major role in various practical situations such as surgical infections and water treatment. We consider a non-linear PDE model of biofilm growth subject to initial and Dirichlet boundary conditions, and the inverse coefficient problem of recovering the unknown parameters in the model from extra measurements of quantities related to the biofilm and substrate. By addressing and analysing this inverse problem we provide reliable and robust reconstructions of the primary physical quantities of interest represented by the diffusion coefficients of substrate and biofilm, the biomass spreading parameters, the maximum specific consumption and growth rates, the biofilm decay rate and the half saturation constant. We give particular attention to the constant coefficients involved in the leading-part non-linearity, and present a uniqueness proof and some numerical results. In the course of the numerical investigation, we have identified extra data information that enables improving the reconstruction of the eight-parameter set of physical quantities associated to the model of biofilm growth.

math.AP↗

arXiv2020

Optimal recovery of a radiating source with multiple frequencies along one line

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

math.FA↗

arXiv2020

Recovering a variable exponent

We consider an inverse problem of recovering the non-linearity in the one dimensional variable exponent $p(x)$-Laplace equation from the Dirichlet-to-Neumann map. The variable exponent can be recovered up to the natural obstruction of rearrangements. The main technique is using a Müntz-Szász theorem after reducing the problem to determining a function from its $L^p$-norms.

math.AP↗

arXiv2019

Boundary determination for hybrid imaging from a single measurement

We recover the conductivity $\sigma$ at the boundary of a domain from a combination of interior and boundary data, with a single quite arbitrary measurement, in AET or CDII. The argument is elementary and local. More generally, we consider the variable exponent $p(\cdot)$-Laplacian as a forward model with the interior data $\sigma |\nabla u|^q$, and find out that single measurement specifies the boundary conductivity when $p-q \ge 1$, and otherwise the measurement specifies two alternatives. We present heuristics for selecting between these alternatives. Both $p$ and $q$ may depend on the spatial variable $x$, but they are assumed to be a priori known. We illustrate the practical situations with numerical examples.

math.NA↗

arXiv2019

Variable exponent Calderón's problem in one dimension

We consider one-dimensional Calderón's problem for the variable exponent $p(\cdot)$-Laplace equation and find out that more can be seen than in the constant exponent case. The problem is to recover an unknown weight (conductivity) in the weighted $p(\cdot)$-Laplace equation from Dirichlet and Neumann data of solutions. We give a constructive and local uniqueness proof for conductivities in $L^\infty$ restricted to the coarsest sigma-algebra that makes the exponent $p(\cdot)$ measurable.

math.AP↗

arXiv2019

Expected characteristic in Tunnels & Trolls character creation, with generalizations

In the roleplaying game Tunnels & Trolls the characteristics of player characters are determined by rolling dice in the following manner: First, one rolls three dice and calculates their sum. If the three dice all give the same result, another three dice are rolled and added to the total. This is continued until the three dice no longer match. We calculate the average result of the stochastic sum: 10 + 4/5. We also consider a generalized dice rolling scheme where we roll an arbitrary number of dice with arbitrary number of sides. This generalization is motivated by various exotic dice that are used in many roleplaying games. We calculate the expectation, and how much it differs from the situation where we only roll the set of dice once, with no rerolling and adding. As the number of dice increases, or the number of sides the dice have increases, this difference approaches zero, unless there are two dice (with the number of sides increasing), in which case the difference approaches one.

math.PR↗

arXiv2017

Superconductive and insulating inclusions for linear and non-linear conductivity equations

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $σ\colonΩ\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.

math.AP↗

arXiv2017

Monotonicity and enclosure methods for the p-Laplace equation

We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.

math.AP↗

arXiv2016

Calderón's problem for p-Laplace type equations

We investigate a generalization of Calderón's problem of recovering the conductivity coefficient in a conductivity equation from boundary measurements. As a model equation we consider the p-conductivity equation with p strictly between one and infinity, which reduces to the standard conductivity equation when p equals two, and to the p-Laplace equation when the conductivity is constant. The thesis consists of results on the direct problem, boundary determination and detecting inclusions. We formulate the equation as a variational problem also when the conductivity may be zero or infinity in large sets. As a boundary determination result we recover the first order derivative of a smooth conductivity on the boundary. We use the enclosure method of Ikehata to recover the convex hull of an inclusion of finite conductivity and find an upper bound for the convex hull if the conductivity within an inclusion is zero or infinite.

math.AP↗

arXiv2015

Enclosure method for the p-Laplace equation

We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.

math.AP↗

arXiv2014

Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary

We recover the gradient of a scalar conductivity defined on a smooth bounded open set in $\mathbb{R}^d$ from the Dirichlet to Neumann map arising from the $p$-Laplace equation. For any boundary point we recover the gradient using Dirichlet data supported on an arbitrarily small neighbourhood of the boundary point. We use a Rellich-type identity in the proof. Our results are new when $p \neq 2$. In the $p = 2$ case boundary determination plays a role in several methods for recovering the conductivity in the interior.

math.AP↗