Department of Mathematics

Spectrum analysis can detect frequency related structures in a time series $\{Y_t\}_{t\in\mathbb{Z}}$, but may in general be an inadequate tool if asymmetries or other nonlinear phenomena are present. This limitation is a consequence of the way the spectrum is based on the second order moments (auto and cross-covariances), and alternative approaches to spectrum analysis have thus been investigated based on other measures of dependence. One such approach was developed for univariate time series in Jordanger and Tjostheim (2017), where it was seen that a local Gaussian auto-spectrum $f_{v}(\omega)$, based on the local Gaussian autocorrelations $\rho_v(\omega)$ from Tjostheim and Hufthammer (2013), could detect local structures in time series that looked like white noise when investigated by the ordinary auto-spectrum $f(\omega)$. The local Gaussian approach in this paper is extended to a local Gaussian cross-spectrum $f_{kl:v}(\omega)$ for multivariate time series. The local cross-spe...

This thesis will consider the performance of the cross-validation copula information criterion, xv-CIC, in the realm of finite samples. The theory leading to the xv-CIC will be outlined, and an analysis will be conducted on an assorted collection of bivariate one-parameter copula models. The restriction to the bivariate case is not a grave one, since more complex d-variate samples can be broken down into a study of conditioned bivariate samples, by the methodology of regular vine-copulas, the pair copula construction and stepwise-semiparametric estimation of parameters. As a by-product of our analysis, we can give an advice with regard to the selection of model selection method in the semiparametric realm. Acknowledgements First and foremost, I would like to thank Prof. Dag Tjøstheim, Ph.D., for his excellent supervision during the writing of this thesis. Furthermore, I would also like to thank both my fellow Master’s Degree students and the staff at the Department of Mathematics, f...

The spectral distribution $f(\omega)$ of a stationary time series $\{Y_t\}_{t\in\mathbb{Z}}$ can be used to investigate whether or not periodic structures are present in $\{Y_t\}_{t\in\mathbb{Z}}$, but $f(\omega)$ has some limitations due to its dependence on the autocovariances $\gamma(h)$. For example, $f(\omega)$ can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega)$ can be an inadequate tool when $\{Y_t\}_{t\in\mathbb{Z}}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations $\gamma_{v}(h)$ introduced in Tj{\o}stheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density $f_{v}(\omega)$ that is presented in this paper. A key feature of $f_{v}(\omega)$ is that it coincides with $f(\omega)$ for Gaussian ti...

The ordinary spectrum is restricted in its applications, since it is based on the second-order moments (auto- and cross-covariances). Alternative approaches to spectrum analysis have been investigated based on other measures of dependence. One such approach was developed for univariate time series by the authors of this paper using the local Gaussian auto-spectrum based on the local Gaussian auto-correlations. This makes it possible to detect local structures in univariate time series that look similar to white noise when investigated by the ordinary auto-spectrum. In this paper, the local Gaussian approach is extended to a local Gaussian cross-spectrum for multivariate time series. The local Gaussian cross-spectrum has the desirable property that it coincides with the ordinary cross-spectrum for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if the ordinary spectrum is flat, then peaks and...

We propose a solution strategy for parameter estimation, where we combine adaptive
multiscale estimation (AME) and level-set estimation (LSE). The approach is applied to the
nonlinear inverse problem of recovering a coefficient function in a system of differential equations
from spatially sparsely distributed measurement data. The specific equations considered in this paper
describe two-phase porous-media flow where a coefficient function defining absolute permeability
(fluid conductivity) is estimated based on fluid pressure observations in wells. This inverse problem
is known to be ill-posed.
The spatial variability of the sought coefficient function is unknown and will typically vary within
the porous medium. Due to limited information in the available data, mainly coarse-scale features of
the existing variability in the coefficient function will be attainable. In AME, one starts out with a
single parameter representation of the sought function, whereafter the domain is successively divided
into finer rectangular zones, each representing a constant parameter value of the coefficient function.
The strong restrictions on the zone geometry may lead to overparameterization. LSE is a method
for moving curves and enables adjustments of the zone structure into more general geometries. In
order to perform well, LSE requires a reasonable starting point. In this paper, we have developed
a methodology to combine AME and LSE, where at each step either refinements or deformation of
the zone structure may be conducted, depending on which method promises the better result. The
combined approach seems promising with respect to recovering coarse-scale features of the sought
coefficient functions with a low number of parameters.

We consider the inverse problem of permeability estimation for two-phase porous-media flow. The novel approach is based on regularization by zonation, where the geometry and size of the regions are chosen adaptively during the optimization procedure. To achieve this, we have utilized level-set functions to represent the permeability. The available data are sparsely distributed in space; hence, it is reasonable to confine the estimation to coarse-scale structures. The level-set approach is able to alter the boundaries between regions of different permeability without strict restrictions on their shape; however, when the data are sparse, a reasonable initial guess for the permeability is required. For this task, we use adaptive multiscale permeability estimation, which has the potential of identifying main permeability variations. These are described by a piecewise constant function, where the constant values are attained on rectangular zones. In the current work, we develop a level-set corrector strategy, assuming adaptive multiscale permeability estimation as a predictor.

We present a novel solution algorithm for 3D parameter identification based on low frequency electromagnetic data. With focus on large-scale applications such as monitoring of subsea oil production, CO2 sequestration, and geothermal systems, the proposed solution algorithm is designed to meet challenges related to low parameter sensitivity, nonuniqueness of the inverse solutions, nonlinearity in the mapping from the data to the parameter space, and costly numerical simulations. Motivated by earlier investigations on the relation between sensitivity, nonlinearity and scale, the proposed solution approach is based on a reduced, composite parameter representation. Though a reduced representation restricts the solution space, flexibility with respect to which parameter functions that can be represented is obtained by facilitating the estimation of the structure and smoothness of the representation itself. Moreover, the resolution of the parameter function is detached from the computational grid and determined as part of the estimation. The performance of the proposed solution algorithm is illustrated through numerical examples for identification of underground electric conductivity changes from time-lapse electromagnetic observations.

Level-set methods are popular for identifying piecewise constant structures. We propose an approach inspired by the level-set idea to identify coarse scale features of a scalar field, where the transitions between different regions can be both sharp and smooth. The nonlinear and coarse reparameterization structure provides regularization on the inverse problem, which is important if the quality of the available information is poor.

In our identification strategy, the resolution of the representation of the parameter function is refined gradually; for several inverse problems, the nonlinearity of the problem is correspondingly carefully increased. Hence, when using a gradient-based optimization routine, the approach may reduce the risk of getting trapped in a local minimum of the objective function.

We demonstrate the identification strategy for estimation of fluid conductivity in a porous medium based on sparsely distributed, transient data. The methodology is well suited for determining channels and barriers as well as smooth structures based on limited information.

In the last decade, the use of level-set functions has gained increasing popularity in solving inverse problems involving the identification of a piecewise constant function. Normally, a fine-scale representation of the level-set functions is used, yielding a high number of degrees of freedom in the estimation. In contrast, we focus on reparameterization of the level-set functions on a coarse scale. The number of coefficients in the discretized function is then reduced, providing necessary regularization for solving ill-posed problems. A coarse representation is also advantageous to reduce the computational work in solving the estimation problem.

We consider identification of absolute permeability (hydraulic conductivity) based on time series of pressure data in sparsely distributed wells for two-phase porous-media flow. For this problem, it is impossible to recover all details of the parameter function. On the other hand, a coarser, approximate recovery may be sufficient for many applications. We propose a novel solution approach, based on reparametrization, for such approximate identification of the parameter function. We use a nonlinear, composite representation, which is detached from the computational grid, allowing for a flexible representation of the parameter function at many resolution levels. This is utilized in a sequential multi-level estimation of the parameter function, starting at a coarse resolution, which is then gradually refined. The composite representation is designed to allow for smooth as well as sharp transitions between regions of nearly constant parameter value. Moreover, it facilitates the estimation also of the structure and smoothness of the parameter function itself. As a limiting case, the chosen representation is reduced to a zonation with implicit representation of the interior boundaries that is equivalent to a level-set representation. A motivation for the selected representation and the multi-level estimation is presented in terms of an analysis of sensitivity and nonlinearity. Numerical examples demonstrate identification of coarse-scale features of reference permeability distributions with varying degree of smoothness. Comparisons show how the multi-level strategy stabilize the identification and avoid local minima of the objective function compared to a single-level strategy.

We consider a discontinuous Galerkin scheme for computing transport in heterogeneous media. An efficient solution of the resulting linear system of equations is possible by taking advantage of a priori knowledge of the direction of flow. By arranging the elements in a suitable sequence, one does not need to assemble the full system and may compute the solution in an element-by-element fashion. We demonstrate this procedure on boundary-value problems for tracer transport and time-of-flight.

We continue the work that was initiated in (K. H. Karlsen, K.-A. Lie, and N. H. Risebro. A fast marching method for reservoir simulation. Comp. Geo., 4(2) (2000)185–206) on a marching method for simulating two-phase incompressible immiscible flow of water and oil in a porous medium. We first present an alternative derivation of the marching method that reveals a strong connection to modern streamline methods. Then, through the study of three numerical test cases we present two deficiencies: (i) the original marching algorithm does not always compute the correct solution of the underlying difference equations, and (ii) the method gives largely inaccurate arrival times in the presence of large jumps within the upwind difference stencil. As a remedy of the first problem, we present a new advancing-front method, which is faster than the original marching method and guarantees a correct solution of the underlying discrete linear system. To cure the second problem, we present two adaptive strategies that avoid the use of finite-difference stencils containing large jumps in the arrival times. The original marching method was introduced as a fast tool for simulating two-phase flow scenarios in heterogeneous formations. The new advancing-front method has limited applicability in this respect, but may rather be used as a fast and relatively accurate method for computing arrival times and derived quantities in heterogeneous media.

In this work we consider models of multi-phase flow which do include capillary forces.
We also allow for three phases. In particular we shall investigate a streamline front tracking
method (SFTM) [1]. This method is based on calculating streamlines locally around grid
points. The hyperbolic part of the problem is then calculated using front tracking. Since the
front tracking method requires a Riemann solver, we do currently only consider a triangular
model for three phase flow, i.e., the advective flux of the gas phase does only depend on
gas-saturation and not on the other saturations. To account for capillary effects we use
operator splitting [6]. The hyperbolic part of the solution is projected onto a regular grid,
and a complete solution in each time step is obtained by solving a parabolic equation.
The paper is organized as follows: In Section 2 we state a standard model for twoand
three-phase flow. In Section 3 the streamline front tracking method (SFTM) [1] is
developed. A modified method of characteristics (MMOC) [5] is also briefly discussed
and we we give an outline of a fast marching method (FMM) [7]. These methods are used
in the two-phase case as comparisons with the SFTM-method proposed here. In Section 4
we discuss a possible extension of the SFTM-method to three phase flow. This extension is
based on a triangular approximation of the fully coupled three-phase problem, and the socalled
H-set method is used to solve associated Riemann problems. Some consequences
of the triangular approximation are also discussed. Finally, in Section 5 we present some
numerical experiments for two- and three-phase flow to demonstrate the methodology, and
a summary and conclusions are given in Section 6.

We uncover novel features of three-dimensional natural convection in porous media by investigating convection in an annular porous cavity contained between two vertical coaxial cylinders. The investigations are made using a linear stability analysis, together with high-order numerical simulations using pseudospectral methods to model the nonlinear regime. The onset of convection cells and their preferred planform are studied, and the stability of the modes with respect to different types of perturbation is investigated. We also examine how variations in the Rayleigh number affect the convection modes and their stability regimes. Compared with previously published data, we show how the problem inherits an increased complexity regarding which modes will be obtained. Some stable secondary modes or mixed modes have been identified and some overlapping stability regions for different convective modes are determined.