| 1. |
- Berg, Jens, 1982-
(author)
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Stable and High-Order Finite Difference Methods for Multiphysics Flow Problems
- 2013
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Doctoral thesis (other academic/artistic)abstract
- Partial differential equations (PDEs) are used to model various phenomena in nature and society, ranging from the motion of fluids and electromagnetic waves to the stock market and traffic jams. There are many methods for numerically approximating solutions to PDEs. Some of the most commonly used ones are the finite volume method, the finite element method, and the finite difference method. All methods have their strengths and weaknesses, and it is the problem at hand that determines which method that is suitable. In this thesis, we focus on the finite difference method which is conceptually easy to understand, has high-order accuracy, and can be efficiently implemented in computer software.We use the finite difference method on summation-by-parts (SBP) form, together with a weak implementation of the boundary conditions called the simultaneous approximation term (SAT). Together, SBP and SAT provide a technique for overcoming most of the drawbacks of the finite difference method. The SBP-SAT technique can be used to derive energy stable schemes for any linearly well-posed initial boundary value problem. The stability is not restricted by the order of accuracy, as long as the numerical scheme can be written in SBP form. The weak boundary conditions can be extended to interfaces which are used either in domain decomposition for geometric flexibility, or for coupling of different physics models.The contributions in this thesis are twofold. The first part, papers I-IV, develops stable boundary and interface procedures for computational fluid dynamics problems, in particular for problems related to the Navier-Stokes equations and conjugate heat transfer. The second part, papers V-VI, utilizes duality to construct numerical schemes which are not only energy stable, but also dual consistent. Dual consistency alone ensures superconvergence of linear integral functionals from the solutions of SBP-SAT discretizations. By simultaneously considering well-posedness of the primal and dual problems, new advanced boundary conditions can be derived. The new duality based boundary conditions are imposed by SATs, which by construction of the continuous boundary conditions ensure energy stability, dual consistency, and functional superconvergence of the SBP-SAT schemes.
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| 2. |
- Hellander, Andreas, 1982-
(author)
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Multiscale Stochastic Simulation of Reaction-Transport Processes : Applications in Molecular Systems Biology
- 2011
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Doctoral thesis (other academic/artistic)abstract
- Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution.
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| 3. |
- Ahlkrona, Josefin, 1985-
(author)
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Computational Ice Sheet Dynamics : Error control and efficiency
- 2016
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Doctoral thesis (other academic/artistic)abstract
- Ice sheets, such as the Greenland Ice Sheet or Antarctic Ice Sheet, have a fundamental impact on landscape formation, the global climate system, and on sea level rise. The slow, creeping flow of ice can be represented by a non-linear version of the Stokes equations, which treat ice as a non-Newtonian, viscous fluid. Large spatial domains combined with long time spans and complexities such as a non-linear rheology, make ice sheet simulations computationally challenging. The topic of this thesis is the efficiency and error control of large simulations, both in the sense of mathematical modelling and numerical algorithms. In the first part of the thesis, approximative models based on perturbation expansions are studied. Due to a thick boundary layer near the ice surface, some classical assumptions are inaccurate and the higher order model called the Second Order Shallow Ice Approximation (SOSIA) yields large errors. In the second part of the thesis, the Ice Sheet Coupled Approximation Level (ISCAL) method is developed and implemented into the finite element ice sheet model Elmer/Ice. The ISCAL method combines the Shallow Ice Approximation (SIA) and Shelfy Stream Approximation (SSA) with the full Stokes model, such that the Stokes equations are only solved in areas where both the SIA and SSA is inaccurate. Where and when the SIA and SSA is applicable is decided automatically and dynamically based on estimates of the modeling error. The ISCAL method provides a significant speed-up compared to the Stokes model. The third contribution of this thesis is the introduction of Radial Basis Function (RBF) methods in glaciology. Advantages of RBF methods in comparison to finite element methods or finite difference methods are demonstrated.
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| 4. |
- Amoignon, Olivier, 1969-
(author)
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Numerical Methods for Aerodynamic Shape Optimization
- 2005
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Doctoral thesis (other academic/artistic)abstract
- Gradient-based aerodynamic shape optimization, based on Computational Fluid Dynamics analysis of the flow, is a method that can automatically improve designs of aircraft components. The prospect is to reduce a cost function that reflects aerodynamic performances.When the shape is described by a large number of parameters, the calculation of one gradient of the cost function is only feasible by recourse to techniques that are derived from the theory of optimal control. In order to obtain the best computational efficiency, the so called adjoint method is applied here on the complete mapping, from the parameters of design to the values of the cost function. The mapping considered here includes the Euler equations for compressible flow discretized on unstructured meshes by a median-dual finite-volume scheme, the primal-to-dual mesh transformation, the mesh deformation, and the parameterization. The results of the present research concern the detailed derivations of expressions, equations, and algorithms that are necessary to calculate the gradient of the cost function. The discrete adjoint of the Euler equations and the exact dual-to-primal transformation of the gradient have been implemented for 2D and 3D applications in the code Edge, a program of Computational Fluid Dynamics used by Swedish industries.Moreover, techniques are proposed here in the aim to further reduce the computational cost of aerodynamic shape optimization. For instance, an interpolation scheme is derived based on Radial Basis Functions that can execute the deformation of unstructured meshes faster than methods based on an elliptic equation.In order to improve the accuracy of the shape, obtained by numerical optimization, a moving mesh adaptation scheme is realized based on a variable diffusivity equation of Winslow type. This adaptation has been successfully applied on a simple case of shape optimization involving a supersonic flow. An interpolation technique has been derived based on a mollifier in order to improve the convergence of the coupled mesh-flow equations entering the adaptive scheme.The method of adjoint derived here has also been applied successfully when coupling the Euler equations with the boundary-layer and parabolized stability equations, with the aim to delay the laminar-to-turbulent transition of the flow. The delay of transition is an efficient way to reduce the drag due to viscosity at high Reynolds numbers.
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| 5. |
- Bängtsson, Erik, 1975-
(author)
-
Robust Preconditioners Based on the Finite Element Framework
- 2007
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Doctoral thesis (other academic/artistic)abstract
- Robust preconditioners on block-triangular and block-factorized form for three types of linear systems of two-by-two block form are studied in this thesis. The first type of linear systems, which are dense, arise from a boundary element type of discretization of crack propagation problems. Numerical experiment show that simple algebraic preconditioning strategies results in iterative schemes that are highly competitive with a direct solution method. The second type of algebraic systems, which are sparse, indefinite and nonsymmetric, arise from a finite element (FE) discretization of the partial differential equations (PDE) that describe (visco)elastic glacial isostatic adjustment (GIA). The Schur complement approximation in the block preconditioners is constructed by assembly of local, exactly computed Schur matrices. The quality of the approximation is verified in numerical experiments. When the block preconditioners for the indefinite problem are combined with an inner iterative scheme preconditioned by a (nearly) optimal multilevel preconditioner, the resulting preconditioner is (nearly) optimal and robust with respect to problem size, material parameters, number of space dimensions, and coefficient jumps. Two approaches to mathematically formulate the PDEs for GIA are compared. In the first approach the equations are formulated in their full complexity, whereas in the second their formulation is confined to the features and restrictions of the employed FE package. Different solution methods for the algebraic problem are used in the two approaches. Analysis and numerical experiments reveal that the first strategy is more accurate and efficient than the latter. The block structure in the third type of algebraic systems is due to a fine-coarse splitting of the unknowns. The inverse of the pivot block is approximated by a sparse matrix which is assembled from local, exactly inverted matrices. Numerical experiments and analysis of the approximation show that it is robust with respect to problem size and coefficient jumps.
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| 6. |
- Cheng, Gong, 1986-
(author)
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Numerical ice sheet modeling : Forward and inverse problems
- 2019
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Doctoral thesis (other academic/artistic)abstract
- Ice sheets have strong influence on the climate system. Numerical simulation provides a mathematical tool to study the ice dynamics in the past and to predict their contribution to climate change in the future. Large scale ice sheets behave as incompressible non-Newtonian fluid. The evolution of ice sheet is governed by the conservation laws of mass, momentum and energy, which is formulated as a system of partial differential equations. Improving the efficiency of numerical ice sheet modeling is always a desirable feature since many of the applications have large domain and aim for long time span. With such a goal, the first part of this thesis focuses on developing efficient and accurate numerical methods for ice sheet simulation.A large variety of physical processes are involved in ice dynamics, which are described by physical laws with parameters measured from experiments and field work. These parameters are considered as the inputs of the ice sheet simulations. In certain circumstances, some parameters are unavailable or can not be measured directly. Therefore, the second part of this thesis is devoted to reveal these physical parameters by solving inverse problems.In the first part, improvements of temporal and spatial discretization methods and a sub-grid boundary treatment are purposed. We developed an adaptive time stepping method in Paper I to automatically adjust the time steps based on stability and accuracy criteria. We introduced an anisotropic Radial Basis Function method for the spatial discretization of continental scale ice sheet simulations in Paper II. We designed a sub-grid method for solving grounding line migration problem with Stokes equations in Paper VI.The second part of the thesis consists of analysis and numerical experiments on inverse problems. In Paper IV and V, we conducted sensitivity analysis and numerical examples of the inversion on time dependent ice sheet simulations. In Paper III, we solved an inverse problem for the thermal conductivity of firn pack at Lomonosovfonna, Svalbard, using the subsurface temperature measurements.
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| 7. |
- Edelvik, Fredrik, 1972-
(author)
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Hybrid Solvers for the Maxwell Equations in Time-Domain
- 2002
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Doctoral thesis (other academic/artistic)abstract
- The most commonly used method for the time-domain Maxwell equations is the Finite-Difference Time-Domain method (FDTD). This is an explicit, second-order accurate method, which is used on a staggered Cartesian grid. The main drawback with the FDTD method is its inability to accurately model curved objects and small geometrical features. This is due to the Cartesian grid, which leads to a staircase approximation of the geometry and small details are not resolved at all.This thesis presents different ways to circumvent this drawback, but still take advantage of the benefits of the FDTD method. An approach to avoid staircasing errors but still retain the efficiency of the FDTD method is to use a hybrid grid. A few layers of unstructured cells are used close to curved objects and a Cartesian grid is used for the rest of the domain. For the choice of solver on the unstructured grid two different alternatives are compared: an explicit Finite-Volume Time-Domain (FVTD) solver and an implicit Finite-Element Time-Domain (FETD) solver.The hybrid solvers calculate the scattering from complex objects much more efficiently compared to using FDTD on highly resolved Cartesian grids. For the same accuracy in the solution roughly a factor of 10 in memory requirements and a factor of 20 in execution time are gained.The ability to model features that are small relative to the cell size is often important in electromagnetic simulations. In this thesis a technique to generalize a well-known subcell model for thin wires, in order to take arbitrarily oriented wires in FETD and FDTD into account, is proposed. The method gives considerable modeling flexibility compared to earlier methods and is proven stable. The results show excellent consistency and very good accuracy on different antenna configurations.The recursive convolution method is often used to model frequency dispersive materials in FDTD. This method is used to enable modeling of such materials in the unstructured FVTD and FETD solvers. The stability of both solvers is analyzed and their accuracy is demonstrated by computing the radar cross section for homogeneous as well as layered spheres with frequency dependent permittivity.
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| 8. |
- Ekström, Sven-Erik
(author)
-
A vertex-centered discontinuous Galerkin method for flow problems
- 2016
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Licentiate thesis (other academic/artistic)abstract
- The understanding of flow problems, and finding their solution, has been important for most of human history, from the design of aqueducts to boats and airplanes. The use of physical miniature models and wind tunnels were, and still are, useful tools for design, but with the development of computers, an increasingly large part of the design process is assisted by computational fluid dynamics (CFD).Many industrial CFD codes have their origins in the 1980s and 1990s, when the low order finite volume method (FVM) was prevalent. Discontinuous Galerkin methods (DGM) have, since the turn of the century, been seen as the successor of these methods, since it is potentially of arbitrarily high order. In its lowest order form DGM is equivalent to FVM. However, many existing codes are not compatible with standard DGM and would need a complete rewrite to obtain the advantages of the higher order.This thesis shows how to extend existing vertex-centered and edge-based FVM codes to higher order, using a special kind of DGM discretization, which is different from the standard cell-centered type. Two model problems are examined to show the necessary data structures that need to be constructed, the order of accuracy for the method, and the use of an hp-adaptation scheme to resolve a developing shock. Then the method is further developed to solve the steady Euler equations, within the existing industrial Edge code, using acceleration techniques such as local time stepping and multigrid.With the ever increasing need for more efficient and accurate solvers and algorithms in CFD, the modified DGM presented in this thesis could be used to help and accelerate the adoption of high order methods in industry.
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| 9. |
- Hellander, Stefan, 1985-
(author)
-
Stochastic Simulation of Reaction-Diffusion Processes
- 2013
-
Doctoral thesis (other academic/artistic)abstract
- Numerical simulation methods have become an important tool in the study of chemical reaction networks in living cells. Many systems can, with high accuracy, be modeled by deterministic ordinary differential equations, but other systems require a more detailed level of modeling. Stochastic models at either the mesoscopic level or the microscopic level can be used for cases when molecules are present in low copy numbers.In this thesis we develop efficient and flexible algorithms for simulating systems at the microscopic level. We propose an improvement to the Green's function reaction dynamics algorithm, an efficient microscale method. Furthermore, we describe how to simulate interactions with complex internal structures such as membranes and dynamic fibers.The mesoscopic level is related to the microscopic level through the reaction rates at the respective scale. We derive that relation in both two dimensions and three dimensions and show that the mesoscopic model breaks down if the discretization of space becomes too fine. For a simple model problem we can show exactly when this breakdown occurs.We show how to couple the microscopic scale with the mesoscopic scale in a hybrid method. Using the fact that some systems only display microscale behaviour in parts of the system, we can gain computational time by restricting the fine-grained microscopic simulations to only a part of the system.Finally, we have developed a mesoscopic method that couples simulations in three dimensions with simulations on general embedded lines. The accuracy of the method has been verified by comparing the results with purely microscopic simulations as well as with theoretical predictions.
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| 10. |
- Monokrousos, Antonios
(author)
-
Optimisation and control of boundary layer flows
- 2009
-
Licentiate thesis (other academic/artistic)abstract
- Both optimal disturbances and optimal control are studied by means of numerical simulations for the case of the flat-plate boundary-layer flow. The optimisation method is the Lagrange multiplier technique where the objective function is the kinetic energy of the flow perturbations and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Control is applied to the bypass-transition scenario with high levels of free-stream turbulence. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to the non-modal growth. These so-called streaks are growing in amplitude until they reach high enough energy levels and breakdown into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, within the region that it is active, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced.
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| 11. |
- Nilsson, Martin, 1974-
(author)
-
Fast Numerical Techniques for Electromagnetic Problems in Frequency Domain
- 2003
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Doctoral thesis (other academic/artistic)abstract
- The Method of Moments is a numerical technique for solving electromagnetic problems with integral equations. The method discretizes a surface in three dimensions, which reduces the dimension of the problem with one. A drawback of the method is that it yields a dense system of linear equations. This effectively prohibits the solution of large scale problems. Papers I-III describe the Fast Multipole Method. It reduces the cost of computing a dense matrix vector multiplication. This implies that large scale problems can be solved on personal computers. In Paper I the error introduced by the Fast Multipole Method is analyzed. Paper II and Paper III describe the implementation of the Fast Multipole Method. The problem of computing monostatic Radar Cross Section involves many right hand sides. Since the Fast Multipole Method computes a matrix times a vector, iterative techniques are used to solve the linear systems. It is important that the solution time for each system is as low as possible. Otherwise the total solution time becomes too large. Different techniques for reducing the work in the iterative solver are described in Paper IV-VI. Paper IV describes a block Quasi Minimal Residual method for several right hand sides and Sparse Approximate Inverse preconditioner that reduce the number of iterations significantly. In Paper V and Paper VI a method based on linear algebra called the Minimal Residual Interpolation method is described. It reduces the work in an iterative solver by accurately computing an initial guess for the iterative method. In Paper VII a hybrid method between Physical Optics and the Fast Multipole Method is described. It can handle large problems that are out of reach for the Fast Multipole Method.
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| 12. |
- O'Reilly, Ossian, 1986-
(author)
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Numerical methods for wave propagation in solids containing faults and fluid-filled fractures
- 2016
-
Doctoral thesis (other academic/artistic)abstract
- This thesis develops numerical methods for the simulation of wave propagation in solids containing faults and fluid-filled fractures. These techniques have applications in earthquake hazard analysis, seismic imaging of reservoirs, and volcano seismology. A central component of this work is the coupling of mechanical systems. This aspect involves the coupling of both ordinary differential equations (ODE)(s) and partial differential equations (PDE)(s) along curved interfaces. All of these problems satisfy a mechanical energy balance. This mechanical energy balance is mimicked by the numerical scheme using high-order accurate difference approximations that satisfy the principle of summation by parts, and by weakly enforcing the coupling conditions. The first part of the thesis considers the simulation of dynamic earthquake ruptures along non-planar fault geometries and the simulation of seismic wave radiation from earthquakes, when the earthquakes are idealized as point moment tensor sources. The dynamic earthquake rupture process is simulated by coupling the elastic wave equation at a fault interface to nonlinear ODEs that describe the fault mechanics. The fault geometry is complex and treated by combining structured and unstructured grid techniques. In other applications, when the earthquake source dimension is smaller than wavelengths of interest, the earthquake can be accurately described by a point moment tensor source localized at a single point. The numerical challenge is to discretize the point source with high-order accuracy and without producing spurious oscillations.The second part of the thesis presents a numerical method for wave propagation in and around fluid-filled fractures. This problem requires the coupling of the elastic wave equation to a fluid inside curved and branching fractures in the solid. The fluid model is a lubrication approximation that incorporates fluid inertia, compressibility, and viscosity. The fracture geometry can have local irregularities such as constrictions and tapered tips. The numerical method discretizes the fracture geometry by using curvilinear multiblock grids and applies implicit-explicit time stepping to isolate and overcome stiffness arising in the semi-discrete equations from viscous diffusion terms, fluid compressibility, and the particular enforcement of the fluid-solid coupling conditions. This numerical method is applied to study the interaction of waves in a fracture-conduit system. A methodology to constrain fracture geometry for oil and gas (hydraulic fracturing) and volcano seismology applications is proposed.The third part of the thesis extends the summation-by-parts methodology to staggered grids. This extension reduces numerical dispersion and enables the formulation of stable and high-order accurate multiblock discretizations for wave equations in first order form on staggered grids. Finally, the summation-by-parts methodology on staggered grids is further extended to second derivatives and used for the treatment of coordinate singularities in axisymmetric wave propagation.
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| 13. |
- Persson, Jonas, 1976-
(author)
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Accurate Finite Difference Methods for Option Pricing
- 2006
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Doctoral thesis (other academic/artistic)abstract
- Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance levels. The adaptive second order method is compared to an alternative discretization technique using radial basis functions. This method is not adaptive but shows potential in option pricing for one and several underlying assets. A finite difference method and a Monte Carlo method are applied to a new financial contract called Turbo warrant. A comparison of these two methods shows that for the case considered the finite difference method is superior.
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| 14. |
- Sjöberg, Paul, 1976-
(author)
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Numerical Methods for Stochastic Modeling of Genes and Proteins
- 2007
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Doctoral thesis (other academic/artistic)abstract
- Stochastic models of biochemical reaction networks are used for understanding the properties of molecular regulatory circuits in living cells. The state of the cell is defined by the number of copies of each molecular species in the model. The chemical master equation (CME) governs the time evolution of the the probability density function of the often high-dimensional state space. The CME is approximated by a partial differential equation (PDE), the Fokker-Planck equation and solved numerically. Direct solution of the CME rapidly becomes computationally expensive for increasingly complex biological models, since the state space grows exponentially with the number of dimensions. Adaptive numerical methods can be applied in time and space in the PDE framework, and error estimates of the approximate solutions are derived. A method for splitting the CME operator in order to apply the PDE approximation in a subspace of the state space is also developed. The performance is compared to the most widely spread alternative computational method.
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| 15. |
- Abenius, Erik, 1971-
(author)
-
Direct and Inverse Methods for Waveguides and Scattering Problems in the Time Domain
- 2005
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Doctoral thesis (other academic/artistic)abstract
- Numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment. Different topics for time-domain finite-difference (FDTD) and finite-element (FETD) methods for Maxwell's equations are treated in this thesis. Subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid. A novel model for thin sheets using shell elements is proposed. This approach has the advantage of taking into account discontinuities in the normal component of the electric field, unlike previous models based on impedance boundary conditions (IBCs). Several results are presented to illustrate the capabilities of the shell element approach. Waveguides are of fundamental importance in many microwave applications, for example in antenna feeds. The key issues of excitation and truncation of waveguides are addressed. A complex frequency shifted form of the uniaxial perfectly matched layer (UPML) absorbing boundary condition (ABC) in FETD is developed. Prism elements are used to promote automatic grid generation and enhance the performance. Results are presented where reflection errors below -70dB are obtained for different types of waveguides, including inhomogeneous cases. Excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency-domain mode solver for the vector Helmholtz equation. Huygens surfaces are used in both FDTD and FETD for excitation in waveguide ports. Inverse problems have received an increased interest due to the availability of powerful computers. An important application is non-destructive evaluation of material. A time-domain, minimization approach is presented where exact gradients are computed using the adjoint problem. The approach is applied to a general form of Maxwell's equations including dispersive media and UPML. Successful reconstruction examples are presented both using synthetic and experimental measurement data. Parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem. Gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry.
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| 16. |
- Edelvik, Fredrik
(author)
-
Finite volume solvers for the Maxwell equations in time domain
- 2000
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Licentiate thesis (other academic/artistic)abstract
- Two unstructured finite volume solvers for the Maxwell equations in 2D and 3D are introduced. The solvers are a generalization of FD–TD to unstructured grids and they use a third-order staggered Adams–Bashforth scheme for time discretization. Analysis and experiments of this time integrator reveal that we achieve a long term stable solution on general triangular grids. A Fourier analysis shows that the 2D solver has excellent dispersion characteristics on uniform triangular grids. In 3D a spatial filter of Laplace type is introduced to enable long simulations without suffering from late time instability.The recursive convolution method proposed by Luebbers et al. to extend FD–TD to permit frequency dispersive materials is here generalized to the 3D solver. A better modelling of materials which have a strong frequency dependence in their constitutive parameters is obtained through the use of a general material model.The finite volume solvers are not intended to be stand-alone solvers but one part in two hybrid solvers with FD–TD. The numerical examples in 2D and 3D demonstrate that the hybrid solvers are superior to stand-alone FD–TD in terms of accuracy and efficiency.
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| 17. |
- Edlund, Johan
(author)
-
A parallel, iterative method of moments and physical optics hybrid solver for arbitrary surfaces
- 2001
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Licentiate thesis (other academic/artistic)abstract
- We have developed an MM–PO hybrid solver designed to deliver reasonable accuracy inexpensively in terms of both CPU-time and memory demands. The solver is based on an iterative block Gauss–Seidel process to avoid unnecessary storage and matrix computations, and can be used to solve the radiation and scattering problems for both disjunct and connected regions. It supports thin wires and dielectrica in the MM domain and has been implemented both as a serial and parallel solver.Numerical experiments have been performed on simple objects to demonstrate certain keyfeatures of the solver, and validate the positive and negative aspects of the MM/PO hybrid. Experiments have also been conducted on more complex objects such as a model aircraft, to demonstrate that the good results from the simpler objects are transferrable to the real life situation. The complex geometries have been used to conduct tests to investigate how well parallelised the code is, and the results are satisfactory.
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| 18. |
- Engblom, Stefan
(author)
-
Numerical methods for the chemical master equation
- 2006
-
Licentiate thesis (other academic/artistic)abstract
- The numerical solution of chemical reactions described at the meso-scale is the topic of this thesis. This description, the master equation of chemical reactions, is an accurate model of reactions where stochastic effects are crucial for explaining certain effects observed in real life. In particular, this general equation is needed when studying processes inside living cells where other macro-scale models fail to reproduce the actual behavior of the system considered.The main contribution of the thesis is the numerical investigation of two different methods for obtaining numerical solutions of the master equation.The first method produces statistical quantities of the solution and is a generalization of a frequently used macro-scale description. It is shown that the method is efficient while still being able to preserve stochastic effects.By contrast, the other method obtains the full solution of the master equation and gains efficiency by an accurate representation of the state space.The thesis contains necessary background material as well as directions for intended future research. An important conclusion of the thesis is that, depending on the setup of the problem, methods of highly different character are needed.
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| 19. |
- Engblom, Stefan, 1976-
(author)
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Numerical Solution Methods in Stochastic Chemical Kinetics
- 2008
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Doctoral thesis (other academic/artistic)abstract
- This study is concerned with the numerical solution of certain stochastic models of chemical reactions. Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior. The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions. In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined. A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution. The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice. Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid. Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method. In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed. Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available. Necessary background material is developed in three chapters in this summary. An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics. In a second chapter the actual stochastic models considered are developed in a multi-faceted way. Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon.
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| 20. |
- Hellander, Andreas
(author)
-
Numerical simulation of well stirred biochemical reaction networks governed by the master equation
- 2008
-
Licentiate thesis (other academic/artistic)abstract
- Numerical simulation of stochastic biochemical reaction networks has received much attention in the growing field of computational systems biology. Systems are frequently modeled as a continuous-time discrete space Markov chain, and the governing equation for the probability density of the system is the (chemical) master equation. The direct numerical solution of this equation suffers from an exponential growth in computational time and memory with the number of reacting species in the model. As a consequence, Monte Carlo simulation methods play an important role in the study of stochastic chemical networks. The stochastic simulation algorithm (SSA) due to Gillespie has been available for more than three decades, but due to the multi-scale property of the chemical systems and the slow convergence of Monte Carlo methods, much work is currently being done in order to devise more efficient approximate schemes.In this thesis we review recent work for the solution of the chemical master equation by direct methods, by exact Monte Carlo methods and by approximate and hybrid methods. We also describe two conceptually different numerical methods to reduce the computational time when studying models using the SSA. A hybrid method is proposed, which is based on the separation of species into two subsets based on the variance of the copy numbers. This method yields a significant speed-up when the system permits such a splitting of the state space. A different approach is taken in an algorithm that makes use of low-discrepancy sequences and the method of uniformization to reduce variance in the computed density function.
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| 21. |
- Hörnell, Karl, 1970-
(author)
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Runge–Kutta Time Step Selection for Flow Problems
- 1999
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Doctoral thesis (other academic/artistic)abstract
- Optimality is studied for Runge-Kutta iteration for solving steady-state and time dependent flow problems. For the former type an algorithm for determining locally optimal time steps is developed, based on the fact that the squared norm of the residual produced by an m-stage scheme is a 2m-degree polynomial, the coefficients of which can be computed from scalar products of Krylov subspace vectors.Under certain conditions on the system matrix, the algorithm is guaranteed to work and its time steps will converge to a global optimum. Furthermore, it will outperform the use of any constant step size. The algorithm is modified to work even when those conditions are not satisfied.Experiments are carried out for a set of Euler and Navier-Stokes problems, both on a single and multiple grids. The algorithm can be extended with optimization over all RK coefficients or discrete parameters like the number of stages or multigrid levels. For that purpose, a simple discrete optimization algorithm is suggested.For some time-dependent problems in one dimension it is shown that if the difference operators and the time steps are properly selected, the local accuracy can be made one order higher than the formal order of the difference operators suggests. This idea cannot be fully generalized, but it will work for scalar problems in 2D if it is combined with an alternating flow technique. Finally an error filter is developed that allows standard step size control algorithms for ordinary differential equations to be efficiently applied to partial differential equations involving shocks.
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| 22. |
- Meinecke, Lina, 1986-
(author)
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Stochastic Simulation of Multiscale Reaction-Diffusion Models via First Exit Times
- 2016
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Doctoral thesis (other academic/artistic)abstract
- Mathematical models are important tools in systems biology, since the regulatory networks in biological cells are too complicated to understand by biological experiments alone. Analytical solutions can be derived only for the simplest models and numerical simulations are necessary in most cases to evaluate the models and their properties and to compare them with measured data.This thesis focuses on the mesoscopic simulation level, which captures both, space dependent behavior by diffusion and the inherent stochasticity of cellular systems. Space is partitioned into compartments by a mesh and the number of molecules of each species in each compartment gives the state of the system. We first examine how to compute the jump coefficients for a discrete stochastic jump process on unstructured meshes from a first exit time approach guaranteeing the correct speed of diffusion. Furthermore, we analyze different methods leading to non-negative coefficients by backward analysis and derive a new method, minimizing both the error in the diffusion coefficient and in the particle distribution.The second part of this thesis investigates macromolecular crowding effects. A high percentage of the cytosol and membranes of cells are occupied by molecules. This impedes the diffusive motion and also affects the reaction rates. Most algorithms for cell simulations are either derived for a dilute medium or become computationally very expensive when applied to a crowded environment. Therefore, we develop a multiscale approach, which takes the microscopic positions of the molecules into account, while still allowing for efficient stochastic simulations on the mesoscopic level. Finally, we compare on- and off-lattice models on the microscopic level when applied to a crowded environment.
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| 23. |
- Nilsson, Martin
(author)
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Iterative solution of Maxwell's equations in frequency domain
- 2002
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Licentiate thesis (other academic/artistic)abstract
- We have developed an iterative solver for the Moment Method. It computes a matrix–vector product with the multilevel Fast Multipole Method, which makes the method scale with the number of unknowns. The iterative solver is of Block Quasi-Minimum Residual type and can handle several right-hand sides at once. The linear system is preconditioned with a Sparse Approximate Inverse, which is modified to handle dense matrices. The solver is parallelized on shared memory machines using OpenMP.To verify the method some tests are conducted on varying geometries. We use simple geometries to show that the method works. We show that the method scales on several processors of a shared memory machine. To prove that the method works for real life problems, we do some tests on large scale aircrafts. The largest test is a one million unknown simulation on a full scale model of a fighter aircraft.
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| 24. |
- Sjöberg, Paul
(author)
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Numerical solution of the Fokker–Planck approximation of the chemical master equation
- 2005
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Licentiate thesis (other academic/artistic)abstract
- The chemical master equation (CME) describes the probability for the discrete molecular copy numbers that define the state of a chemical system. Each molecular species in the chemical model adds a dimension to the state space. The CME is a difference-differential equation which can be solved numerically if the state space is truncated at an upper limit of the copy number in each dimension. The size of the truncated CME suffers from an exponential growth for an increasing number of chemical species.In this thesis the chemical master equation is approximated by a continuous Fokker-Planck equation (FPE) which makes it possible to use sparser computational grids than for CME. FPE on conservative form is used to compute steady state solutions by computation of an extremal eigenvalue and the corresponding eigenvector as well as time-dependent solutions by an implicit time-stepping scheme.The performance of the numerical solution is compared to a standard Monte Carlo algorithm. The computational work for a solutions with the same estimated error is compared for the two methods. Depending on the problem, FPE or the Monte Carlo algorithm will be more efficient. FPE is well suited for problems in low dimensions, especially if high accuracy is desirable.
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| 25. |
- Sterner, Erik, 1966-
(author)
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Accuracy and Convergence Studies of the Numerical Solution of Compressible Flow Problems
- 1997
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Doctoral thesis (other academic/artistic)abstract
- The numerical solution of compressible flow problems governed by the Navier-Stokes equations is considered. A finite volume method is used for the discretization in space. Different techniques to accelerate the convergence to a steady state are suggested, and the accuracy of the spatial difference operator is analyzed.By treating one spatial direction implicitly, it is possible to modify an explicit Runge-Kutta time-marching method, leading to a semi-implicit scheme. A thorough investigation of the stability and convergence properties is presented. Moreover, the scheme is used as a smoother in a multigrid method, and is reformulated as a preconditioner for a number of Newton-Krylov methods. The semi-implicit approach is shown to be very effective for meshes with high aspect ratios. For the flow over a flat plate with a thin boundary layer, the number of iterations to reach convergence is independent of the Reynolds number (Re).An alternative approach for accelerating the convergence is to apply an optimal semicirculant approximation of the spatial operator as a preconditioner. Also here, significant speedups are demonstrated for high Re flows.Two problems appearing for solvers used in computational fluid dynamics are examined. Methods for updating the ghost cells in a multigrid multiblock algorithm are studied, and the accuracy of the finite volume method applied to a polar mesh is analyzed. Although polar mesh singularities lead to a reduction of the order of the truncation error, the global error is shown to be of practically the same order as for a uniform mesh.
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