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Renormalization of stochastic continuity equations on Riemannian manifolds 
Galimberti, Luca; Karlsen, Kenneth H. (Journal article / Tidsskriftartikkel / PublishedVersion; Peer reviewed, 2021)
CONVERGENCE OF A MIXED METHOD FOR A SEMI-STATIONARY COMPRESSIBLE STOKES SYSTEM 
Karlsen, Kenneth H.; Karper, Trygve K. (Research report / Forskningsrapport, 2009)
We propose and analyze a finite element method for a semi-stationary Stokes system modeling compressible fluid flow subject to a Navier-slip boundary condition. The velocity (momentum) equation is approximated by a mixed ...
A CONVERGENT NONCONFORMING FINITE ELEMENT METHOD FOR COMPRESSIBLE STOKES FLOW 
Karlsen, Kenneth H.; Karper, Trygve K. (Research report / Forskningsrapport, 2009)
We propose a nonconforming finite element method for isentropic viscous gas flow in situations where convective effects may be neglected. We approximate the continuity equation by a piecewise constant discontinuous Galerkin ...
VISCOSITY SOLUTIONS FOR A SYSTEM OF INTEGRO-PDES AND CONNECTIONS TO OPTIMAL SWITCHING AND CONTROL OF JUMP-DIFFUSION PROCESSES 
Biswas, Imran H.; Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2009)
We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the ...
A SEMILINEAR EQUATION FOR THE AMERICAN OPTION IN A GENERAL JUMP MARKET 
Karlsen, Kenneth H.; Wallin, Olli (Research report / Forskningsrapport, 2008)
We study the pricing of American put and call options in a market with jumps. We extend and make rigorous previous work that characterizes the price as a solution of an integro-differential equation set on the whole domain. ...
ERROR ESTIMATES FOR FINITE DIFFERENCE-QUADRATURE SCHEMES FOR FULLY NONLINEAR DEGENERATE PARABOLIC INTEGRO-PDES 
Biswas, Imran H.; Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2007)
We derive error estimates for finite difference-quadrature schemes approximating viscosity solutions of nonlinear degenerate parabolic integro-PDEs with variable diffusion coefficients. The relevant equations can be viewed ...
WELL-POSEDNESS OF HIGHER-ORDER CAMASSA-HOLM EQUATIONS 
Coclite, Giuseppe M.; Holden, Helge; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2006)
We consider higher-order Camassa--Holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane. We establish the existence of a strongly ...
A LARGE TIME-STEPPING SCHEME FOR BALANCE EQUATIONS 
Karlsen, Kenneth H.; Mishra, Siddhartha; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
We present a well-balanced, large time stepping method for conservation laws with source terms. The numerical method is based on a local reformulation of the balance law as a conservation law with a discontinuous flux ...
SEMI-GODUNOV SCHEMES FOR GENERAL TRIANGULAR SYSTEMS OF CONSERVATION LAWS 
Karlsen, Kenneth H.; Mishra, Siddhartha; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
We consider general triangular systems of conservation laws which arise in applications like multi-phase flows in porous media and are non-strictly hyperbolic. We device simple and efficient finite volume schemes of the ...
WELL-BALANCED SCHEMES FOR CONSERVATION LAWS WITH SOURCE TERMS BASED ON A LOCAL DISCONTINUOUS FLUX FORMULATION 
Karlsen, Kenneth H.; Mishra, Siddhartha; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
We propose and analyse a finite volume scheme of the Godunov type that preserves discrete steady states. The scheme works in resonance regime as well as for problems with discontinuous flux. Moreover, no additional ...
NUMERICAL SCHEMES FOR COMPUTING DISCONTINUOUS SOLUTIONS OF THE DEGASPERIS-PROCESI EQUATION 
Coclite, Giuseppe M.; Karlsen, Kenneth H.; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
Recent work [4] has shown that the Degasperis-Procesi equation is well-posed in the class of (discontinuous) entropy solutions. In the present paper we construct numerical schemes and prove that they converge to entropy ...
ON A TWO-SIDEDLY DEGENERATE CHEMOTAXIS MODEL WITH VOLUME-FILLING EFFECT 
Karlsen, Kenneth H.; Urbano, J. M.; Bendahmane, Mostafa (Research report / Forskningsrapport, 2006)
We consider a fully parabolic model for chemotaxis with volume-filling effect and a nonlinear diffusion that degenerates in a two-sided fashion. We address the questions of existence of weak solutions and of their regularity ...
ERROR ESTIMATES FOR FINITE DIFFERENCE-QUADRATURE SCHEMES FOR A CLASS OF NONLOCAL BELLMAN EQUATIONS WITH VARIABLE DIFFUSION COEFFICIENTS 
Biswas, Imran H.; Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2006)
We derive error estimates for certain approximate solutions of Bellman equations associated to a class of controlled jump-diffusion (Lévy) processes. These Bellman equations are fully nonlinear degenerate integro-PDEs ...
CONVERGENCE OF FINITE VOLUME SCHEMES FOR TRIANGULAR SYSTEMS OF CONSERVATION LAWS 
Karlsen, Kenneth H.; Mishra, Siddhartha; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
We consider non-strictly hyperbolic systems of conservation laws in triangular form, which arise in applications like three-phase flows in porous media. We device simple and efficient finite volume schemes of Godunov type ...
ON THE UNIQUENESS OF DISCONTINUOUS SOLUTIONS TO THE DEGASPERIS-PROCESI EQUATION 
Coclite, Giuseppe M.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2006)
A CONVERGENT FINITE DIFFERENCE SCHEME FOR THE CAMASSA-HOLM EQUATION WITH GENERAL H¹ INITIAL DATA 
Coclite, Giuseppe M.; Karlsen, Kenneth H.; Risebro, Nils Henrik (Research report / Forskningsrapport, 2006)
We suggest a finite dfference scheme for the Camassa-Holm equation that can handle general H1 initial data. The form of the difference scheme is judiciously chosen to ensure that it satisfies a total energy inequality. We ...
ANALYSIS OF A CLASS OF DEGENERATE REACTION-DIFFUSION SYSTEMS AND THE BIDOMAIN MODEL OF CARDIAC TISSUE 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2005)
We prove well-posedness (existence and uniqueness) results for aÊ class of degenerate reaction-diffusion systems. A prototype system belongingÊ to this class is provided by the bidomain model, which is frequently used toÊ ...
A SINGULAR LIMIT PROBLEM FOR CONSERVATION LAWS RELATED TO THE CAMASSA-HOLM SHALLOW WATER EQUATION 
Coclite, Giuseppe M.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2005)
We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a ...
ERROR ESTIMATES FOR APPROXIMATE SOLUTIONS TO BELLMAN EQUATIONS ASSOCIATED WITH CONTROLLED JUMP-DIFFUSIONS 
Jakobsen, Espen R.; Karlsen, Kenneth H.; La Chioma, Claudia (Research report / Forskningsrapport, 2005)
We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are ...
ON AN EXTENDED CLARIFIER-THICKENER MODEL WITH SINGULAR SOURCE AND SINK TERM 
Bürger, Raimund; García, A.; Karlsen, Kenneth H.; Towers, John D. (Research report / Forskningsrapport, 2005)
A well-studied one-dimensional model for the operation of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. ...
H¹-PERTURBATIONS OF SMOOTH SOLUTIONS FOR A WEAKLY DISSIPATIVE HYPERELASTIC-ROD WAVE EQUATION 
Bendahmane, Mostafa; Coclite, Giuseppe M.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2005)
We consider a weakly dissipative hyperelastic-rod wave equation (or weakly dissipative Camassa-Holm equation) describing nonlinear dispersive dissipative waves in compressible hyperelastic rods. By fixed a smooth solution, ...
CONVERGENT DIFFERENCE SCHEMES FOR THE HUNTER-SAXTON EQUATION 
Holden, Helge; Karlsen, Kenneth H.; Risebro, Nils Henrik (Research report / Forskningsrapport, 2005)
ON THE WELL-POSEDNESS OF THE DEGASPERIS-PROCESI EQUATION 
Coclite, Giuseppe M.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2005)
RENORMALIZED SOLUTIONS OF AN ANISOTROPIC REACTION-DIFFUSION-ADVECTION SYSTEM WITH L¹ DATA MODELING THE PROPAGATION OF AN EPIDEMIC DISEASE 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2005)
We prove existence of a renormalized soluion to a system of nonlinear partial differential equations with anisotropic diffusivities and transport effects, supplemented with initial and Dirichlet boundary conditions. The ...
WELLPOSEDNESS OF SOLUTIONS OF A PARABOLIC-ELLIPTIC SYSTEM 
Coclite, Giuseppe M.; Holden, Helge; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We show existence of a unique, regular global solution of the parabolic-elliptic system $u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and $-P_{xx}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data $u|_{t=0} = u_0$. Here ...
UNIQUENESS OF ENTROPY SOLUTIONS FOR DOUBLY NONLINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We consider doubly nonlinear anisotropic degenerate parabolic equations, supplemented with an initial condition and a homogeneous Dirichlet boundary condition. We introduce a notion of entropy solution and prove that the ...
GLOBAL WEAK SOLUTIONS TO A GENERALIZED HYPERELASTIC-ROD WAVE EQUATION 
Coclite, Giuseppe M.; Holden, Helge; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous ...
CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX 
Karlsen, Kenneth H.; Towers, John D. (Research report / Forskningsrapport, 2004)
We give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form $$ u_t+f(k(x,t),u)_x=0, $$ where the coefficient $k(x,t)$ is ...
ANISOTROPIC NONLINEAR ELLIPTIC SYSTEMS WITH MEASURE DATA AND ANISOTROPIC HARMONIC MAPS INTO SPHERES 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We prove existence results for distributional solutions of anisotropic nonlinear elliptic systems with a measure valued right-hand side. The functional setting involves anisotropic Sobolev spaces as well as weak Lebesgue ...
NONLINEAR DEGENERATE INTEGRO-PARTIAL DIFFERENTIAL EVOLUTION EQUATIONS RELATED TO GEOMETRIC LÉVY PROCESSES AND APPLICATIONS TO BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 
Amadori, Anna Lisa; Karlsen, Kenneth H.; La Chioma, Claudia (Research report / Forskningsrapport, 2004)
We prove a comparison principle for unbounded semicontinuous viscosity sub- and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in ...
A NOTE ON MERTON'S PORTFOLIO SELECTION PROBLEM FOR THE SCHWARTZ MEAN-REVERSION MODEL 
Benth, Fred Espen; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We analyze the classical Merton's portfolio optimization problem when the risky asset follows an exponential Ornstein-Uhlenbeck process, also known as the Schwartz mean-reversion dynamics. The corresponding Hamilton-Jacobi-Bellman ...
Closed-form and finite difference solutions to a population balance model of grinding mills 
Bürger, Raimund; Karlsen, Kenneth H.; Towers, John D. (Research report / Forskningsrapport, 2004)
The wear of steel balls in continuously operated grinding mills, used in mineral processing to comminute metalliferous rocks, can be described by a simple population balance model. This model gives rise to a scalar transport ...
CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF INTEGRO-PDES 
Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We present a general framework for deriving continuous dependence estimates for, possibly polynomially growing, viscosity solutions of fully nonlinear degenerate parabolic integro-PDEs. We use this framework to provide ...
A MODEL OF CONTINUOUS SEDIMENTATION OF FLOCCULATED SUSPENSIONS IN CLARIFIER-THICKENER UNITS 
Bürger, Raimund; Karlsen, Kenneth H.; Towers, John D. (Research report / Forskningsrapport, 2004)
The chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in clarifier-thickener units. This model appears ...
QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS WITH TIME-SPACE DEPENDENT DIFFUSION COEFFICIENTS 
Chen, Gui-Qiang; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We study the well-posedness of discontinuous entropy solutions to quasilinear aniso-tropic degenerate parabolic equations with explicit $(t,x)$--dependence: $$ \pt u + \si \pxi f_i(u,t,x)=\sij \pxj\left(\aij(u,t,x)\pxi ...
A CONVERGENCE RATE FOR SEMI-DISCRETE SPLITTING APPROXIMATIONS FOR DEGENERATE PARABOLIC EQUATIONS WITH SOURCE TERMS 
Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2004)
We study a semi-discrete splitting method for computing approximate viscosity solutions of the initial value problem for a class of nonlinear degenerate parabolic equations with source terms. It is fairly standard to prove ...
WELL-POSEDNESS IN BVt AND CONVERGENCE OF A DIFFERENCE SCHEME FOR CONTINUOUS SEDIMENTATION IN IDEAL CLARIFIER-THICKENER UNITS 
Bürger, Raimund; Karlsen, Kenneth H.; Risebro, Nils Henrik; Towers, John D. (Research report / Forskningsrapport, 2003)
We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We ...
A PDE REPRESENTATION OF THE DENSITY OF THE MINIMAL ENTROPY MARTINGALE MEASURE IN STOCHASTIC VOLATILITY MARKETS 
Benth, Fred Espen; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
Under general conditions stated in Rheinländer 30], we prove that in a stochastic volatility market the Radon-Nikodym density of the minimal entropy martingale measure can be expressed in terms of the solution of a semilinear ...
NONLINEAR ANISOTROPIC ELLIPTIC AND PARABOLIC EQUATIONS IN ℝᴺ WITH ADVECTION AND LOWER ORDER TERMS AND LOCALLY INTEGRABLE DATA 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
FRONT TRACKING FOR SCALAR BALANCE EQUATIONS 
Karlsen, Kenneth H.; Risebro, Nils Henrik; Towers, John D. (Research report / Forskningsrapport, 2003)
We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2x2 quasilinear system without a source term, and ...
FAST COMPUTATION OF ARRIVAL TIMES IN HETEROGENEOUS MEDIA 
Berre, Inga; Karlsen, Kenneth H.; Lie, K.-A.; Natvig, Bent (Research report / Forskningsrapport, 2003)
We continue the work that was initiated in [7] on a fast marching-like method for simulating two-phase incompressible immiscible flow of water and oil in a porous medium. The purpose ot the present paper is threefold: (1) ...
A “MAXIMUM PRINCIPLE FOR SEMICONTINUOUS FUNCTIONS” APPLICABLE TO INTEGRO-PARTIAL DIFFERENTIAL EQUATIONS 
Jakobsen, Espen R.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
We formulate and prove a non-local ``maximum principle for semicontinuous functions'' in the setting of fully nonlinear degenerate elliptic integro-partial differential equations with integro operators of second order. ...
ON A DIFFUSIVELY CORRECTED KINEMATIC-WAVE TRAFFIC MODEL WITH CHANGING ROAD SURFACE CONDITIONS 
Bürger, Raimund; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
The well-known Lighthill-Whitham-Richards kinematic traffic flow model for unidirectional flow on a single-lane highway is extended to include both abruptly changing road surface conditions and drivers' reaction time and ...
L¹-FRAMEWORK FOR CONTINUOUS DEPENDENCE AND ERROR ESTIMATES FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONSL¹-FRAMEWORK FOR CONTINUOUS DEPENDENCE AND ERROR ESTIMATES FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 
Chen, Gui-Qiang; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
We develop a general L1-framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply L1-framework ...
L¹ STABILITY FOR ENTROPY SOLUTIONS OF NONLINEAR DEGENERATE PARABOLIC CONVECTION-DIFFUSION EQUATIONS WITH DISCONTINUOUS COEFFICIENTS 
Karlsen, Kenneth H.; Risebro, Nils Henrik; Towers, John D. (Research report / Forskningsrapport, 2003)
We propose a Kruzkov-type entropy condition for nonlinear degenerate parabolic equations with discointinuous coefficients. We establish L1 stability, and thus uniqueness, for weak solutions satisfying the entropy condition, ...
RENORMALIZED ENTROPY SOLUTIONS FOR QUASILINEAR ANISOTROPIC DEGENERATE PARABOLIC EQUATIONS 
Bendahmane, Mostafa; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2003)
We prove well posedness (existence and uniqueness) of renormalized entropy solutions to the Cauchy problem for quasilinear anisotropic degenerate parabolic equations with L1 data. This paperÊ complements the work by Chen ...
ON DERIVATIVES OF CLAIMS IN COMMODITY AND ENERGY MARKETS USING A MALLIAVIN APPROACH 
Benth, Fred Espen; Dahl, Lars O.; Karlsen, Kenneth H. (Research report / Forskningsrapport, 2002)
In this paper we investigate the recently introduced Malliavin approach compared to more classical approaches to find sensitivities of options in commodity and energy markets. The Malliavin approach has been developed in ...
On the existence of optimal controls for a singular stochastic control problem in finance 
Benth, Fred Espen; Karlsen, Kenneth H.; Reikvam, Kristin (Research report / Forskningsrapport, 2000)
OPTIMAL PORTFOLIO SELECTION WITH CONSUMPTION AND NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS WITH GRADIENT CONSTRAINT: A VISCOSITY SOLUTION APPROACH 
Benth, Fred Espen; Karlsen, Kenneth H.; Reikvam, Kristin (Research report / Forskningsrapport, 2000)
OPTIMAL PORTFOLIO MANAGEMENT RULES IN A NON-GAUSSIAN MARKET WITH DURABILITY AND INTERTEMPORAL SUBSTITUTION 
Benth, Fred Espen; Karlsen, Kenneth H.; Reikvam, Kristin (Research report / Forskningsrapport, 2000)
On the accuracy of a numerical method for two-dimensional scalar conservation laws based on dimensional splitting and front tracking 
Karlsen, Kenneth H. (Research report / Forskningsrapport, 1994)
 
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Date Issued2020 - 2021 (1)2000 - 2009 (49)1994 - 1999 (1)Document TypeForskningsrapport (50)Tidsskriftartikkel (1)Author
Karlsen, Kenneth H. (51)
Risebro, Nils Henrik (10)Coclite, Giuseppe M. (9)Bendahmane, Mostafa (8)Jakobsen, Espen R. (7)... View MorePeer ReviewedPeer reviewed (1)
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