bayesian and non bayesian evidential updating abstract four main results are arrived at in this paper 1 closed convex sets of classical probability functions provide a representation of belief that includes the representations provided by shafer probability mass functions as a special case 2 the impact of uncertain evidence can be formally represented by dempster conditioning in shafer s framework 3 the impact of uncertain evidence can be formally represented in the framework of convex sets of classical probabilities by classical conditionalization 4 the probability intervals that result from dempster shafer updating on uncertain evidence are included in and may be properly included in the intervals that result from bayesian updating on uncertain evidence

eigenvalue multiplicities in principal submatrices let be an eigenvalue of an n by n matrix a the allowable patterns of geometric multiplicities of as an eigenvalue of a and its principal submatrices is explored a graphical hierarchy for succinctly reporting the possible patterns is defined special attention is paid to the case in which a is hermitian classical interlacing already imposes much structure on the hierarchies in the hermitian case here all the known constraints some old and some new on the geometric multiplicity hierarchies of hermitian matrices are listed some differences between allowed hierarchies for real symmetric matrices and hermitian matrices are also discussed

the principal rank characteristic sequence of a real symmetric matrix abstract given a vector u r 2 n the principal minor assignment problem asks when is there an n n matrix having its 2 n principal minors given by u this paper explores the following related problem given a sequence r 0 r 1 r n of 0s and 1s does there exist an n n real symmetric matrix that has a principal submatrix of rank k if and only if r k 1 for all 0 k n certain conditions are shown to be necessary in order for this question to have an affirmative answer several families of matrices are constructed to attain certain classes of sequences the problem is solved completely for n 6 and for 7 n 10 in the case of sequences beginning with 010

stability and convergence of finite element approximation schemes for harmonic maps this article discusses stability and convergence of approximation schemes for harmonic maps a finite element discretization of an iterative algorithm due to f alouges is introduced and shown to be stable and convergent in general only on acute type triangulations an break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement numerical break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency

l 2 discretization error bounds for maps into riemannian manifolds we study the approximation of functions that map a euclidean domain omega subset mathbb r d into an n dimensional riemannian manifold m g minimizing an elliptic semilinear energy in a function set h subset w 1 2 omega m the approximation is given by a restriction of the energy minimization problem to a family of conforming finite dimensional approximations s h subset h we provide a set of conditions on s h such that we can prove a priori w 1 2 and l 2 approximation error estimates comparable to standard euclidean finite elements this is done in an intrinsic framework independently of embeddings of the manifold or the choice of coordinates

finite element approximations of harmonic map heat flows and wave maps into spheres of nonconstant radii we prove the existence of weak solutions to the harmonic map heat flow and wave maps into spheres of nonconstant radii weak solutions are constructed as proper limits of iterates from a fully practical scheme based on lowest order conforming finite elements where discrete lagrange multipliers are employed to exactly meet the sphere constraint at mesh points computational studies are included to motivate interesting dynamics in two and three spatial dimensions

a mixed boundary value problem for tft lcd analysis and numerical methods abstract we study mixed boundary value problems for lcd liquid crystal displays with chiral nematic for calculating the orientation of molecules and the light transmission existence results are established for mixed boundary value problem numerical methods for time independent and dependent problems are investigated some numerical results are also given

a new algorithm for computing liquid crystal stable configurations the harmonic mapping case in this article we propose a new algorithm for minimizing the energy of a nematic liquid crystal based on the equal elastic constants oseen frank model the problem reduces to finding harmonic minimizing maps that take values into the unit sphere of bbb r 3 the convergence of this algorithm is proved in a continuous setting then numerous numerical results that show its efficiency are given

finite element approximation of nematic liquid crystal flows using a saddle point structure in this work we propose finite element schemes for the numerical approximation of nematic liquid crystal flows based on a saddle point formulation of the director vector sub problem it introduces a lagrange multiplier that allows to enforce the sphere condition in this setting we can consider the limit problem without penalty and the penalized problem using a ginzburg landau penalty function in a unified way further the resulting schemes have a stable behavior with respect to the value of the penalty parameter a key difference with respect to the existing schemes two different methods have been considered for the time integration first we have considered an implicit algorithm that is unconditionally stable and energy preserving the linearization of the problem at every time step value can be performed using a quasi newton method that allows to decouple fluid velocity and director vector computations for every tangent problem then we have designed a linear semi implicit algorithm i e it does not involve nonlinear iterations and proved that it is unconditionally stable verifying a discrete energy inequality finally some numerical simulations are provided

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

an energy minimization finite element approach for the frank oseen model of nematic liquid crystals this paper outlines an energy minimization finite element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects the method targets minimization of the system free energy based on the frank oseen free energy model solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions this requires proving continuity coercivity and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite element framework error analysis demonstrates that the method constitutes a convergent scheme numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions the resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research

fourier spectral approximation to a dissipative system modeling the flow of liquid crystals we study a fourier spectral method for a dissipative system modeling the flow of liquid crystals we first prove its convergence in a suitable sense and establish the existence of a global weak solution of the original problem and its uniqueness in the two dimensional case then we derive error estimates which exhibit the spectral accuracy of the fourier spectral method we also construct a fully discrete scheme and carry out a complete stability and error analysis for it finally we present some illustrative numerical results

a structure preserving fem for the uniaxially constrained mathbf q tensor model of nematic liquid crystals we consider the one constant landau de gennes model for nematic liquid crystals the order parameter is a traceless tensor field mathbf q which is constrained to be uniaxial mathbf q s mathbf n otimes mathbf n d 1 mathbf i where mathbf n is a director field s in mathbb r is the degree of orientation and d ge 2 is the dimension building on similarities with the one constant ericksen energy we propose a structure preserving finite element method for the computation of equilibrium configurations we prove stability and consistency of the method without regularization and gamma convergence of the discrete energies towards the continuous one as the mesh size goes to zero we design an alternating direction gradient flow algorithm for the solution of the discrete problems and we show that such a scheme decreases the energy monotonically finally we illustrate the method s capabilities by presenting some numerical simulations in two and three dimensions including non orientable line fields

a matricial proof of the symmetric exchange axiom for eigenvalues of principal submatrices of a complex hermitian matrix abstract in c r johnson b kroschel m omladic eigenvalue multiplicities in principal submatrices linear algebra appl 390 2004 111 120 a result constraining the eigenvalues of principal submatrices of complex hermitian matrices based upon matroid theory was given here we give a matricial proof of this result which also enables us to find a generalization of the original result

a behavioral model for linguistic uncertainty abstract the paper discusses the problem of modelling linguistic uncertainty which is the uncertainty produced by statements in natural language for example the vague statement mary is young produces uncertainty about mary s age we concentrate on simple affirmative statements of the type subject is predicate where the predicate satisfies a special condition called monotonicity for this case we model linguistic uncertainty in terms of upper probabilities which are given a behavioural interpretation as betting rates possibility measures and probability measures are special types of upper probability measure we evaluate zadeh s suggestion that possibility measures should be used to model linguistic uncertainty and the bayesian claim that probability measures should be used our main conclusion is that when the predicate is monotonic possibility measures are appropriate models for linguistic uncertainty we also discuss several assessment strategies for constructing a numerical model

a coupled ericksen allen cahn model for liquid crystal droplets abstract we present a model and discretization that couples the ericksen model of liquid crystals with variable degree of orientation to the allen cahn equations with a mass constraint the coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment the total energy consists of the ericksen energy phase field allen cahn energy and a weak anchoring energy that couples the liquid crystal to the diffuse interface we describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects

convergent discretization of heat and wave map flows to spheres using approximate discrete lagrange multipliers we propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres the finite element based schemes preserve a unit length constraint at the nodes by means of approximate discrete lagrange multipliers satisfy a discrete energy law and iterates are shown to converge to weak solutions of the continuous problem comparative computational studies are included to motivate finite time blow up behavior in both cases

numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces this article studies the numerical approximation of harmonic maps mto surfaces i e critical pomts for the dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold an iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations numerical simulations illustrate the performance of the algorithm for curved domains

relaxation methods for liquid crystal problems a point relaxation method for the liquid crystal problem is proposed and analyzed the liquid crystal problem is a minimization problem with a nonconvex local constraint it is proved that the energy of successive iterates is nonincreasing for the point relaxation method with relaxation parameter omega satisfying 0 omega 2 for a class of material constants it is also shown that the difference between successive iterates converges to zero and that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point and which satisfy the constraint numerical results are given which demonstrate the improved rate of convergence for overrelaxation

finite element approximations of the ericksen leslie model for nematic liquid crystal flow the ericksen leslie model describes dynamics of low molar mass nematic liquid crystals where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field bf d omega t rightarrow mathbb s 2 it consists of the navier stokes equations with an extra viscous stress tensor and a convective harmonic map heat flow equation to govern the dynamics of the director field two fully discrete finite element methods for a regularized system using the ginzburg landau regularization and for the limiting ericksen leslie model are proposed and well posedness and related discrete energy laws are established for the regularized model unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure valued solutions of the limiting ericksen leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero computational experiments are also presented to show the importance of balancing numerical and regularization parameters to compare regularized and direct approaches and to show numerically the finite time formation annihilation and evolution of point defects

a convergent and precise finite element scheme for landau lifschitz gilbert equation in this paper we propose a new scheme for the numerical integration of the landau lifschitz gilbert llg equations in their full complexity in particular including stray field interactions the scheme is consistent up to order 2 in time and unconditionally stable it combines a linear inner iteration with a non linear renormalization stage for which a rigorous proof of convergence of the numerical solution toward a weak solution is given when both space and time stepsizes tend to 0 0 a numerical implementation of the scheme shows its performance on physically relevant test cases we point out that to the knowledge of the authors this is the first finite element scheme for the llg equations which enjoys such theoretical and practical properties

constraint preserving implicit finite element discretization of harmonic map flow into spheres discretization of the harmonic map flow into spheres often uses a penalization or projection strategy where the first suffers from the proper choice of an additional parameter and the latter from the lack of a discrete energy law and restrictive mesh constraints we propose an implicit scheme that preserves the sphere constraint at every node enjoys a discrete energy law and unconditionally converges to weak solutions of the harmonic map heat flow

a convergent and constraint preserving finite element method for the p harmonic flow into spheres an explicit fully discrete finite element method which satisfies the nonconvex side constraint at every node is developed for approximating the p harmonic flow for p in 1 infty convergence of the method is established under certain conditions on the domain and mesh computational examples are presented to demonstrate finite time blow ups and qualitative geometric changes of weak solutions of the p harmonic flow

on multiplicities of eigenvalues of a boundary value problem on a snowflake graph abstract we find a vector majorizing any ordered vector of multiplicities of eigenvalues of a spectral problem generated by the stieltjes string recurrence relations on a snowflake graph i e a graph obtained by attaching edges to pendant vertices of a star graph

lower bound for the energy of bloch walls in micromagnetics we study a two dimensional nonconvex and nonlocal energy in micromagnetics defined over s2 valued vector fields this energy depends on two small parameters and varepsilon penalizing the divergence of the vector field and its vertical component respectively our objective is to analyze the asymptotic regime beta ll varepsilon ll 1 through the method of convergence finite energy configurations tend to become divergence free and in plane in the magnetic sample except in some small regions of typical width varepsilon called bloch walls where the magnetization connects two directions on s2 we are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions for one dimensional transition layers we show by convergence analysis that the exact line density of the energy is quadratic in the jump size we expect the same behaviour for the two dimensional model in order to prove that we investigate the concept of entropies in the prototype case of a periodic strip we establish a quadratic lower bound for the energy with a non optimal constant then we introduce and study a special class of lipschitz entropies and obtain lower bounds coinciding with the one dimensional limit in some particular cases finally we show that entropies are not appropriate in general for proving the expected sharp lower bound

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

mesh optimization for singular axisymmetric harmonic maps from the disc into the sphere we describe in a mathematical setting the singular energy minimizing axisymmetric harmonic maps from the unit disc into the unit sphere then we use this as a test case to compute optimal meshes in presence of sharp boundary layers for the well posedness of the continuous minimizing problem we introduce a lower semicontinuous extension of the energy with respect to weak convergence in bv and we prove that the extended minimization problem has a unique singular solution we then show how a moving finite element method in which the mesh is an unknown of the discrete minimization problem obtained by finite element discretization mimics this geometric point of view finally we present numerical computations with boundary layers of zero thickness and we give numerical evidence of the convergence of the method this last aspect is proved in another paper

a projection gradient method for computing ground state of spin 2 bose einstein condensates in this paper a projection gradient method is presented for computing ground state of spin 2 bose einstein condensates bec we first propose the general projection gradient method for solving energy functional minimization problem under multiple constraints in which the energy functional takes real functions as independent variables we next extend the method to solve a similar problem where the energy functional now takes complex functions as independent variables we finally employ the method into finding the ground state of spin 2 bec the key of our method is by constructing continuous gradient flows cgfs the ground state of spin 2 bec can be computed as the steady state solution of such cgfs we discretized the cgfs by a conservative finite difference method along with a proper way to deal with the nonlinear terms we show that the numerical discretization is normalization and magnetization conservative and energy diminishing numerical results of the ground state and their energy of spin 2 bec are reported to demonstrate the effectiveness of the numerical method

solving variational problems and partial differential equations that map between manifolds via the closest point method abstract maps from a source manifold m to a target manifold n appear in liquid crystals color image enhancement texture mapping brain mapping and many other areas a numerical framework to solve variational problems and partial differential equations pdes that map between manifolds is introduced within this paper our approach the closest point method for manifold mapping reduces the problem of solving a constrained pde between manifolds m and n to the simpler problems of solving a pde on m and projecting to the closest points on n in our approach an embedding pde is formulated in the embedding space using closest point representations of m and n this enables the use of standard cartesian numerics for general manifolds that are open or closed with or without orientation and of any codimension an algorithm is presented for the important example of harmonic maps and generalized to a broader class of pdes which includes p harmonic maps improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods harmonic and p harmonic maps are computed for a variety of numerical examples in these examples we denoise texture maps diffuse random maps between general manifolds and enhance color images

the ericksen model of liquid crystals with colloidal and electric effects we present a robust discretization of the ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields the total energy consists of the ericksen energy a weak anchoring or penalized dirichlet energy to model colloids and an electrical energy for a given electric field we describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non trivial defect patterns such as the saturn ring defect we conclude with a rigorous proof of the convergence of our discrete energy to the continuous energy

finite element approximation of large bending isometries a finite element scheme for the approximation of large isometric deformations with minimal bending energy is devised and analyzed the convergence to a stationary point and energy decreasing property of an iterative algorithm for the numerical solution of the scheme is proved numerical experiments illustrate the performance of the iteration and show that the discretization leads to accurate approximations for large vertical loads and compressive boundary conditions

semi implicit approximation of wave maps into smooth or convex surfaces a semi implicit scheme using linear finite elements for the approximation of wave maps into smooth or convex surfaces is devised and its stability is analyzed convergence is established for the case of the unit sphere as a target manifold which is unconditional in the case of 2 1 minkowski space numerical experiments illustrate the theoretical results

hyperdeterminantal relations among symmetric principal minors abstract the principal minors of a symmetric n n matrix form a vector of length 2 n we characterize these vectors in terms of algebraic equations derived from the 2 2 2 hyperdeterminant

numerical methods for minimization problems constrained to s 1 and s 2 in this paper we propose numerical methods for minimization problems constrained to s1 and s2 by our technique based on the angle formulation standard numerical difficulties are easily overcome applications to computations of harmonic maps denoising of directional data and of color images are presented in two and three dimensions

a new sobolev gradient method for direct minimization of the gross pitaevskii energy with rotation in this paper we improve traditional steepest descent methods for the direct minimization of the gross pitaevskii gp energy with rotation at two levels we first define a new inner product to equip the sobolev space h 1 and derive the corresponding gradient second for the treatment of the mass conservation constraint we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods the descent method with these two new ingredients is studied theoretically in a hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the gp energy the new method is implemented in both finite difference and finite element two dimensional settings and is used to compute various complex configurations with vortices of rotating bose einstein condensates the new sobolev gradient method shows better numerical performances compared to classical l 2 or h 1 gradient methods especially when high rotation rates are considered

weak series bv convergence of a moving finite element method for singular axisymmetric harmonic maps we prove the convergence of a moving finite element method for the minimization of a relaxed dirichlet energy among axisymmetric maps from the disc into the sphere the optimal mesh and the discrete minimizer exist for a discrete relaxed energy similarly to the true solution the discrete minimizer has a boundary layer because of the consistency error introduced by the discretization of the energy this discrete minimizer is nonconforming we show that it converges to the solution of the continuous problem in an appropriate bv space the proof is done for three different piecewise linear finite element discretizations

a renormalized newton method for liquid crystal director modeling we consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material in the presence of pointwise unit vector constraints and coupled electric fields the numerical solution of such equations by lagrange newton methods leads to linear systems with a double saddle point form for which we have previously proposed a preconditioned nullspace method as an effective solver a ramage and e c gartland jr siam j sci comput 35 2013 pp b226 b247 here we propose and analyze a modified outer iteration renormalized newton method in which the orientation variables are normalized onto the constraint manifold at each iterative step this scheme takes advantage of the special structure of these problems and we prove that it is locally quadratically convergent the renormalized newton method bears some resemblance to the truncated newton method of

normal matrices and their principal submatrices of co order one abstract let a be a normal matrix v be any of its indices a v be the matrix obtained from a by deleting the v th row and column and be an eigenvalue of a v in our paper we construct the eigenspace of a associated with from that of a v in particular it is shown that if there is a unique jordan block of size strictly greater than one in the part of the jordan form of a v corresponding to then the geometric multiplicity of decreases by one under the transition from a v to a in other words the typical change of the spectral properties holds for the results obtained are applied to circulant matrices moreover in appendix to our paper we consider almost regular tournament matrices as principal submatrices of co order one of regular tournament matrices in particular it is observed that the brualdi li tournament matrix b 2 n of order 2n is permutationally similar to a principal submatrix of co order one of the circulant matrix of order 2n 1 with the first row 0 1 1 n 0 0 n as a consequence of this fact the weak brualdi li conjecture is formulated for principal submatrices of co order one of the adjacency matrices of cayley tournaments

the enhanced principal rank characteristic sequence abstract the enhanced principal rank characteristic sequence epr sequence of a symmetric n n matrix is a sequence l 1 l 2 l n where l k is a s or n according as all some or none of its principal minors of order k are nonzero such sequences give more information than the 0 1 pr sequences previously studied where basically the k th entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero various techniques including the schur complement are introduced to establish that certain subsequences such as nan are forbidden in epr sequences over fields of characteristic not two using probabilistic methods over fields of characteristic zero it is shown that any sequence of a s and s s ending in a is attainable and any sequence of a s and s s followed by one or more n s is attainable additional families of attainable epr sequences are constructed explicitly by other methods for real symmetric matrices of orders 2 3 4 and 5 all attainable epr sequences are listed with justifications

a probabilistic and statistical view of fuzzy methods fuzzy set theory has primarily been associated with control theory and with the representation of uncertainty in applications in artificial intelligence more recently fuzzy methods have been proposed as alternatives to traditional statistical methods in statistical quality control linear regression and forecasting among other areas we review some basic concepts of fuzzy methods point out some philosophical and practical problems and offer simpler alternatives based on traditional probability and statistical theory applications in control theory and statistical quality control serve as our primary examples

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

on the alleged superiority of probabilistic representation of uncertainty the author identifies some points of his agreement or disagreement with the position taken by laviolette and seaman 1994 in their debate paper in particular the author argues that some statements regarding the superiority of probabilistic representation of uncertainty which are made by laviolette and seaman in their paper are untenable

a saddle point approach to the computation of harmonic maps in this paper we consider numerical approximations of a constraint minimization problem where the object function is a quadratic dirichlet functional for vector fields and the interior constraint is given by a convex function the solutions of this problem are usually referred to as harmonic maps the solution is characterized by a nonlinear saddle point problem and the corresponding linearized problem is well posed near strict local minima the main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions iterative schemes of newton type for the discrete nonlinear saddle point problems are investigated and mesh independent preconditioners for the iterative methods are proposed

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

an adaptive homotopy multi grid method for molecule orientations of high dimensional liquid crystals the liquid crystal molecule orientation is arranged by minimizing the so called oseen frank energy functional for a better understanding of these complicated orientation singularities simplified models resulting from specific choices of elastic constants are always of interest in this paper a pseudo newton method together with a multi grid linear system solver or preconditioner is used to compute the orientation of liquid crystal molecules based on a simplified oseen frank energy functional the penalty method is used to deal with the unit length constraint of liquid crystal molecules the newton and multi grid methods do not converge when some parameters are small a homotopy algorithm combined with mesh refinement strategies in order to deal with small parameter cases is studied and is found to be very robust in computing the solution of the model the method is implemented to compute the orientation of liquid crystal molecules in domains of typical shapes and with various rotational boundary conditions in 2d and 3d interesting singularity patterns are observed