optimized schwarz method with complete radiation transmission conditions for the helmholtz equation in waveguides in this paper we present a nonoverlapping optimized schwarz algorithm with a high order transmission condition for the helmholtz equation posed in waveguides we introduce the new high order transmission conditions based on the complete radiation boundary conditions crbcs that have been developed for high order absorbing boundary conditions we obtain the convergence rate of the algorithm in terms of reflection coefficients of crbcs which decrease exponentially with the order of crbcs it will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the schwarz algorithm also this algorithm can be employed efficiently for a preconditioner in gmres implementations based on the substructured form as in m j gander f magoules and f nataf siam j sci comput 24 2002 pp 38 60 finally numerical examples confirming the theory will be presented

cloaking via change of variables for the helmholtz equation the transformation optics approach to cloaking uses a singular change of coordinates which blows up a point to the region being cloaked this paper examines a natural regularization obtained by 1 blowing up a ball of radius rather than a point and 2 including a well chosen lossy layer at the inner edge of the cloak we assess the performance of the resulting near cloak as the regularization parameter tends to 0 in the context of dirichlet and neumann boundary measurements for the time harmonic helmholtz equation since the goal is to achieve cloaking regardless of the content of the cloaked region we focus on estimates that are uniform with respect to the physical properties of this region in three space dimensions our regularized construction performs relatively well the deviation from perfect cloaking is of order in two space dimensions it does much worse the deviation is of order 1 log in addition to proving these estimates we give numerical examples demonstrating their sharpness some authors have argued that perfect cloaking can be achieved without losses by using the singular change of variable based construction in our regularized setting the analogous statement is false without the lossy layer there are certain resonant inclusions depending in general on that have a huge effect on the boundary measurements 2010 wiley periodicals inc

pseudospectra for the wave equation with an absorbing boundary for systems which can be described by ut au with a highly nonnormal matrix or operator a the spectrum of a may describe the behavior of the system poorly one such operator arises from the one dimensional wave equation on a finite interval with a homogeneous dirichlet condition at one end and a linear damping condition at the other in this paper the pseudospectra norm of the resolvent of this operator are computed in an energy norm using analytical techniques and computations with discrete approximations when the damping condition is perfectly absorbing the pseudospectra are half planes parallel to the imaginary axis and in other cases they are periodic in the imaginary direction and approximate strips of finite thickness the nonnormality of the operator is related to the behavior of the system and the limitations of spectral analysis

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

a krylov stability corrected coordinate stretching method to simulate wave propagation in unbounded domains the krylov subspace projection approach is a well established tool for the reducedorder modeling of dynamical systems in the time domain in this paper we address the main issues obstructing the application of this powerful approach to the time domain solution of exterior wave problems we use frequency independent perfectly matched layers to simulate the extension to infinity pure imaginary stretching functions based on zolotarev s optimal rational approximation of the square root are implemented leading to perfectly matched layers with a controlled accuracy over a complete spectral interval of interest a new krylov based solution method via stabilitycorrected operator exponents is presented which allows us to construct reduced order models roms that respect the delicate spectral properties of the original scattering problem the roms are unconditionally stable and are based on a renormalized bi lanczos algorithm we give a theoretical foundation of our method and illustrate its performance through a number of numerical examples in which we simulate two dimensional electromagnetic wave propagation in unbounded domains including a photonic waveguide example the new algorithm outperforms the conventional finitedifference time domain method for problems on large time intervals

complete radiation boundary conditions for the helmholtz equation i waveguides we consider the use of complete radiation boundary conditions for the solution of the helmholtz equation in waveguides a general analysis of well posedness convergence and finite element approximation is given in addition methods for the optimization of the boundary condition parameters are considered the theoretical results are illustrated by some simple numerical experiments

near optimal perfectly matched layers for indefinite helmholtz problems a new construction of an absorbing boundary condition for indefinite helmholtz problems on unbounded domains is presented this construction is based on a near best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval designed with the help of a classical result by zolotarev using krein s interpretation of a stieltjes continued fraction this interpolant can be converted into a three term finite difference discretization of a perfectly matched layer pml which converges exponentially fast in the number of grid points the convergence rate is asymptotically optimal for both propagative and evanescent wave modes several numerical experiments and illustrations are included

finding adapted solutions of forward backward stochastic differential equations method of continuation the notion of bridge is introduced for systems of coupled forward backward stochastic differential equations fbsdes for short this notion helps us to unify the method of continuation in finding adapted solutions to such fbsdes over any finite time durations it is proved that if two fbsdes are linked by a bridge then they have the same unique solvability consequently by constructing appropriate bridges we obtain several classes of uniquely solvable fbsdes

backward stochastic dynamics on a filtered probability space we demonstrate that backward stochastic differential equations bsde may be reformulated as ordinary functional differential equations on certain path spaces in this framework neither ito s integrals nor martingale representation formulate are needed this approach provides new tools for the study of bsde and is particularly useful for the study of bsde with partial information the approach allows us to study the following type of backward stochastic differential equations with y t on a general filtered probability space f f t p where b is a d dimensional brownian motion l is a prescribed nonlinear mapping which sends a square integrable m to an adapted process l m and m a correction term is a square integrable martingale to be determined under certain technical conditions we prove that the system admits a unique solution y m in general the associated partial differential equations are not only nonlinear but also may be nonlocal and involve integral operators

on inexact preconditioners for nonsymmetric matrices inexact versions of the block triangular preconditioners for nonsymmetric matrices of block two by two structures in m f murphy g h golub and a j wathen siam j sci comput 21 2000 pp 1969 1972 and i c f ipsen siam j sci comput 23 2001 pp 1050 1051 are presented and the two preconditioners for symmetric block two by two matrices in c durazzi and v ruggiero numer linear algebra appl 10 2003 pp 673 688 are extended to general nonsymmetric matrices moreover we precisely describe the spectral properties of the preconditioned matrices and the finite step termination properties of the preconditioned krylov subspace iteration methods with an optimal or galerkin property with respect to these preconditioners several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners

solution of forward backward stochastic differential equations in this paper we study the existence and uniqueness of the solution to forward backward stochastic differential equations without the nondegeneracy condition for the forward equation under a certain monotonicity condition we prove the existence and uniqueness of the solution to forward backward stochastic differential equations

on semi convergence of the parameterized generalized mhss method for singular complex linear systems for solving a class of complex symmetric singular linear systems we propose a parameterized generalized mhss pgmhss iteration method and investigate the semi convergence conditions by analyzing the spectrum of the iteration matrix then we analyze the optimal iteration parameters that minimize the upper bound of the semi convergence factor numerical experiments are used to test the feasibility and the effectiveness of the pgmhss iteration method

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

a lanczos model order reduction technique to efficiently simulate electromagnetic wave propagation in dispersive media in this paper we present a krylov subspace model order reduction technique for time and frequency domain electromagnetic wave fields in linear dispersive media starting point is a self consistent first order form of maxwell s equations and the constitutive relation this form is discretized on a standard staggered yee grid while the extension to infinity is modeled via a recently developed global complex scaling method by applying this scaling method the time or frequency domain electromagnetic wave field can be computed via a so called stability corrected wave function since this function cannot be computed directly due to the large order of the discretized maxwell system matrix krylov subspace reduced order models are constructed that approximate this wave function we show that the system matrix exhibits a particular physics based symmetry relation that allows us to efficiently construct the time and frequency domain reduced order models via a lanczos type reduction algorithm the frequency domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well numerical experiments for two and three dimensional configurations illustrate the performance of the proposed reduction method

quadratic bsdes with convex generators and unbounded terminal conditions in briand and hu probab theory relat fields 136 4 604 618 2006 the authors proved an existence result for bsdes with quadratic generators with respect to the variable z and with unbounded terminal conditions however no uniqueness result was stated in that work the main goal of this paper is to fill this gap in order to obtain a comparison theorem for this kind of bsdes we assume that the generator is convex with respect to the variable z under this assumption of convexity we are also able to prove a stability result in the spirit of the a priori estimates stated in karoui et al math finance 7 1 1 71 1997 with these tools in hands we can derive the nonlinear feynman kac formula in this context

well posedness of backward stochastic differential equations with general filtration abstract this paper is addressed to the well posedness of some linear and semilinear backward stochastic differential equations with general filtration without using the martingale representation theorem the point of our approach is to introduce a new notion of solution i e the transposition solution which coincides with the usual strong solution when the filtration is natural but it is more flexible for the case of general filtration than the existing notion of solutions a comparison theorem for transposition solutions and a pontryagin type stochastic maximum principle are also presented

absorbing boundary conditions for time harmonic wave propagation in discretized domains while many successful absorbing boundary conditions abcs are developed to simulate wave propagation into unbounded domains most of them ignore the effect of interior discretization and result in spurious reflections at the artificial boundary we tackle this problem by developing abcs directly for the discretized wave equation specifically we show that the discrete system mesh can be stretched in a non trivial way to preserve the discrete impedance at the interface similar to the perfectly matched layers pml for continuous wave equation the stretch is designed to introduce dissipation in the exterior resulting in a pml type abc for discrete media the paper includes detailed formulation of the new discrete abc along with the illustration of its effectiveness over continuous abcs with the help of error analysis and numerical experiments for time harmonic problems the improvement over continuous abcs is achieved without any computational overhead leading to the conclusion that the discrete abcs should be used in lieu of continuous abcs

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

a general stochastic maximum principle for optimal control problems the maximum principle for nonlinear stochastic optimal control problems in the general case is proved the control domain need not be convex and the diffusion coefficient can contain a control variable

solving complex valued linear systems via equivalent real formulations most preconditioned iterative methods apply to both real and complex valued linear systems at the same time most iterative linear solver packages available today focus exclusively on real valued systems or deal with complex valued systems as an afterthought by recasting the complex problem in a real formulation a real valued solver can be applied to the equivalent real system on one hand real formulations have been dismissed due to their unfavorable spectral properties on the other hand using an equivalent preconditioned real formulation can be very effective we give theoretical and experimental evidence that an equivalent real formulation is useful in a number of practical situations furthermore we show how to use the advanced features of modern solver packages to formulate equivalent real preconditioners that are computationally efficient and mathematically identical to their complex counterparts the effectiveness of equivalent real formulations is demonstrated by solving ill conditioned complex valued linear systems for a variety of large scale applications moreover the circumstances under which certain equivalent real formulations are competitive is more clearly delineated

an efficient finite difference scheme for electromagnetic logging in 3d anisotropic inhomogeneous media we consider a problem of computing the electromagnetic field in 3d anisotropic media for electromagnetic logging the proposed finite difference scheme for maxwell equations has the following new features based on some recent and not so recent developments in numerical analysis coercivity i e the complete discrete analogy of all continuous equations in every grid cell even for nondiagonal conductivity tensors a special conductivity averaging that does not require the grid to be small compared to layering or fractures and a spectrally optimal grid refinement minimizing the error at the receiver locations and optimizing the approximation of the boundary conditions at infinity all of these features significantly reduce the grid size and accelerate the computation of electromagnetic logs in 3d geometries without sacrificing accuracy

a 2 5d finite difference algorithm for elastic wave modeling using near optimal quadratures abstractwe have developed a 2 5d finite difference algorithm to model the elastic wave propagation in heterogeneous media in 2 5d problems it is assumed that the elastic properties of models are invariant along a certain direction therefore we can convert the 3d problem into a set of 2d problems in the spectral domain the 3d solutions are then obtained by applying a numerical integration in the spectral domain usually the quadrature points used in the numerical integration scheme are sampled from the real axis in the spectral domain the convergence of this quadrature can be very slow especially at high frequencies we have applied the optimal quadrature scheme for the spectral integration this is equivalent to transforming the contour of integration from the real axis into a path in the complex plane our numerical studies have indicated more than 10 times of reduction in the number of quadrature points compared with sampling along the real axis in the spectral domain this scheme alleviates the

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

an mhss like iteration method for two by two linear systems with application to fde optimization problems abstract in this paper we exploit the numerical solvers for a fractional differential equations fde optimization problem with constraints given by fractional elliptic state equations the discretized linear system can be rewritten as a two by two linear system by making use of the strategy of modified hermitian and skew hermitian splitting mhss iteration method proposed by bai we propose an mhss like iteration method comparing with the mhss iteration method the mhss like method only requires one to solve the two by two linear systems by a sparse cholesky factorization and a fast fourier transform hence the mhss like method has less workload than the mhss method the convergence properties and the quasi optimal parameters are analyzed in detail numerical examples are used to testify the efficiency of the new method

relationship between backward stochastic differential equations and stochastic controls a linear quadratic approach it is well known that backward stochastic differential equations bsdes stem from the study on the pontryagin type maximum principle for optimal stochastic controls a solution of a bsde hits a given terminal value which is a random variable by virtue of an it additional martingale term and an indefinite initial state this paper attempts to explore the relationship between bsdes and stochastic controls by interpreting bsdes as some stochastic optimal control problems more specifically associated with a bsde a new stochastic control problem is introduced with the same dynamics but a definite given initial state the martingale term in the original bsde is regarded as the control and the objective is to minimize the second moment of the difference between the terminal state and the terminal value given in the bsde this problem is solved in a closed form by the stochastic linear quadratic lq theory developed recently the general result is then applied to the black scholes model where an optimal mean variance hedging portfolio is obtained explicitly in terms of the option price finally a modified model is investigated where the difference between the state and the expectation of the given terminal value at any time is taken into account

hybrid finite difference integral equation solver for 3d frequency domain anisotropic electromagnetic problems the modeling of the controlled source electromagnetic csem and single well and crosswell electromagnetic em configurations requires fine gridding to take into account the 3d nature of the geometries encountered in these applications that include geological structures with complicated shapes and exhibiting large variations in conductivities such as the seafloor bathymetry the land topography and targets with complex geometries and large contrasts in conductivities such problems significantly increase the computational cost of the conventional finite difference fd approaches mainly due to the large condition numbers of the corresponding linear systems to handle these problems we employ a volume integral equation ie approach to arrive at an effective preconditioning operator for our fd solver we refer to this new hybrid algorithm as the finite difference integral equation method fdie this fdie preconditioning operator is divergence free and is based on a magnetic field formulation similar to the lippman schwinger ie method this scheme allows us to use a background elimination approach to reduce the computational domain resulting in a smaller size stiffness matrix furthermore it yields a linear system whose condition number is close to that of the conventional lippman schwinger ie approach significantly reducing the condition number of the stiffness matrix of the fd solver moreover the fd framework allows us to substitute convolution operations by the inversion of banded matrices which significantly reduces the computational cost per iteration of the hybrid method compared to the standard ie approaches also well established fd homogenization and optimal gridding algorithms make the fdie more appropriate for the discretization of strongly inhomogeneous media some numerical studies are presented to illustrate the accuracy and effectiveness of the presented solver for csem single well and crosswell em applications

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

a class of efficient parameterized shift splitting preconditioners for block two by two linear systems abstract based on the effective constructing an equivalent block two by two linear systems we establish a class of efficient parameterized shift splitting eps preconditioners for solving block two by two linear systems theoretical analysis shows that the eps iteration method is unconditionally convergent and then we study the spectral properties of the corresponding preconditioned matrix by removing one of the shift term of the eps preconditioners two classes of local eps preconditioners are presented and the eigenvalues distribution of the preconditioned matrices are also studied numerical experiments are provided to demonstrate the feasibility and effectiveness of the proposed preconditioners

optimized schwarz methods optimized schwarz methods are a new class of schwarz methods with greatly enhanced convergence properties they converge uniformly faster than classical schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical schwarz methods if the overlap is of the order of the mesh parameter which is often the case in practical applications they achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem we analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new finite element tearing and interconnect feti variants

fully coupled forward backward stochastic differential equations and applications to optimal control existence and uniqueness results of fully coupled forward backward stochastic differential equations with an arbitrarily large time duration are obtained some stochastic hamilton systems arising in stochastic optimal control systems and mathematical finance can be treated within our framework

mean field stackelberg games aggregation of delayed instructions in this paper we consider an n player interacting strategic game in the presence of a endogenous dominating player who gives direct influence on individual agents through its impact on their control in the sense of stackelberg game and then on the whole community each individual agent is subject to a delay effect on collecting information specifically at a delay time from the dominating player the size of his delay is completely known by the agent while to others including the dominating player his delay plays as a hidden random variable coming from a common fixed distribution by invoking a noncanonical fixed point property we show that for a general class of finite n player games each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an epsilon nash equilibrium for the corresponding finite n player game second we provide with explicit solutions a comprehensive study on the corresponding linear quadratic mean field games of

zolotarev quadrature rules and load balancing for the feast eigensolver the feast method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization this relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function we propose improved rational approximants leading to feast variants with faster convergence in particular when using rational approximants based on the work of zolotarev numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues the new approach improves both convergence robustness and load balancing when feast runs on multiple search intervals in parallel

bsde with quadratic growth and unbounded terminal value in this paper we study the existence of solution to bsde with quadratic growth and unbounded terminal value the main idea consists in using a localization procedure together with a priori bounds

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

application of a complete radiation boundary condition for the helmholtz equation in locally perturbed waveguides abstract this paper deals with an application of a complete radiation boundary condition crbc for the helmholtz equation in locally perturbed waveguides the crbc one of efficient high order absorbing boundary conditions has been analyzed in straight waveguides in hagstrom and kim 2019 in this paper we apply crbc to the helmholtz equation posed in locally perturbed waveguides and establish the well posedness of the problem and convergence of crbc approximate solutions the new crbc proposed in this paper improves the one studied in hagstrom and kim 2019 in two aspects the first one is that the new crbc involves more damping parameters with the same computational cost as that of crbc in hagstrom and kim 2019 which results in 50 smaller reflection errors the second one is that the new crbc takes a neumann terminal condition of three term recurrence relations of auxiliary variables instead of a dirichlet terminal condition used in hagstrom and kim 2019 so that it can treat cutoff modes effectively finally we present numerical experiments illustrating the convergence theory

second order backward stochastic differential equations and fully nonlinear parabolic pdes for a d dimensional diffusion of the form dxt xt dt xt dwt and continuous functions f and g we study the existence and uniqueness of adapted processes y z and a solving the second order backward stochastic differential equation 2bsde dyt f t xt yt zt t dt z dxt t 0 t dzt at dt t dxt

parameterized rotated block preconditioning techniques for block two by two systems with application to complex linear systems in this paper based on the rotated block triangular preconditioning technique we present a class of parameterized rotated block preconditioners to accelerate krylov subspace iterative methods with coefficient matrices of nonsymmetric sub blocks the parameterized rotated block preconditioners can be seen as a generalization of the rotated block triangular preconditioners the eigen properties of the corresponding preconditioned matrices are analyzed numerical experiments are used to demonstrate the feasibility and effectiveness of the parameterized rotated block preconditioners