lagrangian optimization of two description scalar quantizers in this paper we study the problem of optimal design of balanced two description fixed rate scalar quantizer 2dsq under the constraint of convex codecells using a graph based approach to model the problem we show that the minimum expected distortion of the 2dsq is a convex function of the number of codecells in the side quantizers this property allows the problem to be solved by lagrangian minimization for which the optimal lagrangian multiplier exists given a trial multiplier we exploit a monotonicity of the objective function and develop a simple and fast dynamic programming technique to solve the parameterized problem to further improve the algorithm efficiency we propose an rd guided search strategy to find the optimal lagrangian multiplier in our experiments on distributions of interest for signal compression applications the proposed algorithm improves the speed of the fastest algorithm so far by a factor of o k logk where k is the number of codecells in each side quantizer we also assess the impact on the optimality of the convex codecell constraint using a published performance analysis of 2dsq at high rates we show that asymptotically this constraint does not preclude optimality for l 2 distortion measure when channels have a higher than 0 12 loss rate

harmonic analysis of stochastic equations and backward stochastic differential equations the bmo martingale theory is extensively used to study nonlinear multi dimensional stochastic equations in mathcal r p p in 1 infty and backward stochastic differential equations bsdes in mathcal r p times mathcal h p p in 1 infty and in mathcal r infty times overline l infty rm bmo with the coefficients being allowed to be unbounded in particular the probabilistic version of fefferman s inequality plays a crucial role in the development of our theory which seems to be new several new results are consequently obtained the particular multi dimensional linear cases for stochastic differential equations sdes and bsdes are separately investigated and the existence and uniqueness of a solution is connected to the property that the elementary solutions matrix for the associated homogeneous sde satisfies the reverse holder inequality for some suitable exponent p 1 finally some relations are established between kazamaki s quadratic critical exponent b m of a bmo martingale m and the spectral radius of the stochastic integral operator with respect to m which lead to a characterization of kazamaki s quadratic critical exponent of bmo martingales being infinite

ell_ 2 optimized predictive image coding with ell_ infty bound in many scientific medical and defense applications of image video compression an l error bound is required however pure l optimized image coding colloquially known as near lossless image coding is prone to structured errors such as contours and speckles if the bit rate is not sufficiently high moreover most of the previous l based image coding methods suffer from poor rate control in contrast the l2 error metric aims for average fidelity and hence preserves the subtlety of smooth waveforms better than the l error metric and it offers fine granularity in rate control but pure l2 based image coding methods e g jpeg 2000 cannot bound individual errors as the l based methods can this paper presents a new compression approach to retain the benefits and circumvent the pitfalls of the two error metrics a common approach of near lossless image coding is to embed into a dpcm prediction loop a uniform scalar quantizer of residual errors the said uniform scalar quantizer is replaced in the proposed new approach by a set of context based l2 optimized quantizers the optimization criterion is to minimize a weighted sum of the l2 distortion and the entropy while maintaining a strict l error bound the resulting method obtains good rate distortion performance in both l2 and l metrics and also increases the rate granularity compared with jpeg 2000 the new method not only guarantees lower l error for all bit rates but also it achieves higher psnr for relatively high bit rates

monotonicity based fast algorithms for map estimation of markov sequences over noisy channels in this correspondence we study algorithmic approach to solving the problem of maximum a posteriori map estimation of markov sequences transmitted over noisy channels which is also known as the map decoding problem for the class of memoryless binary channels that produce independent substitution and erasure errors the map sequence estimation problem can be formulated and solved as one of the longest path in a weighted directed acyclic graph but for algorithm efficiency we transform the graph problem to one of matrix search if the underlying matrix is totally monotone then the complexity of map sequence estimation can be greatly reduced we give a sufficient condition for the matrix induced by map sequence estimation to be totally monotone which is indeed the case if the input sequence is gaussian markov under this condition the complexity of map decoding can be reduced from o n sup 2 m to o nm where n is the size of source alphabet and m is the length of input sequence furthermore for markov sequences of fixed length code we propose a block parsing strategy to reduce the complexity of map sequence estimation to o m n sup 2 m logm or to o m nm logm depending on if the total monotonicity holds another significance of this correspondence lies in the applicability of the presented algorithmic approach which has been thoroughly studied in computer science literature to many other discrete optimization problems encountered in both source and channel coding ranging from optimal multiresolution and multiple description quantizer design to context quantization for minimum conditional entropy and to optimal packetization with uneven error protection

a simple proof of indefinite linear quadratic stochastic optimal control with random coefficients we consider the indefinite linear quadratic optimal control problem for stochastic systems with random coefficients where the corresponding cost parameters need not be definite matrices although the solution to this problem was obtained previously via the stochastic maximum principle or by solving the stochastic hamilton jacobi bellman hjb equation our approach is simple and direct specifically we develop a direct approach also known as the completion of squares method to characterize the explicit optimal solution and the optimal cost the corresponding optimal solution is linear characterized in terms of the stochastic riccati differential equation srde and the linear backward stochastic differential equation bsde in our approach the completion of squares method together with the srde and the bsde allows us to construct an equivalent cost functional that is quadratic in the control u provided that an additional positive definiteness condition in terms of the srde holds then the optimal control and the associated cost can be obtained by eliminating the quadratic term of u in the equivalent cost functional the additional positive definiteness condition is induced due to the indefiniteness of the cost parameters and the dependence of the control on the diffusion term we verify the optimal solution by solving the stochastic hjb equation

mean variance hedging and stochastic control beyond the brownian setting we show for continuous semimartingales in a general filtration how the mean variance hedging problem can be treated as a linear quadratic stochastic control problem the adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process and we give necessary and sufficient conditions for the solvability of these generalized stochastic riccati equations motivated from mathematical finance this paper takes a first step toward linear quadratic stochastic control in more general than brownian settings

image quantization using reaction diffusion equations in this paper we present an image quantization model based on a reaction diffusion partial differential equation the quantized image is given by the asymptotic state of this equation existence and uniqueness of the solution are proved in the framework of viscosity solutions we introduce an l infty stable algorithm in order to compute numerically the solution of the equation and some experimental results are shown a new energy functional based on the classical lloyd method is used to compute the quantizer codewords

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

solvability conditions for indefinite linear quadratic optimal stochastic control problems and associated stochastic riccati equations a linear quadratic optimal stochastic control problem with random coefficients and indefinite state control weight costs is usually linked to an indefinite stochastic riccati equation sre which is a matrix valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown either the optimal control problem or the sre is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined in this paper by introducing a notion of subsolution for the sre we derive several novel sufficient conditions for the existence and uniqueness of the solution to the sre and for the solvability of the associated optimal stochastic control problem

efficient adaptive algorithms and minimax bounds for zero delay lossy source coding zero delay lossy source coding schemes are considered for both individual sequences and random sources performance is measured by the distortion redundancy which is defined as the difference between the normalized cumulative mean squared distortion of the scheme and the normalized cumulative distortion of the best scalar quantizer of the same rate that is matched to the entire sequence to be encoded by improving and generalizing a scheme of linder and lugosi weissman and merhav showed the existence of a randomized scheme that for any bounded individual sequence of length n achieves a distortion redundancy o n sup 1 3 logn however both schemes have prohibitive complexity both space and time which makes practical implementation infeasible in this paper we present an algorithm that computes weissman and merhav s scheme efficiently in particular we introduce an algorithm with encoding complexity o n sup 4 3 and distortion redundancy o n sup 1 3 logn the complexity can be made linear in the sequence length n at the price of increasing the distortion redundancy to o n sup 1 4 spl radic logn we also consider the problem of minimax distortion redundancy in zero delay lossy coding of random sources by introducing a simplistic scheme and proving a lower bound we show that for the class of bounded memoryless sources the minimax expected distortion redundancy is upper and lower bounded by constant multiples of n sup 1 2

optimal estimation over the collision channel consider a distributed sensing system that comprises two sensors each observing a random variable and a remote estimator the goal of the remote estimator is to produce estimates of the random variables based on information transmitted to it by the sensors the random variables are independent and information is transferred from the sensors to the estimator via a collision channel which can only convey a single packet each sensor has the authority to decide what and when to transmit and simultaneous transmissions result in a collision event to be detected at the estimator in our formulation there is no communication between the sensors which precludes the use of coordinated strategies our results characterize the structure of policies at the sensors and the remote estimator that are optimal with respect to a mean squared error criterion more specifically we show that there exist optimal policies at the sensors that use deterministic threshold strategies to decide when to transmit this structural result is independent of the distributions of the observed random variables in our analysis we prove that the computation of a person by person optimal threshold policy can be recast as a one bit optimal quantization problem for which the cost is nonuniform across representation symbols based on that observation we provide an iterative procedure akin to the lloyd max algorithm that can be used to compute locally optimal solutions in the gaussian case our iterative method converged to an optimal solution in all numerical examples we have tried we provide several examples that show the optimality of asymmetric threshold policies even when the overall framework is symmetric such as when both random variables are gaussian with zero mean and the same variance

tracking the best quantizer an algorithm is presented for online prediction that allows to track the best expert efficiently even when the number of experts is exponentially large provided that the set of experts has a certain additive structure as an example we work out the case where each expert is represented by a path in a directed graph and the loss of each expert is the sum of the weights over the edges in the path these results are then used to construct universal limited delay schemes for lossy coding of individual sequences in particular we consider the problem of tracking the best scalar quantizer that is adaptively matched to the source sequence with piecewise different behavior a randomized algorithm is presented which can perform on any source sequence asymptotically as well as the best scalar quantization algorithm that is matched to the sequence and is allowed to change the employed quantizer for a given number of times the complexity of the algorithm is quadratic in the sequence length but at the price of some deterioration in performance the complexity can be made linear analogous results are obtained for sequential multiresolution and multiple description scalar quantization of individual sequences

necessary conditions for optimal control of stochastic systems with random jumps a maximum principle is proved for optimal controls of stochastic systems with random jumps the control is allowed to enter into both diffusion and jump terms the form of the maximum principle turns out to be quite different from the one corresponding to the pure diffusion system the word pure here means the absence of the jump term in calculating the first order coefficient for the cost variation only a property for lebesgue integrals of scalar valued functions in the real number space cal r is used this shows that there is no essential difference between deterministic and stochastic systems as far as the derivation of maximum principles is concerned

on properties of locally optimal multiple description scalar quantizers with convex cells it is known that the generalized lloyd method is applicable to locally optimal multiple description scalar quantizer mdsq design however it remains unsettled when the resulting mdsq is also globally optimal we partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed rate mdsq of convex cells under trushkin s sufficient conditions for the uniqueness of locally optimal fixed rate single description scalar quantizer this result holds for fixed rate multiresolution scalar quantizer mrsq of convex cells as well thus the well known log concave probability density function pdf condition can be extended to the multiple description and multiresolution cases

quantization as histogram segmentation optimal scalar quantizer design in network systems an algorithm for scalar quantizer design on discrete alphabet sources is proposed the proposed algorithm can be used to design fixed rate and entropy constrained conventional scalar quantizers multiresolution scalar quantizers multiple description scalar quantizers and wyner ziv scalar quantizers the algorithm guarantees globally optimal solutions for conventional fixed rate scalar quantizers and entropy constrained scalar quantizers for the other coding scenarios the algorithm yields the best code among all codes that meet a given convexity constraint in all cases the algorithm run time is polynomial in the size of the source alphabet the algorithm derivation arises from a demonstration of the connection between scalar quantization histogram segmentation and the shortest path problem in a certain directed acyclic graph

on convergence of lloyd s method i although lloyd s method i for optimal quantization was proposed more than thirty years ago and has been frequently referred to in the literature its convergence has so far not been shown this correspondence proves that lloyd s method i converges for a large class of error measures if the density function is continuous positive and defined on a finite interval the proof is done by modeling the behavior of a continuous optimization algorithm by a finite state machine

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

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

stochastic hamilton jacobi bellman equations this paper studies the following form of nonlinear stochastic partial differential equation begin gathered d phi t mathop inf v in u left frac 1 2 sum i j left sigma sigma right ij x v t partial x i x j phi t x sum i b i x v t partial x i phi t x l x v t right hfill qquad qquad left sum i j sigma ij x v t partial x i psi j t x right dt psi t x dw t quad phi t x h x hfill end gathered where the coefficients sigma ij b i l and the final datum h may be random the problem is to find an adapted pair phi psi x t uniquely solving the equation the classical hamilton jacobi bellman hjb equation can be regarded as a special case of the above problem an existence and uniqueness theorem is obtained for the case where sigma does not contain the control variable v an optimal control interpretation is given the linear quadratic case is discussed as well

optimal bi level quantization and its application to multilevel quantization a special form of optimal quantization optimal bilevel quantization is studied a fixed point method is embedded in a search scheme to find all the locally optimal bilevel quantizers resulting in an algorithm for computing the globally optimal bilevel quantizer some interesting relations between the optimal bilevel quantizer and the mean and the median of the density function p x are explored efficient algorithms for computing optimal bilevel quantizers are proposed the application of these results to optimal quantization in general is discussed

general linear quadratic optimal stochastic control problems with random coefficients linear stochastic hamilton systems and backward stochastic riccati equations consider the minimization of the following quadratic cost functional j u e langle mx t x t rangle e int 0 t langle q sx s x s rangle langle n su s u s rangle ds where x is the solution of the following linear stochastic control system eq dx t a tx t b tu t dt sum i 1 d c t ix t d t iu t dw t i cr x 0 h cr u is a square integrable adapted process the problem is conventionally called the stochastic lq the abbreviation of linear quadratic problem we are concerned with the following general case the coefficients a b ci di q n and m are allowed to be adapted processes or random matrices we prove the existence and uniqueness result for the associated riccati equation which in our general case is a backward stochastic differential equation with the generator the drift term being highly nonlinear in the two unknown variables this solves bismut and peng s long standing open problem for the case of a brownian filtration which was initially proposed by the french mathematician j m bismut in seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 we also provide a rigorous derivation of the riccati equation from the stochastic hamilton system this completes the interrelationship between the riccati equation and the stochastic hamilton system as two different but equivalent tools for the stochastic lq problem there are two key points in our arguments the first one is to connect the existence of the solution of the riccati equation to the homomorphism of the stochastic flows derived from the optimally controlled system actually we establish their equivalence as a consequence we can construct solutions to a sequence of suitably modified riccati equations in terms of the associated stochastic hamilton systems and the optimal controls the second key point is to establish a new type of a priori estimate for solutions of riccati equations with which we show that the sequence of constructed solutions has a limit which is a solution to the original riccati equation

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

dynamic programming for general linear quadratic optimal stochastic control with random coefficients we are concerned with the linear quadratic optimal stochastic control problem where all the coefficients of the control system and the running weighting matrices in the cost functional are allowed to be predictable but essentially bounded processes and the terminal state weighting matrix in the cost functional is allowed to be random under suitable conditions we prove that the value field mathbb v t x omega t x omega in 0 t times r n times omega is quadratic in x and has the following form mathbb v t x langle mathbb k tx x rangle where mathbb k is an essentially bounded nonnegative symmetric matrix valued adapted processes using the dynamic programming principle dpp we prove that mathbb k is a continuous semimartingale of the form mathbb k t mathbb k 0 int 0 t dk s int 0 t sum i 1 d l s i dw s i t in 0 t with k being a continuous process of bounded variation and the stochastic integral int 0 cdot sum i 1 d l i s dw i s being a bound

on the design of optimal noisy channel scalar quantizer with random index assignment the general approach in a noisy channel scalar quantizer design is an iterative descent algorithm which guarantees only a locally optimal solution while sufficient conditions under which the local optimum becomes a global optimum are known in the noiseless channel case such sufficient conditions were not derived for the noisy counterpart moreover efficient globally optimal design techniques for general discrete distributions in the noiseless case exist however they seem not to extend to the noisy scenario when a fixed index assignment is assumed recently the design of noisy channel scalar quantizer with random index assignment ria was proposed using a locally optimal iterative algorithm in this paper we derive sufficient conditions for the uniqueness of a local optimum which thus guarantee the global optimality of the solution these sufficient conditions are satisfied for a log concave probability density function which is additionally symmetric around its mean furthermore we show that assuming an ria the globally optimal design for general discrete sources can also be carried out efficiently

quantizer monotonicities and globally optimal scalar quantizer design new monotonicity properties of optimal scalar quantizers are discussed these monotonicities reveal a globally optimal scalar quantizer structure depending on the probability mass functions and on the number of quantizer levels by incorporating the monotone quantizer structure into a dynamic programming process the time complexities of previous algorithms for designing globally optimal scalar quantizers can be significantly reduced for very general classes of distortion measures

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

upper bounds on empirically optimal quantizers in designing a vector quantizer using a training sequence ts the training algorithm tries to find an empirically optimal quantizer that minimizes the selected distortion criteria using the sequence in order to evaluate the performance of the trained quantizer we can use the empirically minimized distortion that we obtain when designing the quantizer several upper bounds on the empirically minimized distortions are proposed with numerical results the bound holds pointwise i e for each distribution with finite second moment in a class from the pointwise bounds it is possible to derive the worst case bound which is better than the current bounds for practical training ratio spl beta the ratio of the ts size to the codebook size it is shown that the empirically minimized distortion underestimates the true minimum distortion by more than a factor of 1 1 m where m is the sequence size furthermore through an asymptotic analysis in the codebook size a multiplication factor 1 1 e sup spl beta spl beta spl ap 1 1 spl beta for an asymptotic bound is shown several asymptotic bounds in terms of the vector dimension and the type of source are also introduced

minimization of risk and linear quadratic optimal control theory this article is concerned with the optimal control problem for the linear stochastic system xt x t 0 asxs bsus fs ds t 0 d i 1 ci s xs di s us gi s dwi s with the convex risk functional j u em xt et 0 g t xt u t dt in order to guarantee the existence of an optimal control without any weak compactness assumption on the admissible control set we assume that the risk function m is coercive and thatd 1 d di is uniformlypositive rather than to assume like in the control literature that the running risk function g is coercive with respect to the control variable in this new setting the running risk function g maybe independent of the control variable and therefore the so called singular linear quadratic lq stochastic control problem is included a rigorous theoryis developed for the general stochastic lq problem with random coefficients and the bounded mean oscillation martingale theoryis used to account for the concerned integrability it plays a crucial role in the following exposition a to connect the stochastic lq problem to two associated backward stochastic differential equations bsdes one is an n n symmetric matrix valued nonlinear riccati bsde and the other is an n dimensional linear bsde with unbounded coefficients b to show that the latter bsde has an adapted solution pair of the suitablynecessaryregularity this seems to be the first application in a stochastic lq theory of the bmo martingale theory which roots in harmonic analysis furthermore with the help of an a priori estimate on the risk functional existence and uniqueness of the solutions of backward stochastic riccati differential equations bsrdes in the singular case is reduced to the regular case via a perturbation method and then a new existence and uniqueness result on bsrdes is obtained for the singular case

a kind of optimal investment problem under inflation and uncertain time horizon abstract in this paper we study a kind of optimal investment problem under inflation and uncertain time horizon it can be generally formulated into a stochastic optimal control problem in particular for the constant relative risk aversion utility we employ the method of completion of squares to give an explicit form of optimal portfolio and maximum utility by the solution of a stochastic riccati equation whose wellposedness is obtained and also of significance in its own right the most distinguishing result of our work is that the randomness of exit time actually affects not only the optimal portfolio but also the maximum utility in the case of stochastic market parameters moreover we present several numerical examples to show the application of theoretical results and further discuss the influence of inflation and random time horizon from the economic viewpoint

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

open loop and closed loop solvabilities for stochastic linear quadratic optimal control problems this paper is concerned with a stochastic linear quadratic lq optimal control problem the notions of open loop and closed loop solvabilities are introduced a simple example shows that these two solvabilities are different closed loop solvability is established by means of solvability of the corresponding riccati equation which is implied by the uniform convexity of the quadratic cost functional conditions ensuring the convexity of the cost functional are discussed including the issue of how negative the control weighting matrix valued function r cdot can be finiteness of the lq problem is characterized by the convergence of the solutions to a family of riccati equations then a minimizing sequence whose convergence is equivalent to the open loop solvability of the problem is constructed finally some illustrative examples are presented

a random matrix model of relaxation we consider a two level system coupled to a general n level system via a random matrix we derive an integral representation for the mean reduced density matrix t of in the limit n and we identify a model of which possesses some of the properties expected for macroscopic thermal reservoirs in particular it yields the gibbs form for we also consider an analog of the van hove limit and obtain a master equation markov dynamics for the evolution of t on an appropriate time scale

universality of quantum brownian motion abstract are markovian master equations for quantum brownian motion independent of model assumptions used in the derivation and thus universal with the aim of answering this question we use a random band matrix model for the system bath interaction to derive markovian master equations for the time evolution of one dimensional quantum systems weakly coupled to a heat bath we study in detail two simple systems the harmonic oscillator and the two level system our results are in complete agreement with those of earlier models like the caldeira leggett model and in the large band limit with the agarwal equations both with and without rotating wave approximation this proves the universality of these master equations

a multigrid approach to the scalar quantization problem a multigrid framework for the one dimensional scalar fixed rate quantization problem is presented the framework is based on the lloyd max iterative process which is a central building block in many quantization algorithms this process iteratively improves a given initial solution and generally converges to a local minimum of the quantization distortion contrary to the classical lloyd max process the convergence of the multigrid algorithm is practically independent of the number of representation levels sought using this approach a local minimum is reached at the cost of just a few lloyd max iterations the complexity of the proposed method is o n operations for the continuous case and 3n o n for the discrete case where n is the number of representation levels sought and n is the size of the discrete probability density function in addition to its independent attributes this work is a precursor to the more important vector quantization problem for which a multiscale framework is also being developed

dynamics of a simple quantum system in a complex environment we present a theory for the dynamical evolution of a quantum system coupled to a complex many body intrinsic system environment by modelling the intrinsic many body system with parametric random matrices we study the types of effective stochastic models which emerge from random matrix theory using the feynman vernon path integral formalism we derive the influence functional and obtain either analytical or numerical solutions for the time evolution of the entire quantum system we discuss thoroughly the structure of the solutions for some representative cases and make connections to well known limiting results particularly to brownian motion kramers classical limit and the caldeira leggett approach

design of optimal scalar quantizer for sequential coding of correlated sources this paper addresses the design of a sequential scalar quantizer ssq for finite alphabet correlated sources in the fixed rate fr and entropy constrained ec cases the optimization problem is formulated as the minimization of a weighted sum of distortions and rates the proposed solution is globally optimal for the class of ssqs with convex cells and is based on solving the minimum weight path mwp problem in the ec case respectively a length constrained mwp problem in the fr case in a series of weighted directed acyclic graphs the asymptotic time complexity is o k 1 2 k 2 2 where k 1 and k 2 are the respective sizes of the alphabets of the two sources additionally it is proved that by applying the proposed algorithms to discretizations of correlated sources with continuous joint probability density function the performance approaches that of the optimal ec ssq respectively fr ssq with convex cells for the original sources as the accuracy of the discretization increases extensive experiments performed with correlated gaussian sources validate the effectiveness in practice of the proposed approach in approximating the optimal ssq for the case of continuous alphabet sources