extreme events take place frequently in both nature and society . for instance , the recurrence of floods , droughts , earth quakes and economic recession are all examples of extreme events .the consequences of extreme events to life and property are often enormous and hence it is desirable to study their properties and questions related to their predictability .interestingly , all of these extreme events are also non - equilibrium phenomena and studying the extreme value statistics in them will lead to a better understanding of the models and the phenomenology of non - equilibrium statistical physics .thus , there is an increasing interest in the physics literature to understand a broad range of issues and phenomena connected with the occurance of extreme events and their dynamics .  in the classical extreme value theory ,the limiting distribution for the extreme maximal values in sequences of independent and identically distributed random variables can be one of the frchet , gumbel or weibull distribution depending on the behavior of the tail of the probability density .this has been empirically verified in many cases of practical interest .many new applications continue to be discovered , for example , the recent one being the distribution of extreme components of the eigenmodes of quantum chaotic systems .in contrast to the questions about the distribution of extrema , one of the problems being addressed in the last few years is the distribution of the returns intervals for the extreme events when the underlying time series displays long memory .this is primarily motivated by the fact that many of the natural and socio - economic phenomena , e.g. , daily temperature , dna sequences , river run - off , earth quakes , stock markets etc . ,display long memory or long range correlation .long memory implies slowly decaying auto correlation function of the power law type such that the system does not exhibit typical time scales . in this case , the intervals between extreme events are likely to be correlated as well . on the contrary, it is known that for an uncorrelated time series , intervals between extreme events are also uncorrelated and are exponentially distributed .the question is how the presence of long range correlation modifies the return interval distribution of extreme events ?a definite answer to this question would shed new light on many problems across various disciplines .return interval distributions are interesting and useful for several reasons , the most important being that many problems in diverse fields can be formulated in terms of return interval statistics with wide ranging applications .for instance , the problem of recurrence time interval between earthquakes above a given magnitude , x - ray solar flare recurrences , statistics of acoustic emission from rock fractures , inter arrival packet times on computer and cellular networks and the classical problem of poincare recurrences in hamiltonian systems can all be formulated as extreme event questions involving return interval distribution . in a non - stationary time series , it is often difficult to reliably estimate its temporal statistical properties such as the autocorrelations or higher order correlations . thus , return interval distributions are also a useful tool to characterize temporal properties of such systems .let denote a sequence of random variable , where is the time index .we will call an event extreme if where is some threshold value .the return interval is the time between successive occurance of extreme events . with respect to threshold , we have a well - defined series of return intervals , , .this is schematically shown in fig [ scheme ] .if the random variables are uncorrelated , then the return intervals are also uncorrelated and they are exponentially distributed as in order to use later , we also define the average return interval dependent on threshold to be in contrast to an uncorrelated time series , a long range correlated series has an autocorrelation function that displays power law of the form , where denotes the temporal average and is the auto correlation exponent . the work done in the last few years show that the long range correlation does indeed affect the return interval distribution of extreme events .empirical results in a series of papers have shown that , in the presence of long range correlation , the return interval distribution becomes a stretched exponential given by , with scaled return intervals being defined as . both and are constants that depend on .they can be fixed by normalizing both the probability and the average return interval to unity .it has also been shown that the return intervals themselves are long range correlated .as a function of time .,width=226 ]  however , an analytical justification for the stretched exponential distribution in eq . [ rtd1 ] is still lacking and the main contribution of this paper is to partly fill this void . in this context , it must be noted that deviations from the stretched exponential distribution in eq .[ rtd1 ] have been noted for return intervals shorter ( ) than the average . for short return intervals , i.e , , empirical results display a power law with the exponent  , which is not explained by eq .[ rtd1 ] . while the return interval distribution is expected to depend on the threshold , the stretched exponential form does not explicitly reveal this dependence .this paper addresses these questions using a combination of analytical and numerical results .firstly , from theoretical arguments , we obtain an approximate expression for the return interval distribution , which modifies eq .[ rtd1 ] from a purely stretched exponential form to a product of power law and stretched exponential .secondly , we systematically study the dependence of return interval distribution on the threshold and show that our analytical result holds good in the limit of .in general , the return interval distribution depends on the value of threshold .recently , in the study of global seismic activity above some magnitude , the distribution of recurrence times have been shown to follow a scaling ansatz of the form where the function is the gamma distribution .in fact , this scaling relation seems to hold good for forest fire occurance intervals , tsunami inter event times and ion channel currents in voltage dependent anion channels in the cell .the analytical distribution obtained in this paper might shed light on this scaling found in a variety of systems .in the next section , we obtain an analytical expression for the return interval distribution and in the subsequent section we present our numerical results .further , we systematically study the dependence of the return interval distribution on the threshold used to define the extreme event .finally , we present our conclusions .the starting point of our approach is to transform the given long range correlated time series with autocorrelation exponent into a binary sequence with 1 at positions of extreme events and 0 elsewhere .thus , we obtain new binary sequence defined by , we use the empirical result that for a long range correlated time series with the autocorrelation exponent , the return intervals are also long range correlated with the same exponent .thus , our probability model is the statement that given an extreme event at time , the probability to find an extreme event at time is given by , where is the hurst exponent and is the normalization constant that will be fixed later . we have also used the well - known relation between hurst exponent and autocorrelation exponent ; .equation [ probmodel ] implies that after an extreme event it is highly probable to expect the next event to be an extreme one too ; and this is a reasonable proposition for a persistent time series .notice also that for an uncorrelated time series .this leads to in eq .[ probmodel ] becoming independent of , as would be expected for an uncorrelated time series .further support for this proposition comes from the theorem due to newell and rosenblatt obtained in the context of zero crossing probabilities for gaussian processes .it states that for a separable gaussian stationary process with mean , the probability that for is as , where .next we calculate the probability that given an extreme event at time , no extreme event occurs in the interval . for this, we divide the interval into sub - intervals indexed by and we calculate this probability in each of the intervals . for the sub - interval , using eq .[ probmodel ] , the probability of extreme event is given by ,  \nonumber\end{aligned}\ ] ] after simplifying this expression , the probability that no extreme event occurs in the sub - interval is given by , \ ] ] at this point , we make an approximation and assume that the probability of no extreme event occurance in each sub - interval to be an independent event .then , the probability that no extreme event occurs in any of the sub - intervals in is simply the product of probabilities ,   the required probability is simply the product of with the probability that an extreme event takes place in the infinitesimal interval beyond .this can be assembled together as ,                  \left[1-\phi_{m , r } ( 2^{-\gamma}+1 ) \right ] \label{prob1 } \\     &  &          \left[1-\phi_{m , r } ( 3^{-\gamma}+2^{-\gamma } ) \right ] \ldots\ldots \nonumber \\     &  &  \left[1-\phi_{m , r }  ( m^{-\gamma}+(m-1)^{-\gamma } ) \right ] ~ a ~r^{-(1-\gamma ) } dr \nonumber\end{aligned}\ ] ] where , the value of can be arbitrarily large and the eq .[ prob1 ] can be simplified and rewritten as , where is the generalized harmonic number . in order to take the limit , we note that using this eq . [ limh ] in eq .[ prob2 ] and taking the limit , we obtain the following result for the distribution of return intervals ; the constant will be fixed by normalization as follows ; we demand that the total probability and the average return interval be normalized to unity . however , the distribution in eq . [ rtd2 ] is already normalized and hence eq . [ normint ] will be used to determine the value of .the requirement that is equivalent to transforming the return intervals in units of . performing the integrals above, the normalized distribution in the variable turns out to be , ^{\gamma } ~r^{-(1-\gamma ) } ~e^{-\left [ \gamma\left(\frac{1+\gamma}{\gamma}\right ) \right]^{\gamma } r^{\gamma } } \label{rtd3}\ ] ] where is the gamma function .first , we discuss some of the salient features of this distribution .the case defines the crossover to short range or uncorrelated time series . if we put in the distribution in eq .[ rtd3 ] above , we recover the exponential distribution , . in the region , i.e. , for the return intervals much below the average, the dominant behavior can be seen by taking logarithm on both sides of eq .[ rtd3 ] leading to , where we have used ^{\gamma}$ ] . for ,the second term dominates the distribution and thus we obtain a power law with an exponent ; this power law behavior with exponent for short return intervals has already been noted in the numerical results presented in ref .thus , our approach analytically shows the emergence of a power law regime for short in contrast to the stretched exponential distribution . on the other hand , for , the logarithmic term in eq .[ logp ] can be dropped and the return interval distribution behaves essentially like a stretched exponential distribution , thus , stretched exponential is a good approximation for .this partly explains why a pure stretched exponential distribution as in eq .[ rtd1 ] deviates , for , from the simulated return interval distributions in the earlier works .finally , we also note that eq .[ rtd3 ] can also be derived by other methods without actually discretising the interval as we have done .  as shown above , the return interval distribution in eq .[ rtd3 ] does reproduce the empirical results already known in the literature but is nevertheless approximate in the following sense .it is known that there exist correlations among the return intervals and they are particularly strong as .thus , every return interval depends on the value of previous return interval .this is also well documented in the literature as the conditional probability to find return interval , given that the previous return interval was  .this conditional probability shows interesting features and deviates from the case of uncorrelated return intervals .equation [ rtd3 ] does not take into account these correlations among intervals and in fact is derived on the assumption that return intervals are independent .this is a gross approximation though in the absence any other definitive model for the correlations among intervals this is a simple and analytically tractable choice .based on this argument , one can expect eq .[ rtd3 ] to describe the return interval statistics in the regime where the correlations are not highly dominant , for  .secondly , note that even though threshold plays a crucial role as we will describe in the next section , it does not play any role in eq .threshold is related to such that higher the value of , larger is , though it is not a linear relation .thus , the theoretical arguments leading to eq . [ rtd3 ] would best describe an asymptotic limit of or .( solid lines ) for long range correlated time series .the threshold is with average return interval for all the cases shown above.,width=529 ]  using eq .[ rtd3 ] in practice can lead to strong divergence for . from a physical standpoint , this represents a problem that can be understood based on the fact that there can not be zero return intervals , but they can be arbitrarily small . by definition , and if is the shortest return interval then its corresponding scaled version would be . if the original signal is sampled at equal time intervals , can be scaled to unity and the shortest scaled return interval would be . the modification of eq . [ rtd3 ] should be done by replacing the lower limit in the integrals in eqs [ normint0 ] and [ normint ] by instead of 0 .this also reflects the general idea that all power laws in practice have a lower bound and the return interval distribution like the eq .[ rtd3 ] that displays a power law type regime will necessarily have a lower cut off .we will go back to eqn .[ rtd2 ] and rewrite the return interval distribution as where and are constants that would now depend on both and the average return interval . as usual , both these constants will be fixed by demanding that probability and average return interval normalize to unity .this leads to the following set of integrals ;  where , and is the incomplete gamma function .the algebraic equations to be solved for and are transcendental in nature and closed form solution does not seem possible except for some special values . by further manipulation of eqns .[ renormint1 ] and [ renormint2 ] , we obtain if is the solution of eq [ trans1 ] for a definite , then the constants can be obtained as , in the simulations shown in this paper , we have numerically solved for constants and in eq .[ rtd4 ] for various values of using eqns [ trans1 ] and [ trans2 ] .in this section , we display the numerical results for the return interval distribution of long range correlated time series drawn from a gaussian distribution with zero mean and unit variance . the long range correlated data was generated using the fourier filtering technique .we generate data points for each values of and then compute their return interval distribution .the numerical results are displayed in fig [ distrib1 ] as log - log plot for along with the theoretical distributions given in eqns .[ rtd3 ] and [ rtd4 ] .the agreement with the theoretical distribution is good and as expected gets better as .similar good agreement is also obtained for the values of not shown here .the simulated results in fig [ distrib1 ] does not cover a larger range in because of the large value of threshold chosen corresponding to an average return interval of . to over come this problem, we will need extremely large sequences of random time series .as we have argued in the previous section , the theoretical distribution can be expected to agree with the data when threshold or equivalently the average return interval is large .thus , as we reduce below 2.5 , there are deviations from the theoretical distribution which are systematically studied in the next section .  in fig [ plaw ] ,we show the power law regime indicated by eq .[ power ] . in this figure ,we focus on the region where we expect the power law to appear . for each value of in fig[ plaw ] , we have drawn a straight line ( shown in red ) with the slope . quite clearly, the numerical data show a remarkably good agreement with the theoretical slope . as , the power law regime holds good in a larger range of ; for instance , see the case of . on the other hand , as seen in the case of , the power law region becomes shorter and stretched exponential regime begins to dominate as .this is an indication that the return interval distribution makes a transition from predominantly ( stretched ) exponential behavior to predominantly power law type curve as .it must be pointed out that the agreement with theoretically expected slope is reached only for .this is to be expected since the derived distributions in eqns [ rtd3 ] and [ rtd4 ] do not take into account the correlations among the return intervals . in the next section ,we study how the slope in the power law regime changes with threshold in the numerically simulated long range correlated data .  .a straight line with slope is shown as solid ( red ) line for comparison .for all the cases , corresponding to .,width=283 ]in this section , we will empirically examine the relation between the return interval distribution , especially in the power law regime , and the threshold that define the extreme events .intuitively , we can expect that if the threshold is higher , extreme events will be fewer and hence the return intervals will be longer . thus , larger leads to larger average return intervals . herewe address the question of how the return interval distributions in eqns [ rtd3 ] and [ rtd4 ] are modified by changes in threshold value .one clear indication is that , approximately for , the simulated return interval distributions deviate systematically from eqns [ rtd3 ] and [ rtd4 ] , in particular for . to study this , we plot the return interval distribution for the simulated data in a log - log plotas shown in fig [ distrib1 ] and measure the slope in a linear region for for various values of .the result is displayed in fig [ rtdvsq ] for .it is seen that as increases , the initial part of the distribution , i.e , or , is closer to being a straight line with slope .a similar behavior is seen for all the values of of our interest.   plotted for various values of threshold .a straight line with slope is shown as solid ( blue ) line for comparison .note that as increases , the initial part of the distribution moves closer to a slope of .,width=309 ]  in order to see this variation of the slope of the initial part of the distribution with , we plot in fig [ slopeq](a ) the measured slope against the threshold for various values of .the slope is measured in the linear region in log - log plot for . for a given value of , the slope increases monotonically to reach a saturation value of as .once again we point out that this is in agreement with our expectation that the weakly correlated regime would agree with the distribution obtained in eq .[ rtd3 ] and [ rtd4 ] . for the gaussian distributed data that we use , at , the average return interval is . beyond with datapoints the number of returns intervals are not sufficient for reliable statistics .all this would imply that in order to take into account the effect of , the power law proposed in eq .[ power ] could be modified as with the restriction , suggested by the numerical results in fig [ slopeq](a ) , that as .clearly , the measured slope is simply given by .thus , we can directly visualize the function if we plot as a function of .this is shown in fig [ slopeq](b ) .as we anticipated , the function tends towards unity as .the autocorrelation exponent controls the rate at which the limiting value of unity is reached .we believe that the behavior displayed in fig [ slopeq](a , b ) is related to a more fundamental question of how the auto - correlation exponent of a long range correlated time series changes if it is subjected to thresholding such as the one we have applied using eq .[ threshold ] . obviously , every time we choose a subset of events from a larger set , such as the extreme events , implicitly such thresholding is applied . since the power law regime varies with andif the distribution has to remain normalized , then the stretched exponential part would also be modified . however, this might be difficult to visualize numerically .the central premise of this section is to show that eqns [ rtd3 ] and [ rtd4 ] represent return interval distributions in the limit when the threshold or average return interval is large .we have shown through simulations the dependence of return interval distributions on threshold .this explains why we have chosen to illustrate our result in fig [ distrib1 ] .thus , in principle , the exact return interval distribution should depend on , especially for short return intervals , i.e , .   in the power law regime as a function of for ( circles ) , 0.3 ( squares ) , 0.5 ( triangles ) and 0.7 ( plus ) .( b ) the function as a function of for same values of as in ( a).,width=309 ]apart from corrections arising due to dependence on , the return interval distribution derived in this paper suffers due to approximation arising from assumptions of independence of return intervals .this assumption makes the analysis tractable but does not reflect the reality since we know that the intervals are indeed correlated . in this section ,we argue that the deviations from the numerical simulations evident in fig [ distrib1 ] can be attributed to the presence of correlations in the return interval data .we do this by simulating the probability process in eq .[ probmodel ] that forms the basis for the analytical result in eqn [ rtd3 ] and [ rtd4 ] .if the simulated data agrees with the analytical result , then we could attribute the deviations seen in fig [ distrib1 ] to the correlations present in the return intervals .compared with the theoretical distribution ( solid line ) given in eq [ rtd4].,width=309 ]  in order to numerically simulate the probability process in eq .[ probmodel ] , we first determine the constant by normalizing it in the region and . the normalized probability distribution corresponding to eq .[ probmodel ] is where .we generate a random number from a uniform distribution at every and compare it with the value of .a random number is accepted as an extreme event if at any given value of .if , then it is not an extreme event . by this procedure, we generate a series of extreme events following eq .[ probmodel ] .we then compute the return intervals and its distribution after scaling it by the average return interval . in fig .[ pmodel ] , we show the return interval distribution obtained by simulating our probability process along with the distribution given by eq [ rtd4 ] .the agreement with the theoretical distribution is excellent , including for the values of not shown here .hence , if the long range correlated data had independent return intervals , then we would have obtained nearly perfect agreement with eq [ rtd3 ] and [ rtd4 ] .this implies that the remaining disagreement between the theoretical and numerical results seen in fig [ distrib1 ] can be attributed to the presence of correlations among the return intervals . on the other hand ,if the probability process in eq [ probmodel ] was an incorrect assumption , it may not have been possible to obtain the results displayed in fig [ distrib1 ] .we have studied the distribution of return intervals for the extreme events in long range correlated time series .an approximate analytical expression for this distribution has been obtained starting from the empirically established fact that returns intervals are long range correlated .this distribution is a product of a power law and a stretched exponential and explains the observed power law for short return intervals . for large return intervals ,the distribution is dominated by a stretched exponential decay .the works reported earlier have empirically proposed stretched exponential form for the return interval distribution which is now shown to be valid in the domain of large return intervals .further , we have also carefully studied the role played by the threshold or equivalently the average return interval in the return time statistics .we show that it modifies the return interval distribution , especially in the power law regime of short return intervals .we believe that the results obtained in this paper explains most of the empirically observed features in the return time distributions of long range correlated time series . in the simulations reported in this work ,we have used gaussian distributed random numbers . as studied in ref . , it is natural to ask if the exponential or power law distributed data would modify the results of this paper .we expect that the functional form of the distribution in eq [ rtd4 ] would not be modified though the normalization constants and might change due to their dependence on the threshold .the question of verifying the results of this paper with a measured time series is underway and would be reported elsewhere .  as pointed out before , the inter - event time distribution has applications across many disciplines .hence , it appears in different settings in different areas . in the statistical literature , a related problem of zero crossings , i.e ,the probability that for has been considered . under certain conditions , for a stationary gaussian process , the upper bound for zero crossing probability is shown to be a stretched exponential .this result does not strictly apply to the case of recurrence interval statistics because the zero crossing probability does not make statements about occurrence or non - occurance of another zero crossing after the interval .a return interval , by definition , requires two crossings separated by an interval with no crossings .finally , we would like to remark that the analytical distribution obtained in this paper appears to be related to the universal scaling form proposed recently in the context of earth quakes but appears to be more generally valid .thus it is likely that the exact return interval distribution might incorporate corrections to the one obtained in this paper . indeed ,if the exact distribution is known , it will also become possible to determine the precise time scales over which power law and exponential decay operate .this , in turn , should help address questions of hazard estimation for extreme events more carefully and , needless to say , this has enormous interest in the insurance industry and as a tool for decision support system .a. bunde , j. f. eichner , s. havlin , and j. w. kantelhardt , physica a * 330 * , 1 ( 2003 ) ; m. s. santhanam and h. kantz , physica a * 345 * , 713 ( 2005 ) ; a. bunde , j. f. eichner , j. w. kantelhardt and s. havlin , phys . rev . lett . * 94 * , 048701 ( 2005 ) ; m. i. bogachev , j. f. eichner and a. bunde , physlett . * 99 * , 240601 ( 2007 ) .                    for the classical extreme value theory ,it is known that when the correlations are not strong enough , then the distribution of extreme values converges to that of the independent , and identically distributed random variables , namely the frchet , gumbell and weibull distributions .see , j. pickands , trans . am .soc . * 145 * , 75 ( 1969 ) ; s. m. berman , ann .* 35 * 502 ( 1964 ) .

Abstract: the distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines . it is well known that many physical processes in nature and society display long range correlations . hence , in the last few years , considerable research effort has been directed towards studying the distribution of return intervals for long range correlated time series . based on numerical simulations , it was shown that the return interval distributions are of stretched exponential type . in this paper , we obtain an analytical expression for the distribution of return intervals in long range correlated time series which holds good when the average return intervals are large . we show that the distribution is actually a product of power law and a stretched exponential form . we also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events .


in , shannon defined the _ zero error capacity _ of a memoryless channel with finite input and output alphabets by the supremum of achievable rates of codes which have no decoding error .there are several reasons to study the zero error capacity  .firstly , the zero error capacity indicates the supremum of the achievable rates at which no errors can be tolerated and it thus indicates one of fundamental properties of the channel .secondly , the zero error capacity suggests a property of codes with extremely low decoding error probability on an environment which does not allow to lengthen the code length enough .shannon proposed the zero error capacity and , in addition , showed the equivalence to the maximum independent set problem on an undirected graph called a confusion graph  .exact evaluation of the zero error capacity of a channel is a challenging problem .no general solution that gives the exact value of the zero error capacity has been known .therefore , there have been many studies for deriving upper and lower bounds of the zero error capacity on general channels .for example , shannon showed that the logarithm of the independence number of a given confusion graph is a lower bound of the zero error capacity of the corresponding channel  .he also presented an upper bound based on the logarithm of the optimal value of a linear programming ( lp ) problem .unfortunately , this upper bound is not so tight for many problems in general .for example , this lp upper bound is larger than the zero error capacity of the channel corresponding to the cycle graph with 5-edges , i.e. , the pentagon graph .this means the lp upper bound is useless to determine the exact value of the zero error capacity of the pentagon graph .lovsz showed a tighter upper bound than the lp upper bound .the value of this upper bound is called the _ lovsz number _ , which can be computed by solving a semi - definite programming problem .it is well known that the lovsz number is equal to the value of the trivial lower bound in the case of the pentagon graph , i.e. , this means that the lovsz number is equal to the exact value of the zero error capacity .however , the zero error capacity of the cycle graph with 7-edges remains to be open .it is known that , if a confusion graph is a perfect graph , the upper and lower bounds shown by shannon coincide with each other .therefore , the zero error capacity of the channel corresponding to a perfect graph can be exactly evaluated  .  in the nne channels ,a transmitted symbol is a -tuple of elements in .it is assumed that only one element error to a nearest neighbor element in a transmitted symbol can occur .figure [ fig : nne_channel ] presents the channel transition graph of the nne channel in which a transmitted symbol is a -tuple of elements in , i.e , .for example , from the adjacency relations of the symbols shown in fig .[ fig : nne_channe_adj ] , the symbol can be erroneously received as or , which are the nearest neighbors of .the nne channels can be considered as a special type of limited magnitude error channels , and it is closely related to error models for flash memories .when , the confusion graphs of the nne channels are perfect graphs .therefore the zero error capacity can be exactly evaluated in such cases . however , if , the zero error capacity of these channels has been unknown .  )-tuple of elements in , i.e. , .black arrows in ( b ) represent possible symbol transitions . ,title="fig:",width=113 ] [ fig : nne_channel ]  ) -tuple of elements in , i.e. , .black arrows in ( b ) represent possible symbol transitions . ,title="fig:",width=113 ] [ fig : nne_channe_adj ]  in this paper , we will derive a lower bound of the zero error capacity of the nne channels based on a result of the perfect lee codes .furthermore , an upper bound of the zero error capacity will be derived from a feasible solution of an lp problem defined based on the confusion graph .the outline of this paper is as follows . in section 2, we will introduce some definitions and basic results on the zero error capacity and lee codes which are used in this paper . in section 3, the nne channel will be defined .afterwards , we will show upper and lower bounds on the zero error capacity of the nne channels . finally , a concise expression for the zero error capacity will be shown .in this section , we introduce several definitions and notation to be used in this paper .   a discrete memoryless channel ( dmc ) is defined by the condition probability that is the probability of receiving when is sent over .the symbols and represent the finite input and output alphabets , respectively .the _ confusion graph _ of the channel is defined as follows .let be a dmc .the undirected graph is defined by the graph is called the confusion graph of .figure [ fig : nne_graph ] shows the confusion graph of the channel shown in fig .[ fig : nne_channel ] ., width=151 ]  the strong product is used in the definition of the zero error capacity .the definition of the strong graph product is given as follows .let and be undirected graphs .their strong product is defined as the -times power of is denoted by    we denote the _ independence number _ of an undirected graph by . the independence number is the size of the maximum independent set .shannon defined the zero error capacity of the discrete memoryless channel as follows .let be a dmc .the _ zero error capacity _ is defined by   it is straightforward to show that the zero error capacity can be written as where the capacity of the confusion graph is defined by {\alpha({g}^n)}.\end{aligned}\ ] ]  from the above definitions , is clearly a lower bound on the capacity of . in this paper , we refer to this lower bound as the _ shannon s lower bound_.  the following theorem is known as an upper bound on the capacity of a graph  [the : lp_upperbound ] let be the set of all the cliques in where is a given undirected graph .consider the following lp problem : where is a real - valued variable corresponding to the clique .the optimal value of the above lp problem is an upper bound on the capacity of .  in this paper, we call this bound the _ lp upper bound_.   we introduce definitions of the lee distance and the lee codes in this subsection according to  . in the following sections , the residue class ring modulo is denoted by , and the coset including integer is denoted by ] .let be an element in .the absolute value on is defined as   based on the definition of the absolute value on , the lee distance is defined as follows .the distance function is given by where , .the distance defined by is called the _ lee distance_.we further introduce some definitions in order to define the perfect lee codes .assume that and are given .the lee sphere is defined by   [ def : lee_code ] let be a subset of . if holds for any  , then is said to be an -lee error correcting code .the perfect lee code is one of the important tools for the proof of our main results .let be an -lee error correcting code of length .if holds , then is called a _ perfect lee code _ , which is denoted by .  for given ,it is clear that has codewords .the following theorem proved by albdaiwi et al . states the condition on and that guarantees the existence of .[the : perfect_lee ] for a given , assume that the prime factors of are .a perfect lee code exists if and only if .  from theorem [ the : perfect_lee ], we immediately have the following corollary .[ cor : lee_perfect ] a perfect lee code exists for any .  as an example , fig .[ fig : lee_sphere ] shows a perfect lee code of length over .the code has 5-codewords such as where represent the cosets , \ldots , [ 4]_5 ] be a dmc .if holds for any ^d ] is transmitted , the receiver side receives a symbol contained in ^d \mid d_{l_1}({\boldsymbol{x } } , { \boldsymbol{y } } ) \leq 1 \right\}.   \end{aligned}\ ] ] from the definition of confusion graphs , ^d } w({\boldsymbol{y}}|{\boldsymbol{x}}_1)w({\boldsymbol{y}}|{\boldsymbol{x}}_2 ) \ne 0     \iff \{{\boldsymbol{x}}_1 , { \boldsymbol{x}}_2\ } \in e   \end{aligned}\ ] ] holds for any ^d ] satisfying the following two conditions : and for any ( ) .let be a perfect lee code where  ( ) is a multiple of .the existence of such a perfect lee code is guaranteed by corollary [ cor : lee_perfect ] .let be the set defined as ,\\         u_i \in \set{x_i + [ j]_q | j \in [ 0,n-1 ] }       \end{array } \!\!\!\ !     } ,   \end{aligned}\ ] ] where , . from the definition of ,it is clear that the identity = n^d   \end{aligned}\ ] ] holds for all .the function  ] is defined as where .finally , ^d ] , the following inequality holds due to lemma [ lem : psi_inequality ] .this inequality directly leads to the definitions of distance function can transform this inequality to the inequalities ( [ eq : v_exist_3 ] ) and ( [ eq : v_exist_4 ] ) imply the claim of this lemma .  as an example of lemma [ lem : lower_bound ] , fig .[ fig : lower_bound ] shows an instance of ^ 2 ] are denoted by , respectively .the subset equals to shown in fig .[ fig : lower_bound1 ] .therefore the subset corresponding to equals to shown in fig .[ fig : lower_bound2 ] .this subset satisfies the conditions in lemma [ lem : lower_bound ] .^2 ] satisfing the conditions in lemma [ lem : lower_bound ] .black circles are the codewords of the perfect lee code . ,title="fig:",width=151 ] [ fig : lower_bound2 ]  from the above lemmas , we are ready to prove the following theorem stating a lower bound on the capacity of the nne graphs .this theorem is our main contribution in this paper .[ the : lower_bound ] let be the confusion graph of the -nne channel .the inequality holds .let be a subset of ^d ] be the function defined as ^d \mid d_{l_1}({\boldsymbol{x}},{\boldsymbol{y } } ) \leq 1 \right\}.   \end{aligned}\ ] ] for any vertex in ^d$ ] , an adjacency clique in is called an inner adjacency clique if is an inner vertex ; otherwise is called an outer adjacency clique .we show an example of inner and outer vertices and an adjacency clique . on the confusion graph of the -nne channel shown in fig .[ fig : nne_graph ] , the vertex is an inner vertex because . on the other hand ,the vertices are outer vertices because equal to , respectively .hence , the adjacency clique in is an outer adjacency clique and consists of the vertices .the following lemma states the existence of a certain feasible solution of the lp problem ( [ eq : lp_shannon_main ] ) .[ lem : feasible ] let be the confusion graph of the -nne channel .let be the function defined as the function is a feasible solution of the lp problem ( [ eq : lp_shannon_main ] ) .for any outer vertex , it is trivial that holds .since the size of an inner adjacency clique is , the inequality ( [ eq : constraints ] ) holds for any inner vertex as well .therefore , the function satisfies constraints of the lp problem ( [ eq : lp_shannon_main ] ) .  from the above lemmas, we can prove the following theorem for an upper bound on the capacity of the nne graphs .[ the : upper_bound ] let be the confusion graph of the -nne channel .the inequality holds where   since the number of all the inner vertices in is , the feasible solution in lemma [ lem : feasible ] provides the objective function value equal to .the lp upper bound leads to the claim of the theorem .   from theorem[ the : lower_bound ] and theorem [ the : upper_bound ] , we can prove the following theorem stating a concise expression for the zero error capacity of the nne channels .[ the : zero_error ] let be the -nne channel . for any , the zero error capacity can be written as   let be a confusion graph of . from theorem [ the : lower_bound ] , the inequality ( [ eq : lower_bound ] ) holds .this inequality can be rewritten as from theorem [ the : upper_bound ] , the inequality holds .therefore , can be transformed as follows : since the inequality holds for any , can be written as from ( [ eq : o_lower_bound ] ) and ( [ eq : o_upper_bound ] ) , we thus have and an expression for the zero error capacity :   the error term in the above expression vanishes at the asymptotic limit .in this paper , we studied the zero error capacity of the nne channels .we derived a lower bound of the capacity of the nne graphs based on the idea that the zero error communication problem over the nne channels can be regarded as a packing problem of -dimensional polyominos .the perfect lee codes play a crucial role for proving the existence of a dense packing .moreover , we showed an upper bound of the capacity of the nne graphs from a certain feasible solution of the lp problem given by shannon .finally , from these upper and lower bounds , we obtained a concise expression of the zero error capacity of the nne channels with an asymptotically ( i.e. , ) vanishing error term .shannon , `` the zero error capacity of a noisy channel , '' ire transactions on information theory , vol.2 , no.3 , pp.819 , 1956 .l. lovsz , `` on the shannon capacity of a graph , '' ieee trans .information theory , vol.25 , no.1 , pp.17 , 1979 . j. korner , and a. orlitsky , `` zero - error information theory , '' ieee trans .information theory , vol.44 , no.6 , pp.22072229 , 1998 . s. jain , k. nam , and k. lee , `` on some perfect codes with respect to lee metric , '' linear algebra and its applications , vol.405 , pp.104120 , 2005 .b. albdaiwi , p. horak , and l. milazzo , `` enumerating and decoding perfect linear lee codes , '' designs , codes , and cryptography , vol.52 , no.2 , pp.155162 , 2009 .p. horak , `` on perfect lee codes , '' discrete mathematics , vol.309 , no.18 , pp.55515561 , 2009 .r. ahlswede , h. aydinian , l.h .khachatrian , and l.m.g.m .tolhuizen , `` on q - ary codes correcting all unidirectional errors of a limited magnitude , '' arxiv.org , http://arxiv.org/pdf/cs/0607132v1.pdf , 2006 .cassuto , y. , schwartz , m. , bohossian , v. , and bruck , j. ( 2010 ) .`` codes for asymmetric limited - magnitude errors with application to multilevel flash memories '' .ieee transactions on information theory , 56(4 ) , 15821595 .t. klve , `` systematic single limited magnitude error correcting codes for flash memories , '' ieee trans .information theory , vol.57 , no.7 , pp.44774487 , 2011 . n. elarief , `` limited magnitude error detecting codes over , '' ieee trans . computers , vol.62 , no.5 , pp.984989 , 2013 .

Abstract: this paper studies the zero error capacity of the nearest neighbor error ( nne ) channels with a multilevel alphabet . in the nne channels , a transmitted symbol is a -tuple of elements in . it is assumed that only one element error to a nearest neighbor element in a transmitted symbol can occur . the nne channels can be considered as a special type of limited magnitude error channels , and it is closely related to error models for flash memories . in this paper , we derive a lower bound of the zero error capacity of the nne channels based on a result of the perfect lee codes . an upper bound of the zero error capacity of the nne channels is also derived from a feasible solution of a linear programming problem defined based on the confusion graphs of the nne channels . as a result , a concise formula of the zero error capacity is obtained using the lower and upper bounds .


a diffusion process is deemed nonlocal when the associated underlying lvy process does not only consist of brownian motions .the features of nonlocal diffusion has been verified experimentally to be present in a wide variety of applications , including , e.g. , contaminant flow in groundwater , plasma physics , and the dynamics of financial markets .a comprehensive survey of nonlocal diffusion problems is given in . in this work ,we consider a partial - integral diffusion equation ( pide ) representation of linear nonlocal diffusion problems .since it is typically difficult to obtain analytical solutions of such problems , numerical solutions are highly desired in practical applications .there are mainly two types of numerical methods for nonlocal diffusion equations under consideration .the first is the family of deterministic approaches , e.g. , finite - element - type methods and collocation methods , whereas the second can be classified as stochastic approaches , e.g. , continuous - time random walk ( ctrw ) methods .the deterministic approaches are extensions of existing methods for local partial differential equations ( pdes ) that incorporate schemes to discretize the underlying integral operators .the recently developed nonlocal vector calculus provides helpful tools that allow one to analyze nonlocal diffusion problems in a similar manner to analyzing local pdes ; see for details .however , in the context of implicit time - stepping schemes , the nonlocal operator may result in severe computational difficulties coming from the dramatic deterioration of the sparsity of the stiffness matrices required by the underlying linear systems . on the other hand , stochastic approaches , e.g. , ctrw methods ,are based on the relation between nonlocal diffusion and a class of lvy jump processes .although stochastic methods do not require the solution of linear systems , and the simulations of all paths of the random walk can be easily parallelized , they also have some drawbacks .typically , most stochastic methods are sampling - based monte carlo approaches , suffering from slow convergence , thus requiring a very large number of samples to achieve small errors .even worse , in the case of a general nonlinear forcing term , the nonlocal diffusion equation is no longer the master equation of the underlying jump processes .  in this paper , we propose a novel stochastic numerical scheme for the linear nonlocal diffusion problems studied in based on the relationship between the pides and a certain class of backward stochastic differential equations ( bsdes ) with jumps . the existence and uniqueness of solutions for nonlinear bsdes with and without jumpshave been proved in and , respectively . since then , bsdes have become important tools in probability theory , stochastic optimal control and mathematical finance .unlike bsdes driven by brownian motions , there are very few numerical schemes proposed for bsdes with jumps , and most of the schemes are focussed on time discretizations only .for example , a numerical scheme based on picard s method was proposed in , and a forward euler scheme was proposed in  where the convergence rate was proved to be .our focus will be on the bsdes corresponding to nonlocal diffusion equations with _integrable _ jump kernels in unbounded domains .since the pides of interest do not possess local convection and diffusion terms , the corresponding bsdes can be simplified , such that , the underlying lvy processes are merely _ compound poisson processes_. also , motivated by various engineering applications discussed in , our goal is to develop and analyze computationally efficient numerical schemes that can achieve high - order accuracy in both temporal and spatial discretization , rather than focus on the scalability of our schemes in solving extremely high - dimensional problems ( e.g. , in option pricing ) . from this perspective , although the bsdes under consideration is a special case of the equations studied in , high - order fully - discrete schemes and the corresponding error analysis are still missing in the literature .in fact , based on our previous work , it is particularly challenging to recover the classic finite element convergence rates in the spatial discretization , even for the simplified bsdes .thus , the main contributions of this paper are summarized as follows :  * development of stable high - order matrix - free numerical schemes for the bsdes and the corresponding nonlocal diffusion problems ; * analysis of the approximation error of the proposed semi - discrete and fully - discrete schemes , and ; * numerical demonstration of the capabilities of handling multiple types of integrable kernels , e.g. , symmetric , non - symmetric , or singular kernels .specifically , the bsdes will be discretized , in the temporal domain , using a -scheme extended from our previous works for bsdes without jumps .in particular , the cases where , and correspond to forward euler , crank - nicolson and backward euler schemes , respectively . as discussed in ,a quadrature rule adapted to the underlying stochastic processes is critical to approximate all the conditional mathematical expectations in our numerical scheme .thus , we propose a general formulation of the quadrature rule for estimating the conditional expectations with respect to the compound poisson processes , and a specific form of a certain pide can be determined based on regularities of the kernel and forcing term . in [ sec : ex ] , gauss - legendre , gauss - jacobi and newton - cotes rules are substituted into the proposed quadrature rule to approximate different kernels and forcing terms . in addition , since the total number of quadrature points grows exponentially with the number of time steps , we construct a piecewise lagrange interpolation scheme based on a pre - determined spatial mesh , in order to evaluate the integrand of the expectations at all quadrature points .  both theoretical analysis and numerical experimentsshow that our approach is advantageous for linear nonlocal diffusion equations with integrable kernels on unbounded domains . compared to deterministic approaches , e.g. ,finite elements and collocation approaches , _ the proposed methods do not require the solution of dense linear systems_. instead , the pide can be solved independently at different spatial grid points on each time level , making it straightforward to incorporate parallel implementation and adaptive spatial approximation .compared to stochastic approaches , our scheme is more accurate than the ctrw method due to the use of the -scheme for time discretization , the high - order quadrature rule , and piecewise lagrange polynomial interpolation for spatial discretization . in addition , our method can handle a broad class of problems with general nonlinear forcing terms as long as they are globally lipchitz continuous .the outline of the paper is as follows . in [ sec : nlocbsde ] , we introduce the mathematical description of the linear nonlocal diffusion equations and the corresponding class of bsdes under consideration . in [ sec : scheme ] , we propose our numerical schemes for the bsdes of interest , where the semi - discrete and the fully - discrete approximations are presented in [ sec : semi ] and [ sec : fully ] , respectively .error estimates for the proposed scheme are proved in [sec : err ] .extensive numerical examples and comparisons to existing techniques are given in [ sec : ex ] , which are shown to be consistent with the theoretical results and reveal the effectiveness of our approach .finally , several concluding remarks are given in [ sec : con ] .this section is dedicated to describing the nonlocal diffusion problem that is the focus of this paper . in particular , in [ sec : nloc1 ] we introduce the definitions of a general nonlocal operator and diffusion equation .the relationship between the backward kolmogorov equation , which is a generalization of the nonlocal diffusion equation of interest , and the bsde with jumps , is described in [ sec : bsde ] . based on this equivalence , a simplified bsde corresponding to the nonlocal diffusion problem of [ sec : nloc1 ]is also given at the end of [ sec : bsde ] .let us recall a nonlocal operator introduced in .for a function \times \mathbb{r}^d \rightarrow \mathbb{r} ] , the value of at a point requires information about at points ; this is contrasted with local operators , e.g. , the value of at a point requires information about only in an infinitesimal neighborhood of .our interest in the operator is due to its participation in the _ nonlocal diffusion equation _ \times \mathbb{r}^d\\ & u(0,{\bm{x } } ) = u_0({\bm{x } } ) , \text { for } { \bm{x}}\in \mathbb{r}^d,\\ \end{aligned}\right.\ ] ] where is the initial condition and the forcing term may be a _ nonlinear_ function of and .we remark that is a nonlocal cauchy problem defined on the _ unbounded _ spatial domain .also , in the context of kernel functions that are compactly supported in , the initial - boundary value nonlocal problem with _ volume constraints _ ,i.e. , constraints applied on a regions of nonzero measure in , has been well studied .however , the well - posedness of the corresponding bsdes with volume constraints is still an open challenge . as such, we do not impose volume constraints to in this effort .now we discuss the probabilistic representation of the solution of the nonlocal diffusion equation in .the nonlinear feynman - kac formula studied in shows that bsdes driven by brownian motion provides a probabilistic representation of the solutions of a class of second - order quasi - linear parabolic pdes .this result was extended in to the case of bsdes driven by general lvy processes with jumps . due to the inclusion of lvy jumps , the counterpart of a bsde is a partial integral differential equation ( pide ) , i.e. , the backward kolmogorov equation .it turns out that such a pide is a generalization of the nonlocal diffusion equation in . herewe first introduce a general form of a bsde with jumps and the backward kolmogorov equation , then give a simplified form of the bsde corresponding to the nonlocal diffusion equation under consideration .let be a probability space with a filtration for a finite terminal time .we assume the filtration satisfies the usual hypotheses of completeness , i.e. , contains all sets of -measure zero and maintains right continuity , i.e. , .moreover , the filtration is assumed to be generated by two mutually independent processes , i.e. , one -dimensional brownian motion and one -dimensional poisson random measure on  ] is a martingale for all .we also assume that is a -finite measure on satisfying based on the stochastic basis , we introduce the backward stochastic differential equation with jumps where the solution is the quadruplet with , , and .the map \times \mathbb{r}^{d } \rightarrow \mathbb{r}^d ] is the local diffusion coefficient , \times \mathbb{r}^d \times \mathbb{r}^d \rightarrow \mathbb{r}^d ] is the generator of the bsde , and the processes \rightarrow \mathbb{r}^q ] of the backward kolmogorov equation , i.e. , in , is the terminal condition at the time , and the second - order integral - differential operator is of the form with is an integral operator defined as \eta({\bm{e } } ) \lambda(d{\bm{e}}).\ ] ] the functions , , , , in have the same definitions as in the bsde in . under the conditionthat for a fixed , the solution of the bsde for can be represented by where denotes the gradient of with respect to and . by comparing the operator in and the bsde in, we can see that the incremental consists of three components : the drift term , the brownian diffusion , and the jump diffusion .now let us relate the bsde in to the nonlocal diffusion equation in . without loss of generality , we only consider the case when is a scalar function , i.e. , by setting in ; the numerical schemes and analysis presented in this paper can be directly extended to the case of . observing that the operator in is a generalization of the nonlocal operator in, we aim to find a simplified form of the bsde which is the stochastic representation of the nonlocal diffusion equation in .to proceed , we define for and for , satisfying the condition in .due to the integrability assumption of in , the jump diffusion term in is a _compensated compound poisson process_. in this case , the compensated poisson random measure can be represented by where , with being the intensity of poisson jumps and being the probability measure of the jump size satisfying . then , the operator in can be simplified to the nonlocal operator in by setting , i.e. , removing the brownian motion , and where is the characteristic function of the domain . in the context of ,the drift coefficient is a constant vector which indeed helps cancel the first - order derivatives of in . on the other hand , substituting into the definition of , we have \ ; \text { for } i = 1 , \ldots , d,\ ] ] which is the compensator of .hence , is just a compound poisson process under the poisson measure without compensation .then , by defining and , with and being the forcing term and initial condition in respectively , we obtain the bsde corresponding to the nonlocal diffusion equation in according to theorem 2.1 in , the well - posedness of the bsde in requires the generator is globally lipschitz continuous , i.e. , there exists such that for all with and . in this case , there exists a unique solution where is the set of -adapted cdlg processes such that < \infty,\ ] ] and is the set of mappings \times e \rightarrow \mathbb{r} ] is the viscosity solution of satisfying \times \mathbb{r}^d,\ ] ] again , for some real . moreover , if and are bounded and uniformly continuous , then is also bounded and uniformly continuous .now we introduce the following notations which will be used throughout .let  be a -field generated by the stochastic process starting from the time - space point , and set .denote by  ] the conditional mathematical expectation under the -field , i.e. , =\mathbb{e}[\,\cdot\,|\mathcal{f}_s^{t,{\bm{x}}}] ] to denote  ] , we introduce the partition with and . in the time interval ] on both sides of , due to the fact that for is a martingale , we obtain that +\int_{t_n}^{t_{n+1}}\mathbb{e}_{t_n}^{{\bm{x}}}[f(s , { \bm{x}}_s , y_s)]ds,\ ] ] where = y_{t_n}({\bm{x } } ) = u(t - t_n,{\bm{x}}) ] is a deterministic function of  ] and the residual is given by  -\theta f(t_n , { \bm{x}}_{t_n},y_{t_n})\\ & \quad -(1-\theta)\mathbb{e}_{t_n}^{{\bm{x}}}[f(t_{n+1 } , { \bm{x}}_{t_{n+1}},y_{t_{n+1}})]\big\}ds .\end{aligned}\ ] ] next we propose a semi - discrete scheme for the bsde in as follows : given the random variable as the terminal condition , for , under the condition that , the solution is approximated by satisfying + ( 1-\theta)\delta t_n \mathbb{e}_{t_n}^{{\bm{x } } } \big[f(t_{n+1 } , { \bm{x}}_{n+1 } , y_{n+1})\big]\\ & \hspace{2.6cm}+\theta\delta t_n f(t_n , { \bm{x}}_n , y_n),\\ \end{aligned}\right.\ ] ] where . by choosing different values for , we obtain different semi - discrete schemes , e.g. , , and lead to forward euler ( fe ) , backward euler ( be ) and crank - nicolson ( cn ) schemes , respectively .  note that since is the standard compound poisson process , the probability distributions of or any incremental , with , are well known .thus , one does not need to discretize so that , in , denotes the exact solution evaluated at . in this case , in is the local truncation error of the scheme ; a rigorous error analysis of is given in [ sec : err ] .other time - stepping schemes , e.g. , linear multi - step schemes , can be directly used in order to further improve the accuracy of time discretization . note that in the general case where the coefficients and in and are functions of and , another time - stepping scheme is also needed for to discretize , so that an extra local truncation error will be introduced into the semi - discrete scheme .this is beyond the scope of this effort but will be the focus of future work .the semi - discrete scheme is not computationally feasible , because the involved conditional mathematical expectation  ] . to proceed, we partition the -dimensional euclidean space by where for is a partition of the one - dimensional real space , i.e. , such that and . for each multi - index , the corresponding grid point in denoted by .now we turn to constructing a quadrature rule for approximating  ] as an example and the proposed quadrature rule can be directly extended to approximate  ] , denoted by , follows a poisson distribution with intensity , and the size of each poisson jump follows the distribution defined in .thus ,  ] is of order , the conditional mathematical expectation  ] the approximation of  ] .an analogous notation  ] by retaining jumps .next , we also need to approximate the -dimensional integral in for .this can be accomplished by selecting an appropriate quadrature rule base on the properties of and the smoothness of .a straightforward choice is to utilize monte carlo methods by drawing samples from , but this is inefficient because of slow convergence . to enhance the convergence rate , an alternative way is to use the tensor product of high - order one - dimensional quadrature rules , e.g. , newton - cotes , gaussian , etc .. in addition , for multi - dimensional problems , i.e. , , by requiring second - order convergence , i.e. , , sparse - grid quadrature rules can be applied to further improve the computational efficiency . without loss of generality , for , we denote by and the set of weights and points , respectively , of the selected quadrature rule for estimating the -th integral in .then , the approximation of  ] is given by & = \exp(-\lambda\delta t ) \ ; y_{n+1}({\bm{x}}_{{\mathbf{i } } } ) \\ & \quad + \sum_{m=1}^{m_y } \exp(-\lambda \delta t ) \frac{(\lambda \delta t)^m}{m ! } \ ; \sum_{q=1}^{q_m }  w^m_q \ ;  y_{n+1}\big ( { \bm{x}}_{{\mathbf{i } } } + \sum_{k=1}^m\bm{a}^{k , m}_{q}\big ) .\end{aligned}\ ] ]  note that it is possible that the quadrature points in do not belong to the spatial grid .the simplest approach to manage this difficult is to add all quadrature points to the spatial grid at each time stage .however , this will result in an exponential growth of the total number of grid points as the number of time steps increases .thus , we follow the same strategy as in , and construct piecewise lagrange interpolating polynomials based on to approximate at the non - grid points .specifically , at any given point , is approximated by where is a -th order tensor - product lagrange interpolating polynomial . for ,the interpolation points are the closest neighboring points of such that , for and , constitute a local tensor - product sub - grid around .the fully - discrete solution at the interpolation point is given by .note that does not interpolate the semi - discrete solution due to the error .therefore , the fully - discrete scheme of the bsde in is described as follows : given the random variable for , and for , solve the quantities for such that +  ( 1-\theta)\delta t_n \widehat{\mathbb{e}}_{t_n , m_f}^{{\bm{x}}_{{\mathbf{i}}}}\left[f(t_{n+1 } , { \bm{x}}_{n+1 } , { y}_{n+1,p})\right]\\ &  \hspace{3.5cm}+\theta\delta t_n f\left(t_n , { \bm{x}}_{n } , { y}_{n , p}^{{\mathbf{i}}}\right),\\ \end{aligned}\right.\ ] ] where  ] and  ] .as such , there is no need to discretize the nonlocal operator in the fully - discrete scheme , so that stability does not require any cfl - type restriction on the time step ; this is in contrast to classic numerical schemes for which explicit schemes do require a cfl condition .moreover , according to the analysis in , the approximation is stable as long as the semi - discrete scheme is absolutely stable .the total computational cost of the scheme is dominated by the cost of approximating  ] in and with is uniformly lipschitz continuous with respect to and , then the nonlocal diffusion equation in with , and has a unique viscosity solution \times \mathbb{r}) ] , which completes the proof .next we provide the estimate of the local truncation error in .[ err_lem3 ] using lemma [ lem1 ] with , the local truncation error in , for the semi - discrete scheme with can be bounded by \leq c(\delta t)^3 \ ; \text { for } \ ;n = 0 , \ldots , n-1,\ ] ] where the constant depends only on the terminal time , the jump intensity , the upper bounds of , , and their derivatives .the formula shows that the solution of the bsde can be represented by . under lemma[ lem1 ] , if \times \mathbb{r}^2) ] .thus , by defining , we have \times \mathbb{r}) ] , we obtain   ds\\ = & \int_{t_n}^{t_{n+1 } } \mathbb{e}_{t_n}^x \left[f(t_n , x )  + \int_{t_n}^s l^0 f(r , x_r )  dr + \int_{t_n}^s l^1 f(r , x_{r- } ) \tilde{\mu}(de , dr ) \right ] ds\\ = & f(t_n , x ) \int_{t_n}^{t_{n+1 } } ds + \int_{t_n}^{t_{n+1}}\int_{t_n}^s \mathbb{e}_{t_n}^x [ l^0 f(r , x_r ) ] dr ds\\ = & f(t_n , x)\delta t +  \int_{t_n}^{t_{n+1}}\int_{t_n}^s  \mathbb{e}_{t_n}^x\left [ l^0 f(t_n , x ) +  \int_{t_n}^r l^0l^0 f(z , x_z )  dz\right ] dr ds\\ = & f(t_n , x)\delta t + \frac{1}{2 } ( \delta t)^2 l^0 f(t_n , x ) +  \int_{t_n}^{t_{n+1}}\int_{t_n}^s \int_{t_n}^r  \mathbb{e}_{t_n}^x [ l^0l^0 f(z , x_z ) ]  dzdrds .\end{aligned}\ ] ]  similar derivation can be conducted to to obtain ds\\ = & f(t_n , x)\delta t +  ( \delta t)^2 l^0 f(t_n , x ) +  \int_{t_n}^{t_{n+1}}\int_{t_n}^{t_{n+1 } } \int_{t_n}^r  \mathbb{e}_{t_n}^x [ l^0l^0 f(z , x_z ) ]  dzdrds .\end{aligned}\ ] ] therefore , the residual in with can be represented by -\frac{1}{2 } f(t_n , x)-\frac{1}{2}\mathbb{e}_{t_n}^x[f(t_{n+1},x_{t_{n+1}})]ds \\  & = \int_{t_n}^{t_{n+1}}\int_{t_n}^s \int_{t_n}^r \mathbb{e}_{t_n}^x [ l^0l^0 f(z , x_z ) ]  dzdrds\\  & \hspace{1.0cm}- \frac{1}{2 } \int_{t_n}^{t_{n+1}}\int_{t_n}^{t_{n+1 } } \int_{t_n}^r \mathbb{e}_{t_n}^x [ l^0l^0 f(z , x_z ) ]  dzdrds .\end{aligned}\ ] ] then , taking the mathematical expectation of , and using cauchy - schwarz inequality , we have that & \le \mathbb{e}\left[\left|\int_{t_n}^{t_{n+1}}\int_{t_n}^{s}\int_{t_n}^{r } \mathbb{e}_{t_n}^x[l^0l^0f(z , x_{z})]dzdrds\right|\right ] \\ & \quad + \frac{1}{2}\mathbb{e}\left[\left|\int_{t_n}^{t_{n+1 } } \int_{t_n}^{t_{n+1}}\int_{t_n}^{r } \mathbb{e}_{t_n}^x[l^0 l^0f(z , x_{z})]dzdrds\right|\right]\\ & \le ( \delta t)^{\frac{3}{2 } } \sqrt{\int_{t_n}^{t_{n+1}}\int_{t_n}^{s}\int_{t_n}^{r } \mathbb{e}[|l^0l^0f(z , x_{z})|^2]dzdrds}\\ & \quad + \frac{1}{2}(\delta t)^{\frac{3}{2 } } \sqrt{\int_{t_n}^{t_{n+1 } } \int_{t_n}^{t_{n+1}}\int_{t_n}^{r } \mathbb{e}[|l^0 l^0f(z , x_{z})|^2]dzdrds}\\ & \le \frac{3}{2 } \sup_{0\le t \le t } \sqrt{\mathbb{e}[|l^0 l^0f(t , x_{t})|^2 ] } ( \delta t)^3\\ & \le c ( \delta t)^3 , \end{aligned}\ ] ] where , by the definition of , it is easy to see that the constant only depends on the terminal time , the jump intensity the upper bounds of , and their derivatives .we now turn our attention to proving an error estimate for semi - discrete scheme .[ tl ] let and , for , be the exact solution of the bsde and the semi - discrete solution obtained by the scheme , respectively . then, under the same conditions as that for theorem [ err_lem3 ] , for sufficiently small time step , the global truncation error , for , can be bounded by \le c ( \delta t)^2,\ ] ] where the constant is the same as in theorem [ err_lem3 ] .  given and , subtracting from ,we have +  \frac{\delta t}{2 } \mathbb{e}_{t_n}^x [ f(t_{n+1 } , x_{t_{n+1 } } , y_{t_{n+1 } } ) - f(t_{n+1 } , x_{n+1 } , y_{n+1})]\\ & \hspace{1.5 cm } + \frac{\delta t}{2 } [ f(t_{n } , x_{t_{n } } , y_{t_{n } } ) - f(t_{n } , x_{n } , y_{n } ) ] + r_n\\ \end{aligned}\ ] ] due to the lipschitz continuity of and the fact that , , we get the bound + \frac{l\delta t}{2 } \mathbb{e}_{t_n}^x[|e_{n+1}| ] + \frac{l\delta t}{2 } |e_n| + |r_n|,\ ] ] where is the lipschitz constant of and the time step size satisfies . substituting the upper bound of , the above inequality can be rewritten as + \frac{c(\delta t)^3}{1-\frac{l\delta t}{2}}.\ ] ] taking mathematical expectation on both sides , we obtain , by induction , an upper bound of  ] , without loss of generality , we consider a special case stated in the following assumption .[ assump1 ] for and , the quadrature rules , used in are assumed to be the tensor - product of the selected one - dimensional quadrature rule , denoted by .the error in the quadrature rule is represented by where the convergence rate and the constant are determined by the smoothness of and .the same strategy is also applied to  ] and  ] , where the one - dimensional quadrature rule for each example is chosen according to the property of the kernel , the initial condition and the forcing term .our main goal is to test the accuracy and convergence rate of the proposed scheme with respect to and . to this end , according to the analysis in theorem [ err_full ] , we always set the number of quadrature points to be sufficiently large so that the error contributed by the use of quadrature rules is too small to affect the convergence rate .first we consider the following nonlocal diffusion problem in  ] is a characteristic function of the interval  ] .since the density function is uniform with the support  ] .first , we test the convergence rate with respect to . to this end , we set and use piecewise cubic lagrange interpolation ( ) to construct for , so that the time discretization error dominates the total error . for ,the number of time steps is set to , respectively .the numerical results are shown in table [ ex1:t1 ] . as expected , the convergence rate with respect to depends not only on the value of , but also the number of jumps retained in constructing  ] .for example , when , i.e. , no jump is included in  ] , the scheme fails to converge . in general , in order to achieve a -th order convergence ( ) , and must satisfy and .   & & & & & & cr + , , & 1.932e-1 & 1.978e-1 & 2.000e-1 & 2.012e-1 & 2.017e-1 & -0.015 + , , & 7.958e-2 & 3.435e-2 & 1.572e-2 & 7.487e-3 & 3.649e-3 & 1.109 + , , & 3.673e-2 & 1.849e-2 & 9.279e-3 & 4.675e-3 & 2.326e-3 & 0.995 + , , & 2.111e-1 & 2.067e-1 & 2.045e-1 & 2.034e-1 & 2.028e-1 & 0.014 + , , & 4.891e-2 & 2.581e-2 & 1.328e-2 & 6.736e-3 & 3.393e-3 & 0.964 + , , & 6.170e-2 & 3.007e-2 & 1.468e-2 & 7.231e-3 & 3.585e-3 & 1.027 + , , & 2.030e-1 & 2.025e-1 & 2.023e-1 & 2.023e-1 & 2.023e-1 & 0.001 + , , & 2.623e-2 & 1.326e-2 & 6.666e-3 & 3.342e-3 & 1.674e-3 & 0.993 + , , & 3.207e-3 & 8.258e-4 & 2.094e-4 & 5.271e-5 & 1.322e-5 & 1.981 + , , & 3.330e-3 & 8.632e-4 & 2.196e-4 & 5.538e-5 & 1.391e-5 & 1.977 +  we observe in theorem [ err_full ] that the errors with respect to from  ] are of order and , respectively .if the intensity is large , i.e. , the horizon is small , the value of will affect the error decay in the pre - asymptotic region and change the total error up to a constant .such phenomenon is investigated by setting and , and and .the results are shown in figure [ ex1:f1 ] . in figure [ ex1:f1](a ) with , in the case that ( ) , and , the error is actually of order .when one more jump is included , i.e. , and , we observe that the convergence rate remains the same but the total error is significantly reduced , since the error contributed by  ] is reduced to .next , we test the convergence rate with respect to by setting , , , , and .we divide the spatial interval  ] ; and figure [ ex3:f1](b ) illustrates the exact solution and its approximation using 33 spatial grid points for which the error is only 9.745e-04 .further illustration is shown in figure [ ex3:f2](a ) and figure [ ex3:f2](b ) where , for the scheme , the error decay vs. the number of interpolating points and vs. the grid size , respectively , of using uniform and adaptive grids are plotted . for the adaptive case , is the starting grid for the refinement process .we clearly see the half - order and the optimal second - order convergence rates for the uniform and adaptive grid cases , respectively .we also study the convergence with respect to for with the error tolerance for the adaptive grid is set to 0.01 , 0.005 , 0.001 and 0.0005 . in figure [ ex3:f2](c ) , it can be seen that the desired convergence rate with respect to is achieved if the error is smaller than the tolerance ; otherwise , the spatial discretization error will dominate and the total error will not decay as decreases .of course , no amount of refinement , whether adaptive or not , will reduce the norm error , i.e. , it will remain .however , if we omit the single element containing the discontinuity , we see the following from figure [ ex3:f2](d ) . for uniform grid refinement ,the error remains due to pollution of the error into neighboring elements which are all of size .however , for adaptive grid refinement , the error is close to because pollution is to neighboring elements of very small size .in this work , we propose a novel stochastic numerical scheme for linear nonlocal diffusion problems based on the relationship between the pides and a certain class of backward stochastic differential equations ( bsdes ) with jumps . as discussed in [ sec : intro ] , this effort focuses on solving low - dimensional nonlocal diffusion problems with high - order accuracy , where the potential applications will be non - darcy flow in groundwater , or plasma transport in magnetically confined fields .however , our approach can be extended to solve moderately high - dimensional problems by incorporating our previous work on sparse - grid approximation .compared to standard finite element and collocation approaches for approximating linear nonlocal diffusion equations with integrable kernels , our method completely avoids the solution of dense linear systems .moreover , the ability to utilize high - order temporal and spatial discretization schemes , as well as efficient adaptive approximation , and the potential of massively parallel implementation , make our technique highly advantageous .these assets have been verified by both theoretical analysis and numerical experiments .our future efforts will focus on extending the proposed numerical schemes to the case of non - integrable kernels , e.g. , fractional laplacians , where the underlying lvy processes can not be described by compound poisson processes .moreover , the well - posedness of bsdes on bounded domains , with volume constraints , is also critical for studying nonlocal diffusion equations on bounded domains .                  , _ backward stochastic differential equations and quasilinear parabolic partial differential equations _ , in stochastic partial differential equations and their applications , springer berlin heidelberg , berlin / heidelberg , jan .1992 , pp .

Abstract: we propose a novel numerical approach for nonlocal diffusion equations with integrable kernels , based on the relationship between the backward kolmogorov equation and backward stochastic differential equations ( bsdes ) driven by lvy processes with jumps . the nonlocal diffusion problem under consideration is converted to a bsde , for which numerical schemes are developed and applied directly . as a stochastic approach , the proposed method does not require the solution of linear systems , which allows for embarrassingly parallel implementations and also enables adaptive approximation techniques to be incorporated in a straightforward fashion . moreover , our method is more accurate than classic stochastic approaches due to the use of high - order temporal and spatial discretization schemes . in addition , our approach can handle a broad class of problems with general nonlinear forcing terms as long as they are globally lipchitz continuous . rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach .   backward stochastic differential equation with jumps , nonlocal diffusion equations , superdiffusion , compound poisson process , -scheme , adaptive approximation


compressed sensing is one of the most exciting topics of present - day signal processing .signal reconstruction from its low - dimensional representation becomes a possibility , given the sparse nature of the signal and , incoherence and/or restricted isometry property ( rip ) of the sensing / measurement process . in this regard ,a number of approaches can be used , e.g. , basis pursuit ( bp ) , least absolute shrinkage and selection operator ( lasso ) and greedy algorithms . in order to exploit the structure of the signal being sensed , a number of variants of lasso have become popular , e.g. , group lasso ( g - lasso ) , sparse group lasso ( sg - lasso ) and fused lasso ( f - lasso ) , etc . in this letterwe propose new lasso formulations to handle block sparse smooth signals .smooth signals are often encountered in a wide range of engineering and biological fields . in engineering ,signals observed in image processing , control systems and environment monitoring are often smooth or piece - wise smooth . in biology , a similar structureis observed , e.g. , in protein mass spectroscopy .the goal is to recover such structured signals from noisy and/or under - sampled measurements .a related topic is signal smoothing which deals with removing random outliers . apart from being smooth , such signalscan often be represented as sparse in some basis .this sparsity pattern normally varies in terms of the location and block size of the sparse coefficients .the challenge for signal reconstruction is to exploit the block sparsity with varying block sizes , while keeping smoothness intact and using fewer measurements , but all at low complexity . in the cs domain ,signal smoothness has been handled by using a fusion constraint in .the fusion is also known as total variation ( tv ) in the image processing literature .apart from fusion , also proposed an -norm penalty to cater for signal sparsity .however , since most of the signals are block sparse , can not give efficient results . to cater for the block sparsity, one can replace the -norm penalty with a group penalty .although this approach can handle the block sparsity very well , it only offers fixed group sizes and causes complete groups to be zero or nonzero . to avoid elimination of small sets of nonzero elements ,a very small group size is opted but that can make the algorithm inefficient in eliminating large blocks of zero elements . in this regardwe propose to use a moderate group size along with an -norm penalty over the signal , to create sparsity within the groups .thus by using fusion in combination with -norm penalty and a moderate group size , a smooth signal can be reconstructed with high accuracy .the problem of varying group sizes can also be handled by using the concept of latent groups , see and references therein .these are basically overlapping groups , with recurring signal elements in possibly multiple groups .thus , an element lost in one group may resurface through another group after reconstruction .so we also propose to use such latents groups in combination with a fusion constraint to recover block sparse smooth signals with varying block sizes .note that a work on using overlapping groups over the fusion function instead of the signal structure has appeared in , which however requires complete signal samples .instead , we propose overlapping groups and fusion penalties over the actual signal for the under - determined systems .thus , we exploit the actual structure of the signal rather than the difference of elements .further , in order to solve the proposed formulations we derive low - complexity algorithms based on the alternating direction method of multipliers ( admm ) .the reason for using this version of the augmented lagrangian methods is primarily the non - separability of the fusion penalty in terms of the elements of the signal .thus , the general convergence properties of admm can be used to guarantee optimal results for our proposed algorithms . * _ notations_*. matrices are in upper case bold while column vectors are in lower case bold , {i , j} ] is the entry of the vector , is the identity matrix of size , is transpose , is the estimate of , defines an entity , {i}\right|^p)^{1/p}} ] .we can see from ( [ eq : sglf_x ] ) that accounts for group sparsity , for element - wise sparsity and -[\xbf]_{j-1}\|_1 ^ 1 ] , and is the difference matrix with {j , j}=-1 ] , for and {n-1,n-1}=1 ] is used to update } ] in ( [ eq : rho_z ] ) .   forthe latent group fused lasso ( lgf - lasso ) , the signal is segmented into many overlapping groups of certain sizes .in contrast to the disjoints groups , overlapping groups can reselect the elements from other groups .we create overlapping groups through an sub - selection matrices which select rows from an identity matrix .an overlapping group can then be obtained by the relation , , for , where is such that ^t ] . nowthe lagrangian for ( [ eq : olf_admm ] ) can be written as where collects the lagrangian multipliers with sub - vectors for .now the successive approximations for the solution of ( [ eq : glf_admm_lagr ] ) w.r.t . , and can be written as } & = \mathop{\arg \min}\limits_{\small \xbf } \call\left ( \xbf,\tilde{\ubf}^{[n-1]},\zbf^{[n-1]},\rhobf_{\tilde{u}}^{[n-1]},\rhobf_z^{[n-1 ] } \right ) \label{eq : oglf_admm_op_x } \\\tilde{\ubf}^{[n ] } & = \mathop{\arg \min}\limits_{\small \ubf } \call\left ( \xbf^{[n-1]},\tilde{\ubf},\rhobf_{\tilde{u}}^{[n-1 ] } \right ) \label{eq : oglf_admm_op_u } \\ \rhobf_{\tilde{u}}^{[n ] } & = \rhobf_{\tilde{u}}^{[n-1 ] } + c_u(\xbf^{[n]}-\tilde{\ubf}^{[n ] } ) \label{eq : rho_u_til } \end{aligned}\ ] ] whereas , the expressions for the estimates of and are the same as in ( [ eq : glf_admm_op_z ] ) and ( [ eq : rho_z ] ) , respectively .  from ( [ eq : oglf_admm_op_x ] ) , the closed - form expression for the estimate of at the iteration can be derived as } = \left ( \phibf^t\phibf + c_z\dbf^t\dbf + c_u\wbf^t\wbf \right)^{-1 } \notag \\ & \times \left ( \phibf^t\ybf - \dbf^t\rhobf_z^{[n-1 ] } + c_z\dbf^t\zbf^{[n-1 ] } - \wbf^t(\rhobf_u^{[n-1 ] } - c_u \ubf^{[n-1 ] } ) \right ) .\label{eq : oglf_admm_x_cf}\end{aligned}\ ] ] where we can see that as is already known , the matrix inversion part of the estimate can again be obtained off - line and does not need to be estimated for each iteration .  from ( [ eq : oglf_admm_op_u ] ) , the closed - form expression for the estimate of , for , at the iteration can be derived as } = & \left ( \| \wbf_i\xbf^{[n-1 ] } + \frac{\rhobf_{\tilde{u}_i}^{[n-1]}}{c_u } \|_2 ^ 2 - \frac{\lambda_g}{c_u }  \right)_+ \notag \\ & \times \dfrac{\wbf_i\xbf^{[n-1 ] } + \frac{\rhobf_{\tilde{u}_i}^{[n-1]}}{c_u } } { \| \wbf_i\xbf^{[n-1 ] } + \frac{\rhobf_{\tilde{u}_i}^{[n-1]}}{c_u } \|_2 ^ 2 } \label{eq : oglf_admm_ui_cf}\end{aligned}\ ] ] and the estimate for } ] , } ] , } ] , } ] , have all been considered as zero vectors , respectively .note that , a least - squares solution of , can also be considered as a warm - start to speed up the convergence rate .the group size for sgf - lasso , lgf - lasso and g - lasso has been taken as . therefore , the number of groups in sgf - lasso and g - lasso are the same , i.e. , . for lgf - lasso , an overlapping factor of has been used , and therefore the number of overlapping groups of size are .we use a maximum of iterations for each algorithm .we have observed that a tolerance level of between consecutive updates is reached much earlier than this limit , and therefore we stop the algorithm at this stage .figure [ fig : glf_sigrecon_comp_01 ] shows the reconstruction performance of sgf - lasso , lgf - lasso and g - lasso when the signal was sensed with a compression ratio .we can see that the performance of sgf - lasso and lgf - lasso is very close to each other and both are able to recover the smooth transitions of the original signal .sgf - lasso has an edge over lgf - lasso , as it better reconstructs even a very small group of nonzero elements .on the other hand , the performance of g - lasso deteriorates both on the front of smoothness as well as block size .note that in contrast to sgf - lasso and lgf - lasso , is the only sparsity creating parameter for g - lasso .therefore , we increase its value to , which is the minimum to recreate the actual zero blocks .  figure [ fig : glf_sigrecon_mse_01 ] shows the performance comparison of the proposed algorithms through the mean squared error ( mse ) metric against varying compression ratios , where .we can see that the performance improves in general with increasing value of , for .nonetheless , the difference in performance follows the previously observed pattern .sgf - lasso keeps an edge over lgf - lasso , whereas g - lasso remains quite far away .here we would like to mention that sgf - lasso has an edge over lgf - lasso with a lower number of groups .lgf - lasso can have improved performance by increasing the overlapping factor but that would cause a subsequent increase in the computational complexity .= 3.5 in  = 3.5 inin this letter , we have proposed two new lasso formulations , namely , sparse group fused lasso and latent group fused lasso . the former uses element - wise sparsity , group sparsity ( over disjoint groups ) and fusion penalties , whereas the latter combines the fusion penalty with a latent group penalty .both formulations can be used to reconstruct smooth signals from their compressed measurements .we also provide low - complexity solvers for the proposed formulations , based on the alternating direction method of multipliers .we compared the performance of our proposed algorithms with standard group lasso over a smooth test signal .the simulation results confirm the better performance of the proposed algorithms for signal reconstruction against group lasso .similar results were obtained for the mean squared error metric , for varying compression ratios .e. cands , j. romberg and t. tao , `` robust uncertainty principles : exact signal reconstruction from highly incomplete frequency information , '' _ ieee transaction on information theory _ , vol .489 - 509 , february 2006 .

Abstract: we present reconstruction algorithms for smooth signals with block sparsity from their compressed measurements . we tackle the issue of varying group size via group - sparse least absolute shrinkage selection operator ( lasso ) as well as via latent group lasso regularizations . we achieve smoothness in the signal via fusion . we develop low - complexity solvers for our proposed formulations through the alternating direction method of multipliers .   compressed sensing , block sparsity , smoothness , signal reconstruction