work examines the dynamics that results when adaptive nodes are allowed to interact with each other . through cooperation, some interesting behavior occurs that is not observed when the nodes operate independently .for example , if one adaptive agent has worse performance than another independent adaptive agent , can both agents cooperate with each other in such a manner that the performance of _ both _ agents improves ?what if agents are interacting with each other ? can all agents improve their performance relative to the non - cooperative case even when some of them are noisier than others ? does cooperation need to be performed in a centralized manner or is distributed cooperation sufficient to achieve this goal ? starting with two adaptive nodes , we derive analytical expressions for the mean - square performance of the nodes under some conditions on the measurement data .the expressions are then used to compare the performance of various ( centralized and distributed ) adaptive strategies .the analysis reveals a useful fact that arises as a result of the cooperation between the nodes ; it establishes that , by optimizing over the combination weights , diffusion least - mean - squares ( lms ) strategies for distributed estimation can deliver lower excess - mean - square - error ( emse ) than a centralized solution employing traditional block or incremental lms strategies .we first study in some detail the situation involving combinations of two adaptive nodes for which the performance levels can be characterized analytically .subsequently , we extend the conclusion to -node ad - hoc networks .reference provides an overview of diffusion strategies for adaptation and learning over networks .it is worth noting that the performance of diffusion algorithms was already studied in some detail in the earlier works .these works derived expressions for the network emse and mean - square - deviation ( msd ) in terms of the combination weights that are used during the adaptation process .the results in were mainly concerned in comparing the performance of diffusion ( i.e. , distributed cooperative ) strategies with _ non - cooperative _ strategies . in the cooperative case , nodes share information with each other , whereas they behave independently of each other in the non - cooperative case . in the current work ,we are instead interested in comparing diffusion or _ distributed _ cooperative strategies against _ centralized _ ( as opposed to non - cooperative ) strategies . in the centralized framework , a fusion center has access to all data collected from across the network , whereas in the non - cooperative setting nodes have access only to their individual data .therefore , finding conditions under which diffusion strategies can perform well in comparison to centralized solutions is generally a demanding task .we start our study by considering initially the case of two interacting adaptive agents .though structurally simple , two - node networks are important in their own right . for instance , two - antenna receivers are prevalent in many communication systems .the data received by the antennas could either be transferred to a central processor for handling or processed cooperatively and locally at the antennas .which mode of processing can lead to better performance and how ?some of the results in this article help provide answers to these questions .in addition , two - node adaptive agents can serve as good models for how estimates can be combined at master nodes that connect larger sub - networks together .there has also been useful work in the literature on examining the performance of combinations of two adaptive filters .the main difference between two - node adaptive networks and combinations of two adaptive filters is that in the network case the measurement and regression data are fully distributed and also different across nodes , whereas the filters share the same measurement and regression data in filter combinations .for this reason , the study of adaptive networks is more challenging and their dynamics is richer .  the results in this work will reveal that distributed diffusion lms strategies can outperform centralized block or incremental lms strategies through proper selection of the combination weights .the expressions for the combination weights end up depending on knowledge of the noise variances , which are generally unavailable to the nodes .nevertheless , the expressions suggest a useful adaptive construction . motivated by the analysis , we propose an adaptive method for adjusting the combination weights by relying solely on the available data .simulation results illustrate the findings ._ notation _ : we use lowercase letters to denote vectors , uppercase letters for matrices , plain letters for deterministic variables , and boldface letters for random variables .we also use to denote transposition , for conjugate transposition , for matrix inversion , for the trace of a matrix , for the spectral radius of a matrix , for the kronecker product , for stacking the columns of on top of each other , and for constructing a vector by using the diagonal entries of .all vectors in our treatment are column vectors , with the exception of the regression vectors , , which are taken to be row vectors for convenience of presentation .we refer to the two nodes as nodes 1 and 2 .both nodes are assumed to measure data that satisfy a linear regression model of the form : for , where is a deterministic but unknown vector , is a random measurement datum at time , is a random regression vector at time , and is a random noise signal also at time .we adopt the following assumptions on the statistical properties of the data .[ asm : all ] 0.5em  1 .the regression data are temporally white and spatially independent random variables with zero mean and uniform covariance matrix .the noise signals are temporally white and spatially independent random variables with zero mean and variances .the regressors and noise signals are mutually - independent for all and , and .it is worth noting that we do not assume gaussian distributions for either the regressors or the noise signals .we note that the temporal independence assumption on the regressors may be invalid in general , especially for tapped - delay implementations where the regressions at each node would exhibit a shift structure .however , there have been extensive studies in the stochastic approximation literature showing that , for stand - alone adaptive filters , results based on the temporal independence assumption , such as and further ahead , still match well with actual filter performance when the step - size is sufficiently small .thus , we shall adopt the following assumption throughout this work .[ asm : smallstepsize ] the step - sizes are sufficiently small , i.e. , , so that terms depending on higher - order powers of the step - sizes can be ignored , and such that the adaptive strategies discussed in this work are mean - square stable ( in the manner defined further ahead ) .we are interested in the situation where one node is less noisy than the other .thus , without loss of generality , we assume that the noise variance of node 2 is less than that of node 1 , i.e. , the nodes are interested in estimating the unknown parameter .assume initially that each node independently adopts the famed lms algorithm to update its weight estimate ( as illustrated in fig .[ fig : indlms ] ) according to the following rule : \end{aligned}\ ] ] for , where is a positive constant step - size parameter .the steady - state performance of an adaptive algorithm is usually assessed in terms of its mean - square error ( mse ) , emse , and msd , which are defined as follows . if we introduce the error quantities : then the mse , emse , and msd for node are defined as the following steady - state values : where the notation denotes the euclidean norm of its vector argument . substituting expression into the definition for in ,it is easy to verify that the errors are related as follows : for .since the terms and are independent of each other , it readily follows that the mse and emse performance measures at each node are related to each other through the noise variance : therefore , it is sufficient to examine the emse and msd as performance metrics for adaptive algorithms . under assumption [ asm : smallstepsize ] ,the emse and msd of each lms filter in are known to be well approximated by : and for . to proceed , we further assume that both nodes employ the same step - size and observe data arising from the same underlying distribution .[ asm : uniform ] it is assumed that both nodes employ identical step - sizes , i.e. , , and that they observe data arising from the same statistical distribution , i.e. , .  under assumption[ asm : uniform ] , expression confirms the expected result that node 2 will achieve a lower emse than node 1 because node 2 has lower noise variance than node 1 .the interesting question that we would like to consider is whether it is possible to improve the emse performance for _ both _ nodes if they are allowed to cooperate with each other in some manner .the arguments in this work will answer this question in the affirmative and will present _ distributed _ cooperative schemes that are able to achieve this goal , not only for two - node networks but also for -node ad - hoc networks ( see sec .one form of cooperation can be realized by connecting the two nodes to a fusion center , which would collect the data from the nodes and and use them to estimate .fusion centers are generally more powerful than the individual nodes and can , in principle , implement more sophisticated estimation procedures than the individual nodes . in order to allow for a fair comparison between implementations of similar nature at the fusion center and remotely at the nodes ,we assume that the fusion center is limited to implementing lms - type solutions as well , albeit in a centralized manner . in this work, the fusion center is assumed to operate on the data in one of two ways .the first method is illustrated in fig .[ fig : veclms ] and we refer to it as _ block lms_. in this method , the fusion center receives data from the nodes and updates its estimate for according to the following : with a constant positive step - size .the second method is illustrated in fig .[ fig : inclms ] and we refer to it as _ incremental lms_. in this method , the fusion center still receives data from the nodes but operates on them sequentially by incorporating one set of measurements at a time as follows : \\ { \bs{w}}_{i}&={\bs{\phi}}_{i}+\mu'{\bs{u}}_{2,i}^*\left[{\bs{d}}_{2}(i)-{\bs{u}}_{2,i}{\bs{\phi}}_{i}\right]\\ \end{aligned}\right.\ ] ] we see from that the fusion center in this case first uses the data from node 1 to update to an intermediate value , and then uses the data from node 2 to get .method is a special case of the incremental lms strategy introduced and studied in and is motivated by useful incremental approaches to distributed optimization .we observe from and that in going from to , the block and incremental lms algorithms employ two sets of data for each such update ; in comparison , the conventional lms algorithm used by the stand - alone nodes in employs one set of data for each update of their respective weight estimates .we define the emse and msd for block lms and incremental lms as follows : where the a priori error is now a vector : note that in we are scaling the definition of the emse by because the squared euclidean - norm in involves the sum of the two error components from .we shall explain later in sec .vi that in order to ensure a fair comparison of the performance of the various algorithms ( including non - cooperative , distributed , and centralized ) , we will need to set the step - size as ( see ) this normalization will help ensure that the rates of convergence of the various strategies that are being compared are similar .  now , compared to the non - cooperative method where the nodes act individually , it can be shown that the two centralized algorithms and lead to improved mean - square performance ( the arguments further ahead in sec .iv - f establish this conclusion among several other properties ) .specifically , the emse obtained by the two centralized algorithms and will be smaller than the average emse obtained by the two non - cooperative nodes in .the question that we would like to explore is whether distributed cooperation between the nodes can lead to superior performance even in comparison to the centralized algorithms and . to address this question ,we shall consider distributed lms algorithms of the diffusion - type from , and which are further studied in .reference provides an overview of diffusion strategies .adaptive diffusion strategies have several useful properties : they are scalable and robust , enhance stability , and enable nodes to adapt and learn through localized interactions .there are of course other useful algorithms for distributed estimation that rely instead on consensus - type strategies , e.g. , .nevertheless , diffusion strategies have been shown to lead to superior mean - square - error performance in comparison to consensus - type strategies ( see , e.g. , ) .for this reason , we focus on comparing adaptive diffusion strategies with the centralized block and incremental lms approaches .the arguments further ahead will show that diffusion algorithms are able to exploit the spatial diversity in the data more fully than the centralized implementations and can lead to better steady - state mean - square performance than the block and incremental algorithms and , when all algorithms converge in the mean - square sense at the same rate .we shall establish these results initially for the case of two interacting adaptive agents , and then discuss the generalization for -node networks in sec .diffusion lms algorithms are distributed strategies that consist of two steps : updating the weight estimate using local measurement data ( the adaptation step ) and aggregating the information from the neighbors ( the combination step ) . according to the order of these two steps ,diffusion algorithms can be categorized into two classes : combine - then - adapt ( cta ) ( as illustrated in fig .[ fig : difflmscta ] ) : \end{aligned}\right.\ ] ] and adapt - then - combine ( atc ) ( as illustrated in fig .[ fig : difflmsatc ] ) : \\ { \bs{w}}_{k , i}&=a_{1k}{\bs{\psi}}_{1,i}+a_{2k}{\bs{\psi}}_{2,i } \end{aligned}\right.\ ] ] for , where the are positive step - sizes and the denote convex combination coefficients used by nodes 1 and 2 .the coefficients are nonnegative and they satisfy for .we collect these coefficients into a matrix and denote them more compactly by for node 1 and for node 2 : where  ] .we then get .thus , from , the network emse can be rewritten as that is , similarly , the network msd can be rewritten as    for -nodes , the block lms recursion is replaced by \end{aligned}\ ] ] and the incremental lms recursion is replaced by \\ \quad\mbox{set } \;\ ; { \bs{w}}_{i}={\bs{\psi}}_n \\ \mbox{end } \end{cases}\end{aligned}\ ] ] in order for block and incremental lms to converge at the same rate as diffusion atc , we must set their step - sizes to ( compare with ) .following an argument similar to the one presented in appendix [ app : derivationemsevec ] , we can derive the emse and msd for the block lms strategy as and respectively . a similar argument to ( see also expression ( 84 ) in ) leads to the conclusion that the performance of incremental lms can be well approximated by that of block lms for small step - sizes .therefore , and for this reason , we shall not distinguish between block lms and incremental lms in the sequel .observe that the emse expression for atc diffusion lms and for block and incremental lms only differ by a scaling factor , namely , versus .then , atc diffusion would outperform block lms and incremental lms when where is diagonal and given by .we assume that the noise variance of at least one node in the network is different from the other noise variances to exclude the case in which the noise profile is uniform across the network ( in which case would be a scaled multiple of the identity matrix ) .thus , note that , if we select the combination matrix to be _ doubly - stochastic _, i.e. , and , then it is straightforward to see that so that this result means that , for sufficiently small step - sizes and for any doubly - stochastic matrix , the emse performance of atc diffusion and block lms match each other .however , as indicated by , the diffusion lms strategy can employ a broader class of combination matrices , namely , left - stochastic matrices .if we optimize over the larger set of left - stochastic combination matrices and in view of , we would expect where is the optimal combination matrix that solves the following optimization problem : where denotes the set consisting of all left - stochastic matrices whose entries satisfy the conditions in .we show next how to determine left - stochastic matrices that solve .first note that the optimization problem is equivalent to the following non - convex problem : where denotes the nonnegative vector space .we solve this problem in two steps .first we solve the _ relaxed _ problem : since is positive definite and diagonal , the closed - form solution for is given by next , if we can determine a primitive left - stochastic matrix whose right eigenvector associated to eigenvalue 1 coincides with , then we would obtain a solution to . indeed ,note that any primitive left - stochastic matrix can be regarded as the probability transition matrix of an irreducible aperiodic markov chain ( based on the connected topology and condition on the weights ) . in that case , a vector that satisfies would correspond to the stationary distribution vector for the markov chain .now given an arbitrary vector , whose entries are positive and add up to one , it is known how to construct a left - stochastic matrix that would satisfy .a procedure due to hastings was used in to construct such matrices . applying the procedure to our vector given by , we arrive at the following combination rule , which we shall refer to as the hastings rule( we may add that there are many other choices for that would satisfy the same requirement ) : where denotes the cardinality of .it is worth noting that the hastings rule is a _ fully - distributed _ solution each node only needs to obtain the degree - variance product from its neighbor to compute the corresponding combination weight . by using the hastings rule ,the vector in is attained and the emse expression is therefore minimized .the minimum value of is then given by compared to the emse of block and incremental lms and , we conclude that diffusion strategies using the hastings rule achieve a lower emse level under assumption [ asm : smallstepsize ] .this is because , from the cauchy - schwarz inequality , we have when the entries on the diagonal of are not uniform ( as we assumed at the beginning of this subsection ) .  in real applications , where the noise variances are unavailable, each node can estimate its own noise variance recursively by using the following iteration : _ remark _ : in the two - node case , we determined the combination weights by seeking coefficients that essentially minimize the emse expressions and . the argument in appendix [ app : minimizeemsecta ] expressed the emse as the sum of two factors : a dominant factor that depends on and a less dominant factor that depends on higher powers of . in the -node network case , we instead used the small step - size approximation to arrive at expressions and , which correspond only to the dominant terms in the emse and msd expressions and depend on .we can regard and as first - order approximations for the performance of the network for sufficiently small step - sizes .we simulate atc diffusion lms versus block lms over a connected network with nodes .the unknown vector of length is randomly generated .we adopt , for atc diffusion lms , and for block lms .the network topology and the profile of noise variances are plotted in fig .[ fig : profiles ] . for atc algorithms ,we simulate three different combination rules : the first one is the ( left - stochastic ) adaptive hastings rule using and without the knowledge of noise variances , the second one is the hastings rule with the knowledge of noise variance , and the third one is the ( doubly - stochastic ) metropolis rule ( which is a simplified version of the hastings rule ) : we adopted and for the adaptive hastings rule to match the convergence rate of the other algorithms .we also consider the non - cooperative lms case for comparison purposes .the emse learning curves are obtained by averaging over 50 experiments and are plotted in fig .[ fig : emse ] .it can be seen that atc diffusion lms with metropolis weights exhibits almost the same convergence behavior as block lms in transient phase and attains a steady - state value that is less than 1 db worse than block lms . in comparison ,atc diffusion lms using adaptive hastings weights ( where the noise variances are estimated through ) has almost the same learning curve as atc using hastings weights with the knowledge of the noise variances ; both of them are able to attain about 7 db gain over block lms at steady - state .in this work we derived the emse levels for different strategies over lms adaptive networks and compared their performance .the results establish that diffusion lms strategies can deliver lower emse than centralized solutions employing traditional block or incremental lms strategies .we first studied the case of networks involving two cooperating nodes , where closed - form expressions for the emse and msd can be derived .subsequently , we extended the conclusion to generic -node networks and established again that , for sufficiently small step - sizes , diffusion strategies can outperform centralized block lms strategies by optimizing over left - stochastic combination matrices .it is worth noting that although the optimized combination rules rely on knowledge of the noise statistics , it is possible to employ adaptive strategies like to adjust these coefficients on the fly without requiring explicit knowledge of the noise profile in this way , the hastings rule can be implemented in a manner similar to the adaptive relative variance rule . clearly , the traditional block and incremental implementations and can be modified to incorporate information about the noise profile as well . in that case , it can be argued that diffusion strategies are still able to match the emse performance of these modified centralized algorithms .under assumptions [ asm : all][asm : uniform ] , the emse expression for node of the general diffusion strategy is given by eq .( 39 ) from reference ( see also ) : ^\t(i_{n^2m^2 } - { \mc{f}})^{-1}\vecm(e_{kk}\otimes r_{u})\end{aligned}\ ] ] where and is an all - zero matrix except for the entry on the diagonal , which is equal to one . since for atc algorithms , and , and for cta algorithms , and , we know that for both cases .therefore , we get we can reduce into the form , which is more suitable for our purposes , by introducing the eigen - decompositions of and .thus , let denote the eigen - decomposition of , where is unitary and is diagonal with positive entries .let also denote the eigen - decomposition of the real matrix , where is real and invertible and is in the _ real _ jordan canonical form .then , the eigen - decomposition of is given by \nonumber\\ { } & = ( t^{-\t}\otimes{u})[d^{\t}\otimes(i_m-\mu\lambda)](t^{\t}\otimes{u}^*)\end{aligned}\ ] ] and the eigen - decomposition of is then given by (t^{\t}\otimes{u}^*)\}^{\t } \otimes\{(t^{-\t}\otimes{u})[d^{\t}\otimes(i_m-\mu\lambda)](t^{\t}\otimes{u}^*)\}^*\nonumber\\ { } & = { \mc{x}}\{[d^{\t}\otimes(i_m-\mu\lambda)]^{\t}\otimes[d^{\t}\otimes(i_m-\mu\lambda)]^*\}{\mc{x}}^{-1}\nonumber\\ { } & = { \mc{x}}\{[d\otimes(i_m-\mu\lambda)]\otimes[d\otimes(i_m-\mu\lambda)]\}{\mc{x}}^{-1}\nonumber\\ { } & = { \mc{x}}({\mc{g}}\otimes{\mc{g}}){\mc{x}}^{-1}\end{aligned}\ ] ] where we used the facts that are real and is diagonal , and introduced the matrices : then , from , we get \cdot\mu^2\vecm(q^{\t}r_vq\otimes r_u^\t)\nonumber\\ & = \mu^2\vecm\left[(t^{*}\otimes{u}^\t)(q^{\t}r_vq\otimes r_u^\t)(t\otimes{u}^{*\t})\right]\nonumber\\ & = \mu^2\vecm(t^{\t}q^{\t}r_vqt\otimes\lambda)\end{aligned}\ ] ] where we used the fact that is real .likewise , we get \cdot\vecm(e_{kk}\otimes r_{u})\nonumber\\ & = \vecm(t^{-1}e_{kk}t^{-\t}\otimes\lambda)\end{aligned}\ ] ] then , from , the emse expression in can be rewritten as ^\t{\mc{x}}(i_{n^2m^2}-{\mc{g}}\otimes{\mc{g}})^{-1 } { \mc{x}}^{-1}\vecm(e_{kk}\otimes r_{u})\nonumber\\ { } & = \mu^2[\vecm(t^{\t}q^{\t}r_vqt\otimes\lambda)]^{\t}(i_{n^2m^2}-{\mc{g}}\otimes{\mc{g}})^{-1 } \vecm(t^{-1}e_{kk}t^{-\t}\otimes\lambda)\end{aligned}\ ] ] using the fact that in is stable under assumption [ asm : smallstepsize ] , we can further obtain ^{\t } \left(\sum_{j=0}^{\infty}{\mc{g}}^j\otimes{\mc{g}}^j\right ) \vecm(t^{-1}e_{kk}t^{-\t}\otimes\lambda)\nonumber\\ { } & = \mu^2[\vecm(t^{\t}q^{\t}r_vqt\otimes\lambda)]^{\t } \sum_{j=0}^{\infty}\vecm\left[{\mc{g}}^j(t^{-1}e_{kk}t^{-\t}\otimes\lambda){\mc{g}}^{\t j}\right]\nonumber\\ { } & = \mu^2\sum_{j=0}^{\infty}\tr\left[(t^{\t}q^{\t}r_vqt\otimes\lambda){\mc{g}}^j ( t^{-1}e_{kk}t^{-\t}\otimes\lambda){\mc{g}}^{\t j}\right]\end{aligned}\ ] ] where we used the identities and ^\t\vecm(b)$ ] for matrices of compatible dimensions . from , we get \nonumber\\ { } & \quad=\tr\left\{(t^{\t}q^{\t}r_vqt\otimes\lambda)[d^j\otimes(i_m-\mu\lambda)^j ] ( t^{-1}e_{kk}t^{-\t}\otimes\lambda)[d^{\t j}\otimes(i_m-\mu\lambda)^{j}]\right\}\nonumber\\ { } & \quad=\tr\left[t^{\t}q^{\t}r_vqtd^jt^{-1}e_{kk}t^{-\t}d^{\t j } \otimes\,\lambda(i_m-\mu\lambda)^j\lambda(i_m-\mu\lambda)^{j}\right]\nonumber\\ { } & \quad=\tr\left[\lambda(i_m-\mu\lambda)^j\lambda(i_m-\mu\lambda)^{j } \otimes\,t^{\t}q^{\t}r_vqtd^jt^{-1}e_{kk}t^{-\t}d^{\t j}\right]\nonumber\\ { } & \quad=\sum_{m=1}^{m}\lambda_m^2(1-\mu\lambda_m)^{2j } \tr(t^{\t}q^{\t}r_vqtd^jt^{-1}e_{kk}t^{-\t}d^{\t j})\nonumber\\ { } & \quad=\sum_{m=1}^{m}\lambda_m^2(1-\mu\lambda_m)^{2j}[\vecm(t^{\t}q^{\t}r_vqt)]^\t ( d^{j}\otimes d^j)\vecm(t^{-1}e_{kk}t^{-\t})\end{aligned}\ ] ] where we used the identity for square matrices and the fact that is diagonal . substituting back into leads to ^\t \left[\sum_{j=0}^{\infty}(1 - \mu\lambda_m)^{2j}d^{j}\otimes d^j\right ] \vecm(t^{-1}e_{kk}t^{-\t})\nonumber\\ { } & = \mu^2\sum_{m=1}^{m}\lambda_m^2[\vecm(t^{\t}q^{\t}r_vqt)]^{\t } [ i_{n^2 } - ( 1 - \mu\lambda_m)^2d\otimes d]^{-1}{\mathrm{vec}}(t^{-1}e_{kk}t^{-\t})\nonumber\\ { } & \approx\mu^2\sum_{m=1}^{m}\lambda_m^2[\vecm(t^{\t}q^{\t}r_vqt)]^{\t } [ i_{n^2 } - ( 1 - 2\mu\lambda_m)d\otimes d]^{-1}{\mathrm{vec}}(t^{-1}e_{kk}t^{-\t})\end{aligned}\ ] ] where due to assumption [ asm : smallstepsize ] .to minimize the network emse for cta given by , we introduce two auxiliary variables and such that and , where and .the network emse can be rewritten as \end{aligned}\ ] ] where and is given by under assumption [ asm : smallstepsize ] .minimizing expression in closed - form over both variables is generally non - trivial .we exploit the fact that the step - size is sufficiently small to help locate the values of and that approximately minimize the value of . for this purpose , we first substitute into and use assumption [ asm : smallstepsize ] to note that expression writes the emse as the sum of two factors : the first factor is linear in the step - size and depends only on , and the second factor depends on higher - order powers of the step - size . for sufficiently small step - sizes , the first factor is dominant and we can ignore the second factor .doing so allows us to estimate the value of that minimizes .observe that the first factor on rhs of is minimized at because we now substitute back into the original expression for the network emse to find that : \end{aligned}\ ] ] we use this form to minimize the higher - order terms of over the variable .it is obvious that expression is minimized at .the value of emse under and is then given by \end{aligned}\ ] ] similarly , we can employ the same approximate argument to find that the solution essentially minimizes the network msd under assumption [ asm : smallstepsize ] ; the corresponding value of the msd is \end{aligned}\ ] ] the solution translates into , where .  in a similar manner , in order to minimize the network emse of atc given by , we introduce two auxiliary variables and such that and , where and .then , from we have \end{aligned}\ ] ] for which we can again motivate the selection . the value of the network emse at and is then given by start from . to simplify the notation , we rewrite and as where the error recursion is then given by  be an arbitrary positive semi - definite matrix that we are free to choose .using , we can evaluate the weighted square quantity .doing so and taking expectations under assumption [ asm : all ] , we arrive at the following weighted variance relation : where where , in view of assumption [ asm : smallstepsize ] , we are dropping higher - order terms in .let again denote the eigen - decomposition of .we then introduce the transformed quantities : relation is accordingly transformed into where since we are free to choose , or equivalently , , let be diagonal and nonnegative . then , it can be verified that is also diagonal and nonnegative under assumptions [ asm : all][asm : uniform ] so that under assumption [ asm : all ] , the second term on the right - hand side of evaluates to \end{aligned}\ ] ] where therefore , we get when the filter is mean - square stable , taking the limit as of both sides of and selecting , we get likewise , by selecting and taking the limit of both sides of as , we arrive at authors would like to acknowledge useful feedback from ph.d .student z. towfic on sec .vi - 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Abstract: in this work we analyze the mean - square performance of different strategies for distributed estimation over least - mean - squares ( lms ) adaptive networks . the results highlight some useful properties for distributed adaptation in comparison to fusion - based centralized solutions . the analysis establishes that , by optimizing over the combination weights , diffusion strategies can deliver lower excess - mean - square - error than centralized solutions employing traditional block or incremental lms strategies . we first study in some detail the situation involving combinations of two adaptive agents and then extend the results to generic -node ad - hoc networks . in the later case , we establish that , for sufficiently small step - sizes , diffusion strategies can outperform centralized block or incremental lms strategies by optimizing over left - stochastic combination weighting matrices . the results suggest more efficient ways for organizing and processing data at fusion centers , and present useful adaptive strategies that are able to enhance performance when implemented in a distributed manner .   adaptive networks , distributed estimation , centralized estimation , diffusion lms , fusion center , incremental strategy , diffusion strategy , energy conservation .


the goal of this study is to re - assess an example of non - uniqueness of entropy weak solutions for multi - dimensional systems of hyperbolic conservation laws constructed by k.a .pericak - spector and s. spector .the example is associated with the the onset of cavitation from a homogeneously deformed state for the equations of nonlinear elasticity in homogeneous and isotropic elastic media .any attempt to study solutions that lie at the limits of continuum modeling ( like cavities or shear bands ) needs to reckon with the problem of giving a proper definition for such solutions .proposing such a definition is the main premise of the present work .we consider the initial - boundary value problem for the equations of elasticity for a given stretching and some here , stands for the motion , is the deformation gradient , and we employ the constitutive theory of hyperelasticity , that the piola kirchhoff stress is given as the gradient of a stored energy density the homogeneous deformation solves - .the stored energy function determines the constitutive properties of the elastic material and - due to frame indifference - has to be invariant under rotations . for homogeneous and _ isotropic _elastic materials takes the simplified form , where is a symmetric function of the eigenvalues of ; see . in that caseadmits as solutions radially symmetric motions , generated by solving for the amplitude the initial - boundary value problem for the equations of isotropic radial elastodynamics , this problem admits the special solution corresponding to a state of homogeneous deformation .the question arises if additional solutions of - may be constructed by solving the problem - .this idea was pursued by ball using methods from the calculus of variations to construct cavitating solutions in the context of static elasticity .the study was groundbreaking in part because continuum modeling is used to address a problem that at least _ pro - forma _ concerns situations beyond its range of applicability .there exists a critical stretching so that for the only equilibrium solution is the homogeneously deformed state ; by contrast , for there exist non - trivial equilibria corresponding to a cavity in the material and with energy _ less _ than the energy of the homogeneous deformation .the reader is referred to ( and references therein ) for an account of developments concerning cavitating equilibrium solutions in nonlinear elasticity .  in another remarkable development , k.a .pericak - spector and s. spector use the ansatz to construct a self - similar weak solution for the dynamic problem - that corresponds to a spherical cavity emerging at time from a homogeneously deformed state .the solution in is constructed in dimension for polyconvex stored energies of the special form where satisfies the hypotheses the hypothesis refers to polyconvexity , while indicates elasticity of softening type ; more general stored energies were treated in .it is proved in that the self - similar solution has smaller mechanical energy than the associated homogeneously deformed state from where it emerges , and thus provides an example of nonuniqueness of entropy weak solutions ( at least for polyconvex energies ) . as already noted in ,the paradox arises that by opening a cavity the energy of the material decreases , what induces an autocatalytic mechanism for failure .there is a class of problems in material science , such as fracture , cavitation or shear bands , where discontinuous motions emerge from smooth motions via a mechanism of material instabilities .of course after the material breaks or a shear band forms the motion can no longer be described at the level of continuum modeling and microscopic modeling or higher - order regularizing mechanisms have to be taken into account .still , as such structures develop there is expected an intermediate time scale where both types of modeling apply .the premise of this work is to introduce a meaning for solutions at this intermediate scale and to explore its ramifications .the idea is roughly the following , presented here at the level of . given a possibly discontinuous motion we introduce the averaged motions , where is a mollifier , and define to be a singular limiting induced from continuum solution ( in short a _slic_-solution ) if for every choice of the mollifier the smooth approximating family in the limit of small - scale averaging gives in this notion the precise form of regularizing mechanisms is not taken into account , instead it is enforced that they act in a stable way amounting to averaging of the tested solution .moreover , the energy equation for , suggests that , even though in , the power of the `` microscopic forces '' may well have a non - trivial contribution in the limit , which needs to be calculated .the definition of slic - solutions exploits the fact that the momentum equation is a second - order evolution .it has certain conceptual analogies to the approach of weak asymptotic solution employed by danilov and shelkovich in order to define solutions for special systems of conservation laws involving delta shocks as asymptotic limits of smooth approximate solutions .there are also certain differences with the concept of weak asymptotic solution : slic - solutions are generated via averagings of a candidate discontinuous solution , and are natural in a context of discontinuous solutions for second order evolution equations .the latter property provides a mechanical intuition for the definition and suggests its name .we examine the ramifications of this definition in three examples . in section [ sec - fracture ]we consider the equations of one - dimensional elasticity and test a specific example of a motion , towards being a slic - solution .the example is a counterpart ( in the one - dimensional case ) of the cavitating solutions ( for ) of and corresponds to a crack forming out of a homogeneously deformed state at the location at time , in conjunction with two outgoing lax - shocks propagating at .it is more singular than the dynamic cavitating solution in as has a delta - mass at the origin .a definition of the notion of slic - solution is given in definition [ defslicconvx ] , conditions are then provided under which is a slic - solution for in proposition [ propo - slic1d ] , and the energy balance for the approximate solution of the crack is computed in proposition [ propo - eb1d ] .it turns out that can be interpreted as a slic - solution , at the same time there is an energetic cost for creating the crack that is projected in the limiting energy balance equation .  in section [ sec : cavitation ], we consider the dynamically cavitating solution , with as constructed in ( see lemma [ lem : psest ] stating its properties ) . the notion of slic - solution adapted to that exampleis provided in definition [ def : slic3d ] and a natural definition for the energy is given in definition [ def : energy3d ] .the analysis is more cumbersome and is based on detailed estimations of the layers of the approximate solution , but the results parallel those of the one - dimensional example .namely , conditions are given under which the cavitating solutions provide a slic - solution in theorem [ lem : slic3d ] , and the energy balance of the approximate solutions is computed in proposition [ lem : energy3d2 ] . regarding the issue of uniqueness of entropy weak solutions , it turns out that if the solution is construed as an entropy weak solution , then there is non - uniqueness .by contrast , if the solution is construed in the slic - sense , then there is a contribution to the total energy in the process of forming the cavity which results to the energy after the cavity formation being larger than before the cavitation ( proposition [ lem : increase ] ) .this indicates that the notion of entropy weak solutions is inadequate when dealing with discontinuous solutions and in particular it can not account for the work needed to create the cavity .a more discriminating concept of solution has to be employed on strong singularities , and the slic - solution concept is such a possibility .the last example is the riemann problem for the p - system in lagrangean coordinates with vacuum initial data .it has been conjectured that delta - shocks are needed to resolve the riemann problem with vacuum ( * ? ? ? * sec 9.6 ) , a proper definition for such solutions is not available at the hyperbolic level , nevertheless solutions with delta shocks are constructed by using viscous wave fans in the zero - viscosity limit ( _ e.g. _ , , ( * ? ? ? * sec 9.8 ) ) . in section [ sec - vacuum ] , we provide a definition for self - similar slic - solutions ( definition [ def : selfsimslic ] ) which is capable to define riemann solutions with delta - shocks at the hyperbolic level without recourse to a construction method ( theorem [ lem : vac : sol ] ) and to calculate their mechanical energy ( proposition [ lem : vac : energy ] ) .the equation describes longitudinal or shearing motions of one - dimensional elastic bars .we consider loading situations that the bar is subjected to homogeneous deformations far away , that is for some sufficiently large introducing the velocity and the strain , the equation is expressed as a system of conservation laws in what follows we will assume that the stress function satisfies the hypotheses  under the wave speeds are real and is hyperbolic .the hypothesis is appropriate for an elastic material exhibiting softening elastic response and plays an important role in the forthcoming analysis .the hypothesis is applicable in the case of longitudinal motions and is placed to exclude that a finite volume is compressed down to zero . in the sequel we will consider only tensile deformations and this hypothesis will not play any significant role .( in the case of shearing motions is defined for and is removed ) .smooth solutions of satisfy the additional conservation law where is the elastic stored energy function .this equation captures the conservation of mechanical energy , namely , for each subdomain , the rate of mechanical energy is balanced by the fluxes through the boundaries . for elastic bars subjected to homogeneous deformations for , the mechanical energy in is conserved .we are interested in self - similar solutions , with a function of the similarity variable .the function solves the equation alternatively , we may introduce in the self - similar ansatz , and recast it in the form of the riemann problem where and .if is sufficiently large we expect that the elastic bar will break . once the bar breaks , coherence is lost and the hypothesis of continuum implicit in the derivation ofis no longer valid .nevertheless , there will be a transition region from a range of loading where the model is valid to a range where it loses validity , and in some situations one may arguably give some meaning to equation past the transition regime .it is precisely this transition at the onset of fracture that we wish to explore .the scope of this section is to test a class of self - similar solutions describing the onset of fracture in one - space dimension that are generated by the function : where , , and are positive parameters that satisfy , , and are connected through the equations the function is continuous at but discontinuous at the origin .the associated distributions and are determined via :   in the sequel we propose a notion of solution according to which the function may be interpreted as a solution of .several preliminary remarks are in order :  * the solution has several common features and was in fact inspired by the dynamic cavitating solutions in three - dimensions proposed in the important work by spector and pericak - spector .one important difference is that the cavitating solutions in do not involve a delta measure for the strain . in the present one - dimensional situationthe solution may be thought as describing fracture , and accordingly we will call it a crack .* for fixed , there is a one parameter family of functions that satisfy , . *note that and that .accordingly , at time the bar is in a configuration of homogeneous deformation with strain and at rest . * for positive times the function has a jump at the origin associated to a crack that is moving according to .the parameter stands for the speed of the crack . *the singularities at are shocks .the rankine - hugoniot conditions at , are satisfied due to , .this shock belongs to the second characteristic family . in view of hypothesis, the lax shock - admissibility criterion will be satisfied provided that .a similar analysis indicates that the shock at belongs to the first characteristic family and satisfies the rankine - hugoniot conditions and the lax shock - admissibility criterion . *the distribution contains the delta measure and therefore on one hand it is more singular than the usual solution of the riemann problem , on the other hand one needs to give a meaning to .a direct computation , using and the identity in , shows that in the sense of distributions .hence , is satisfied .it remains to give a meaning to .indeed , this is a main task of this section , and will be done through the notion of _ singular limiting induced from continuum solution _ that we introduce in the sequel .  in what follows we test , with defined in , towards being a solution ( interpreted in an appropriate sense ) of .it is expedient to extend the function for by setting .the extended function still denoted by reads the parameters satisfy , and .note that has a discontinuity at , but it is continuous at the shocks due to .for longitudinal deformations of an elastic bar , the motion may be interpreted as a crack emerging at time from a homogeneously deformed state ; the form of the solution in the plane is shown in figure [ fig1](a ) .the equation may also be interpreted as describing elastic shear motions . in this contextrepresents a shear band emerging at and is sketched ( in a graph ) in figure [ fig1](b ) .\(a ) ( b )  for future reference we calculate certain properties for the solution : first , note that for  furthermore , in view of the kinematic compatibility assumption and , the distributional derivatives and are given by observe that as for .there is a class of problems in material science where structures with discontinuous displacement fields emerge via a material instability mechanism .typical examples are development of cracks in fracture , cavitation in elastic response , or formation of shear bands in plastic deformations .such problems lie at the limits of applicability of continuum modeling and the usual concept of weak solutions is in any case inadequate to describe these motions .nevertheless , as the material transitions from a regime where continuum modeling is applicable to a regime that the model has to be modified ( or perhaps atomistic modeling has to be employed ) , it is expected that at the interface both types of modeling have to apply in an intermediate regime .it is further expected that such structures should appear in a small parameter limit of more complex models that incorporate `` higher - order physics '' , and that their appearance occurs in a stable way .the concept of _ singular limiting induced from continuum solution _ ( in short _ slic_-solution ) is an attempt to give meaning to such discontinuous solutions .it is presented here at the level of the equations of nonlinear elasticity . roughly speaking a discontinuous solution ofwill be a slic - solution if it can be obtained as the limit of approximate smooth solutions that are an averaging of .more precisely :  [ defslic ] let where is an open domain in space - time .given a mollifier , with , , , we set . for any , compactly embedded in , the averaged function is well defined for and .we will say that is a _ singular limiting induced from continuum solution _ of if , for any mollifier any and for ,   several remarks are in order .this definition incorporates as a requirement that the singular solution results by a smooth averaging process , but it does not contain any additional information on the approximation process that might induce a mechanism for selecting among singular solutions . one can easily check that the usual concept of weak solutions ( containing shocks ) of the equations of elasticity are also slic - solutions . finally , the concept of slic - solution is quite natural when continuum modeling is viewed as resulting from the discrete , especially in situations where the distance between lattice elements is large but nevertheless matter is present . in that senseit is more natural in a context of shear bands than in a context of cavitation after a hole has fully formed .  the definition of _slic_-solution is cumbersome to use in practice .nevertheless , it provides a theoretical framework to define such singular solutions and to calculate the ramifications on the energy .a difficulty occurs because in the general setting of definition [ defslic ] averaging in both space and time is required . for hyperbolic problems we expect that averaging in space will also induce averaging in time and in many settingsthe double averaging might be avoided .also , in practice we expect to check the definition at only local singularities what simplifies calculations .in particular when checking it for self - similar solutions averaging in space only is adequate .  for certain special systems of conservation laws , like the system of pressureless gas dynamics and the so called keyfitz - kranzer system , there has been suggested to use solutions that take values in measures , a literature going by the name of delta - shocks ( _ e.g. _  , ( * ? ? ?* sec 9.6 ) and references therein ) .danilov and shelkovich have proposed to define delta shocks as asymptotic limits of smooth approximate solutions , a process that they call weak asymptotic solution .this definition has obvious similarities to the concept of slic - solution presented here .there are also two differences : first , the notion of slic - solution is connected via averaging to a possibly discontinuous `` solution candidate '' , second , slic - solutions emerge in a context of discontinuous solutions for second order evolution equations , what provides a mechanical intuition for the definition and suggests the name .next , we turn to the example of the fracturing solution with the objective to test it as a slic - solution for the equations . as already mentionedthe goal is to give a meaning to the solution at . to simplify calculationswe will introduce a variant of definition [ defslic ] that uses a mollification in the space variable only :  [ defslicconvx ] let satisfy for some the monotonicity condition : for a mollifier , , , ( the ball of radius 1 ) , , we let and define the averaged function the function is called a singular limiting induced from continuum ( slic ) solution of provided for any symmetric mollifier as for   to justify that the left hand side of is well defined , observe that in view of and thus for of class , the derivative .the monotonicity condition placed on ensures that for all the definition in the form stated applies to longitudinal motions . in the case that models shearing motionsthen hypothesis is removed and the condition is no longer necessary .definition [ defslic ] using the convolution in formula relies on the special form of and is easier to use in practice .we turn now to the example with , , satisfying , and do some preparatory computations . for , dz \\ & = t \big ( \sigma ( \lambda - \alpha ) - y(0 ) \big ) \phi_n ( x + \sigma t ) + 2 \phi_n ( x ) t y(0 )  + t \big ( \sigma ( \lambda - \alpha ) - y(0 ) \big ) \phi_n ( x - \sigma t ) \\ & \quad  + \lambda \int_{-\infty}^{-\sigma t } \phi_n ( x - z ) dz  + \alpha \int_{-\sigma t}^{\sigma t } \phi_n ( x - z ) dz + \lambda \int_{\sigma t}^{\infty } \phi_n ( x - z ) dz \\ & \stackrel{\eqref{kinem}}{= } 2 \phi_n ( x ) t y(0 ) + \lambda \int_{-\infty}^{-\sigma t } \phi_n ( x - z ) dz  + \alpha \int_{-\sigma t}^{\sigma t } \phi_n ( x - z ) dz + \lambda \int_{\sigma t}^{\infty } \phi_n ( x - z ) dz \end{aligned}\ ] ] and since and , we have the bounds in addition , we compute it is a tedious but straightforward calculation to check that , mainly as a consequence of , the regularity of is of class ( but has a jump discontinuity at ) .one also easily checks that which should be compared to .next , we test whether in provides a slic - solution for .[ propo - slic1d ] if , are satisfied and then defined in is a slic - solution for ._ proof _ due to , the limit is always well defined : indeed , since is monotone increasing , either and , or and exists by .  for be a slic - solution , its regularization should satisfy the approximate equation with in .as the equation is satisfied in a classical sense , and it will be viewed as defining .let be a test function , and consider the distribution associated to .the time - domain is split in two parts : and .the threshold is selected so that the waves in the approximate solution are separated and no longer interact for .  we proceed to estimate the right hand side of starting with the term .observe that and thus using , we next obtain using , and the bound for , , we have   finally , we compute the contribution of the origin :   combining with , , , and , we see that \psi( -\sigma t , t )  - [ \tau(\lambda ) - \tau ( \alpha ) - \sigma y(0 ) ] \psi ( \sigma t , t )  \\ & \qquad \qquad \qquad + 2 y(0 ) t l { \partial}_x \psi(0,t ) \big )  dt \\ & =  - 2 y(0 ) l <  { \partial}_x \delta_{x=0 } , t \psi > \end{aligned}\ ] ] and the latter is zero if and only if .   the function is smooth and satisfies the approximate equation and the associated energy identity even though the product may yield in the limit a nontrivial contribution to the energy . in this subsectionwe compute that contribution .to consider the energy balance , fix a domain containing the entire wave fan of the approximate solution at time .for instance , if we are interested in times , we select .the velocity at the boundary of such a domain vanishes , .the total energy of the wave fan = \int_b \frac{1}{2 } ( v^n)^2 + w ( { \partial}_x y^n )  \ , dx\ ] ] evolves according to the energy balance equation   if , then for the ( rate of ) change of total energy t >  \frac{2}{n\sigma}t >  \frac{2}{n\sigma} ] of the wave fan satisfies & = \left (  \mu_{-\sigma } + \mu_\sigma + p^n_c \right )  \charf_{t > \frac{2}{n \sigma } }  + e^n \charf_{0 < t < \frac{2}{n \sigma } }  \end{aligned}\ ] ] with uniformly bounded .accordingly , the limiting energy satisfies - e [ \lambda x ; b ] =  \int_0^t  \mu_{-\sigma } + \mu_\sigma + 2 ( \tau_\infty - \tau(\alpha ) ) y(0 )  ds = :  \int_0^t  t > 0\ ] ] where and stand for the ( rate of ) energy dissipation of the shocks , is the cost of opening the crack . moreover , the total change of energy .the only thing left to prove is the last statement . indeed , using , , and , we deduce , consider the case of three or more space dimensions . the situation at hand , of a displacement field discontinuous at one point , can be interpreted as cavitation .we begin by describing the multi dimensional setting of the problem . for consider the system of elastodynamics where , subject to the initial and boundary conditions for a given number and some the homogeneously deformed state is a particular solution of this problem . in analogy to the one - dimensional case ( _ cf . _ ) we introduce the velocity and the deformation gradient and observe that can be written as a system of conservation laws the piola kirchhoff stress is given by the assumption of hyperelasticity , with an energy density that is isotropic and homogeneous . for isotropic and frame - indifferent stored energies , for some symmetric function of the eigenvalues , , of the matrix , ( see ) .  in what follows we consider free energies having the special form , where satisfies the hypothesis relates to polyconvexity , while indicates elasticity of softening type .these hypotheses are analogous to the hypothesis in section [ sec - fracture ] placed on the stored energy in the one - dimensional case .we consider radially symmetric motions , then satisfies the differential equation with the initial and boundary conditions obviously , and it is derived in that and where are given by and is the identity matrix .  in , k.a .pericak - spector and s. spector construct a self - similar weak solution of , satisfying for and corresponding to a cavity forming at time .their solution is based on the _ ansatz _ and is constructed under the framework , in ; extensions to more general free energies are carried out in .it is also shown in that the self - similar solution with cavity has smaller total mechanical energy than the associated homogeneously deformed state from where it emerges .the loss of energy is due to the energy dissipation of a shock generated at the onset of cavitation and propagating outwards ahead of the cavity , see ( * ? ? ?* thm 7.2 ) .the cavitating solution has the special property that the cauchy stress at the surface of the cavity vanishes and thereby all integrals involved in defining weak solutions and the energy of the solution are well defined .however , the energy of this solution does not contain any contribution reflecting work which is needed to create the cavity . here, we will propose a different concept of solution and energy which accounts for the energetic cost for creating the cavity .following the ideas in section [ sec - fracture ] , we will study the emergence of the cavitating solution of as the limit of continuous approximate solutions .we start our considerations by defining sequences of approximating functions for the class of spherically symmetric functions in : the function is first extended for by using an odd extension , .consider next a symmetric mollifier satisfying   ] , in particular 4 .the specific volume is strictly increasing in for and .5 .  for any and is   for the proof of lemma [ lem : psest ] the reader is referred to and specifically to proposition 6.2 for assertion 2 ; to propositions 6.2 and 6.3 and ( 6.1 ) for assertions 3 and 5 ; and to proposition 6.5 ( see formulas ( 6.19 ) , ( 6.20 ) of ) for assertions 1 and 4 .let be the approximate solution defined by and , and let us fix the following notation : for every quantity related to the same symbol with superscript denotes the corresponding quantity related to due to , we have we proceed to derive uniform in bounds for the velocity , the eigenvalues and the specific volume of the approximate solutions .[ lem : vest3d ] there exist constants such that for   _ proof_the domain , can be split in two ranges : and . in the former case .so we only consider the range and start with proving the lower bound for .we infer from that which implies , upon using , the monotonicity property of and jensen s inequality , that moreover , combining and yields it thus suffices to obtain a lower bound for .we distinguish the cases and still relying on , the ansatz and lemma [ lem : psest ] . in case we find for we obtain combining , and we have a lower bound for   next , we show an upper bound for always in the range . observe that implies and that proceeding along the lines of the derivation of , we obtain this implies for we have for and for we estimate without using .we rely on and combining , and we see that is bounded from above .  the next lemma establishes a global bound for and bounds for the eigenvalues outside a neighborhood of the cavity .[ lem : bulkest3d ] there exists a constant ( independent of ) such that  1 . for all and 2 . for and 3 . for and   _ proof_using we write also , from lemma [ lem : psest ] and we have , , and hence for .to show the first assertion we distinguish two cases . for have for we find   to prove the second assertion , we note that for it holds such that upper and lower bounds for are upper and lower bounds for .  it remains to prove the third assertion .we have     the next lemma investigates the behaviour of eigenvalues in the vicinity of the cavity . in preparation , we define   [ lem : cavest3d ] for and the following holds  1 . 2 .if then there exist such that for  4 .  if there exist constants such that   _ proof_upon differentiating and using the integration by parts formula , we obtain  w_r(\tilde r , t ) d \tilder. \end{aligned}\ ] ] recall that .then  w_r(\tilde r , t ) d \tilde r \\ & \le \lambda \int_0^\infty  \phi_n(r - \tilde r ) + \phi_n(r + \tilde r ) d \tilde r \\ & = \lambda\end{aligned}\ ] ] and shows assertion 1 . by, it also implies which once integrating and noting that proves assertion 2 .we turn now to assertions 3 and 4 .observe that , since , if there are and such that for .we restrict our computations to the range and use and assertion 1 to infer which shows assertion 3 .  turning now to, we note that the upper bound follows from assertions 1 . and 2 .via the lower bound is more delicate and we distinguish three cases :  * , for some to be selected below ; * , ; * , .  for in the range ( i ) ,we have and the desired bound follows from .next , let be in the range ( ii ) and define so that . using weobtain by selecting sufficiently small , we ensure that and the lower bound follows in range ( ii ) follows from .we next consider in range ( iii ) .using again we have moreover , using assertion 1 ., , the symmetry of and the monotonicity of , we obtain in turn , always in the range , which concludes the proof of assertion 4 .  the hypothesis is necessary to obtain the lower bounds stated in assertions 3 . and 4 .indeed , when the mollifying kernel satisfies then we will show that the approximate solution satisfies for every .this entails that the approximate solution will have regions in space with behavior of large compression intermixed with neighboring regions with behavior of large extensions , which is quite unexpected for a large extension situation as appears in cavitation .the assumption rules out such behavior for the approximants .  to show , note that due to the symmetry the condition implies that for sufficiently small .consider now the region and note that implies  w_r(\tilde r , t ) d \tilde r \nonumber \\ & \le \frac{2 c}{n } w(0,t ) + \sup_{0 < \tilde r \le \frac{1}{n}+ \frac{1}{n^2 } } w_r ( \tilde r , t ) \label{105 } \\ & = \frac{2 c}{n } w(0,t ) + c_n ( t ) \longrightarrow 0 \qquad  \mbox{as } \nonumber\end{aligned}\ ] ] where we have set and used that . integrating gives and thus which shows .   in this sectionwe investigate under which conditions the weak solutions constructed in and exhibiting cavitation are also a slic - solution according to definion [ def : slic3d ] .this turns out to depend on the growth behaviour of for .[ lem : slic3d ] the weak solutions constructed in  extended by for  are slic - solutions provided they are not slic - solutions in case   _ proof _ by construction the solutions from satisfy for all and the cavitating solution is extended to negative values of by setting for .it is easy to check that the extended function ( still denoted by ) is a weak solution on and satisfies the weak form for any .  to investigate whether is a slic - solution it suffices to studywhether to check this , we decompose the integral on the left into three parts by standard properties of convolution converges to in hence , converges to we will now show that the cavitating solution constructed in is an element of and thus and pointwise almost everywhere ; accordingly pointwise almost everywhere .it remains to show that the integrand in is uniformly bounded by an -function .to shorten notation we define and compute h'(v^n(|{\bf x}|,t ) ) \right ) \charf_{\{|{\bfx}| > \frac{1}{n}\ } } \\ &\stackrel{\text{lemma } \ref{lem : vest3d}}{\leq } c\|\nabla { \boldsymbol \psi } \|_\infty \charf_a \left(\lambda_1^n(|{\bf x}|,t)+ \lambda_2^n(|{\bf x}|,t )  + \lambda_1^n(|{\bf x}|,t ) \lambda_2^n(|{\bf x}|,t)^{d-2 } + \lambda_2^n(|{\bf x}|,t)^{d-1 } \right ) \charf_{\{|{\bf x}| > \frac{1}{n}\ } } \\ & \stackrel{\text{lemma } \ref{lem : bulkest3d}}{\leq } c\|\nabla { \boldsymbol \psi } \|_\infty \charf_a \max\left\ {  1 , 1 + \frac{t}{r } , \left (  1 + \frac{t}{r}\right)^{d-2 } , \left ( 1 + \frac{t}{r}\right)^{d-1 } \right\ } \end{split}\ ] ] which is in   hence , holds if and only if .let us start with the case .then , it is sufficient to show that for every  as is diagonalisable this holds provided in and the integrand is bounded by according to lemma [ lem : cavest3d ] and the volume of the integrational domain is proportional to for the fourth limit we find {c(1+t^dn^d ) } } \frac{\sqrt[d]{c(1+t^dn^d)}}{n}r^{d-1 }  dr dt=0 ,  \end{split}\ ] ] by the dominated convergence theorem .the third term can be treated analogously .thus , the first assertion of the lemma is proven .  in case we consider special test functions which satisfy for with  ] for in both cases has the following properties , which we will exploit in the sequel combining and we get where is the surface area of the unit sphere .let us note that the absolute value of the integrand on the right hand side of is bounded by therefore integral and limit commute and we obtain dr \zeta(t)dt\\ & =  \omega_d \int_{-t}^t \int_0 ^ 1  2^{d-1 }  w(0,t)^{d-1 }  \left(\frac{\phi(r)}{r}\right)^{d-1 }  r^{d-1 } \\& \qquad\qquad\qquad \qquad  \lim_{\hat v \rightarrow \infty } \frac{g(\hat v^d)}{\hat v }  2 w(0,t)\sqrt[d]{\phi(r ) \left ( \frac{\phi(r)}{r}\right)^{d-1 } } dr \zeta(t ) dt  \ ; > \ ; 0  \end{split}\ ] ] for a suitable choice of by an analogous calculation we derive a lower bound for the term in , {\phi(r ) \left ( \frac{\phi(r)}{r}\right)^{d-1 } } dr \zeta(t ) dt \ ; > \ ; 0.\end{gathered}\ ] ]   we will now study the energy of the solutions constructed in using our notion of energy , cf . definition [ def : energy3d ] .we will only consider energies in domains with finite volume .let us note first that if is a set with lipschitz boundary containing a neighborhood of the whole wave fan at time t , then where is the unit outer normal to .thus , in the limit there is no work performed by exterior forces .our first result shows that , if the material is not sufficiently weak , cavitating solutions have infinite energy for arbirarily small positive times , although their energy in at time is given by , i.e. is finite .[ lem : energy3d1 ] if the energy of the cavitating solution constructed in in the sense of the definition [ def : energy3d ] satisfies (t)= \infty \quad \text { for every } t>0.\ ] ]  _ proof_for approximate solutions generated by mollifiers with , we fix and as in lemma [ lem : cavest3d ] and recall that satisfies assertions 3 . and 4 . of that lemma .as contains the whole wave fan at time , it also contains for small enough .the non negativity of implies (t ) \geq\lim_{n \rightarrow \infty } \int_{b(n ) } h(v^n(|{\bf x}|,t ) ) d { \bf x }  = \lim_{n \rightarrow \infty } \omega_d \int_0^{{\varepsilon } } h(v^n(\frac{r}{n},t ) ) \frac{r^{d-1}}{n^d } d r.\ ] ] moreover , for , we have by fatou s lemma (t )  & \geq   \liminf_{n \rightarrow \infty } \omega_d \int_0^{{\varepsilon } } h \big((2\delta n ) ^d w(0,t)^d \big ) \frac{r^{d-1}}{n^d } d r  \\ & \ge \omega_d \int_0^{{\varepsilon } } \liminf_{n \rightarrow \infty } \frac{h \big((2\deltan ) ^d w(0,t)^d \big)}{n^d } r^{d-1 } d r = \infty.\end{aligned}\ ] ]    for weaker materials , i.e. , the situation changes and the energy of cavitating solutions becomes finite .[ lem : energy3d2 ] let be finite and let for some contain a neighborhood of the whole wave fan at time . then , the energy of the weak solution from - in the sense of definition [ def : energy3d ] - satisfies ( t)= e[\lambda { \bf x},b](t ) + \frac{t^d \sigma^d \omega_d}{d } j + \frac{t^d \omega_d}{d } r(0)^d l,\ ] ] where   ( \lambda -  w_r(t \sigma-,t ) ) \end{gathered}\ ] ] is the energy dissipation of the outgoing shock ._ proof _ taking into account proposition 7.1 from it is sufficient to show to this end we decompose the integral on the left hand side of into three parts the fact that converges to follows from the boundedness of and its pointwise convergence to . regarding the convergence of we note that due to lemmas [ lem : bulkest3d ] and [ lem : vest3d ] the integrand is bounded by which is integrable .moreover , the integrand pointwise converges to thus , it remains to determine the limit of  the first two summands converge to zero as the integrand is bounded , quadratic in and we rewrite the third summand such that the integrand can be estimated by independent of for fixed thus , by the dominated convergence theorem we get ^ 1 = \frac{\omega_d l  w(0,t)^d}{d}.  \end{split}\ ] ]    our next aim is to show that , in fact , energy is needed to create the cavity .[ lem : increase ] let be given , and a cavitating solution as computed in . then (t ) > 0\ ] ] for any ball containing the whole wave fan at time   before we can prove this we need a preparatory lemma .under the conditions of lemma [ lem : increase ] where is the speed of the shock ._ proof _ this proof heavily relies on the estimates in lemma [ lem : psest ] , in particular the monotonicity of .having said that , we calculate as    _ proof of lemma [ lem : increase ] : _ by proposition [ lem : energy3d2 ] we have (t ) - e[\lambda { \bf x},b](t ) = \frac{t^d \omega_d}{d } d\ ] ] where   ( \lambda -  w_r(t \sigma-,t ) ) \big ] + r(0)^d l.\end{gathered}\ ] ] thus , it is sufficient to show we rewrite using + r(0)^d l.\end{gathered}\ ] ] we note that as is increasing , because is convex .furthermore , because and cf .lemma [ lem : psest ] .we combine , and to obtain + r(0)^d l\\  & = \sigma^d \left [ h(r'(\sigma-)\lambda^{d-1 } ) - h(\lambda^d )  \right ] + r(0)^d l\\  & = \sigma^d h'(\tilde v ) ( r'(\sigma- ) - \lambda)\lambda^{d-1 } + r(0)^d l  \end{split}\ ] ] for some using in that , as and , we find   this section we describe how our notion of slic solution easily extends to the system of gas dynamics in lagrangean coordinates .in particular , we will consider a solution to the riemann problem for the containing a vacuum state .the is given by where denotes specific volume and denotes velocity , for some polytropic gas law , i.e. , with . for we complement with initial data in section 9.6 of it is noted that whether , admits a standard weak solution depends on the sign of in case there exists no standard weak solution but a solution containing a vacuum state , which is given in and has the form , where , , it is noted in that is a distributional solution of provided is understood as a continuous function that vanishes at , and it is referred to an eulerian description for a validation of this solution . moreover , converges pointwise to the initial data for subsequently we will show that our approach can treat this solution in a systematic fashion . to fit into our frameworkwe compute a displacement field such that it is easy to verify that is satisfied for   [ def : selfsimslic ] let be any mollifier satisfying the conditions from definition 2.1 and let denote mollification in . for call a self similar slic solution of if the sequences fulfill for all mollifiers and all .     _proof_the anti symmetry of is inherited by and we may write as follows we will compute expressions for and which are well suited for estimates : due to we find for   to show that is bounded it is sufficient to show that for the specific volume is bounded from below by some positive constant . as and find for that     _ proof_as the constant state and the rarefaction wave in given in are linked continuously it is sufficient to consider test functions with support inside the rarefaction / vacuum wave , i.e. , $ ] .furthermore , observe that pointwise . using lemma [ lem : vac : est ] and the dominated convergence theorem , we obtain     for any the energy of a self similar slic solution inside the wave fan from to is defined by := \lim_{n \rightarrow \infty}\int_{-\bar \xi}^{\bar \xi } w(u_n ) + \frac{1}{2 } ( v_n)^2 d\xi \quad  \text { with } w(u ) = \frac{1}{\gamma(\gamma-1 ) } u^{1-\gamma}.\ ] ]  [ lem : vac : energy ] there is no contribution of the vacuum state to the energy . in particular , =\int_{-\bar \xi}^{\bar \xi } \frac{1}{\gamma(\gamma -1 ) } |\xi|^{\frac{2(\gamma-1)}{\gamma+1 } } +  \frac{2}{(\gamma-1)^2 } \left ( |\xi|^{\frac{\gamma-1}{\gamma+1 } } -w\right)^2 d\xi\ ] ] for any .   *acknowledgements * this research was supported by the eu fp7-regpot project `` archimedes center for modeling , analysis and computation '' .aet is partially supported by the `` aristeia '' program of the greek secretariat for research .

Abstract: in the works of k.a . pericak - spector and s. spector a class of self - similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions . cavitating solutions decrease the total mechanical energy and provide a striking example of non - uniqueness of entropy weak solutions ( for polyconvex energies ) due to point - singularities at the cavity . to resolve this paradox , we introduce the concept of singular limiting induced from continuum solution ( or slic - solution ) , according to which a discontinuous motion is a slic - solution if its averages form a family of smooth approximate solutions to the problem . it turns out that there is an energetic cost for creating the cavity , which is captured by the notion of slic - solution but neglected by the usual entropic weak solutions . once this cost is accounted for , the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state . we also apply the notion of slic - solutions to a one - dimensional example describing the onset of fracture , and to gas dynamics in langrangean coordinates with riemann data inducing vacuum in the wave fan .   [ section ] [ lemma]theorem [ lemma]proposition [ lemma]main theorem [ lemma]corollary [ lemma]definition [ lemma]remark [ lemma]remarks [ lemma]notation  = msbm10  l = msbm7