the study of dynamical processes on complex networks has been one of the most active fields within network science , where the ultimate goal is the precise evaluation of how network topology affects dynamics  .first numerical and analytical approaches to this problem were mainly concerned with the pure effect of the heterogeneity of the degree distribution on the overall network dynamics  .great part of the interest in this subject has been due to the detailed description of the topology of real systems that the network representation offers , a fact that motivates the search for a better understanding of their dynamical behavior as well .early analytical developments were then mostly based on the configuration model  , which is able to construct networks with a given degree distribution . whilst offering a benchmark to study dynamical processes in networks with any kind of degree distribution , the networks constructed by such a model are generally locally tree - like , i.e. with a vanishing density of cycles  . on the other hand , it is known that real networks in turn exhibit a much more sophisticated topology , which encompasses features such as clustering , high - order loops , degree - degree correlations , community organization etc .thus , it is evident that studies based solely on the configuration model oversee topological characteristics that are essential for a thorough analysis of the function of real networks .  in order to overcome this limitation and take into account higher - order topological features ,two strategies are commonly used , namely the adoption of networks constructed through stochastic rewiring algorithms  or variations of the traditional configuration model that allow the creation of networks with tunable clustering or other kinds of subgraphs  . in the former ,one starts with a random tree - like network and switches the edges according to some heuristics in order to obtain a desirable network configuration , which is then used as a substrate for the dynamics under study .although this approach enables a precise control of a given network property , as a function of which the dynamics can be analyzed , other properties are dramatically changed  .the latter makes the assessment of the isolated contribution of a particular topological property to the network dynamics unfeasible , since the results can be potentially influenced by spurious effects generated by the method .this limitation can be surpassed by extensions of the configuration model .however , depending on the subgraph structure modelled , the computational complexity quickly escalates , imposing further constraints on the analysis  .  in this paper, we address the effect of particular structural patterns found in real networks , namely cycles of different lengths , by adopting a different approach . in order to control their occurrence , and therefore evaluate their contribution to the network dynamics ,here we consider tree networks with minimal link addition in a way that the number and the length of the cycles are precisely varied .besides the thorough control of network structures being created , the minimal link addition adopted here opens the path for new strategies intended to enhance the stability of real networks .the reason for that resides in the fact that the creation and rewiring of links are usually costly tasks to be performed in these systems  .a prominent example is that of power grids , whose proper functioning is vital for modern society .since the connectivity pattern of the surroundings of a given dynamical unit strongly influences its stability  , it is crucial that the inclusion of new transmission lines is done in a way to ensure , or even enhance , the local and global stability of the network , while spending minimal amounts of resources .otherwise , counterintuitive dynamical effects like braess paradox lead to certain new links destabilizing synchronization by increasing the largest lyapunov exponent , eventually changing it s sign  .this is deeply related to the appearance of cycles in the network .similar arguments also hold for other spatially embedded man - made , or natural networks with constrained connectivity , for instance transportation or neuronal networks .this paper contributes in this direction by quantifying the impact of cycles created under minimal link addition on the global network behaviour with a focus on stability . in general , however , we expect that our approach translates to a broad class of problems ranging from synchronization to percolation , spreading processes or control theory . regarding the network dynamics , we consider the nodes as identical oscillators operating in periodic or chaotic regimes in the paradigmatic cases of logistic maps and rssler oscillators .we evaluate the stability of the synchronous regimes depending on the variation of the length of cycles in the network topology . by employing the master stability function ( msf ) formalism , we map the problem into a spectral analysis of the laplacian matrix . this spectral approach has also been successfully applied to reveal network - dependent coherence .recently , pade and pereira showed that link additions in directed networks can destabilize the synchronous regime .furthermore , in the case of a removal of links or altered link weights , changes to the synchronous state and it s stability are also found to relate to the laplacian spectrum  , in leading order to the fiedler eigenvector . in this paperwe study undirected networks and link addition , considering the whole stability interval of coupling values instead of only the lower boundary .our results suggest that cycles of length four play a special role in network dynamics .more precisely , we find that the inclusion of links that create these motifs yields networks with higher synchronizability in comparison with cycles of different lengths . furthermore , cycles of length three are found to have a weak effect on the laplacian spectrum and , consequently , on the stability of the synchronized state .interestingly , this peculiar innocuous effect of triadic connections on critical dynamical properties has also been reported in other contexts  .this paper is organized as follows : in the next section we explain how we create different network topologies starting from a tree .we present two different methods ( 1 ) introducing only one cycle of a given length and ( 2 ) adding several random links .moreover , we show in detail how different cycle lengths change the properties of the network that control the synchronizability .thereafter , we study how the number of random links added to the starting tree impacts the synchronization of the network using two numerical models , namely the time discrete logistic map and the continuous rssler system .finally , we state the conclusions and perspectives .. in the single link addition procedure , we distinguish two cases , namely connecting nodes in the same fundamental branch ( `` in '' , ) or in different ( `` out '' , ) , i.e. every path between the nodes before the addition contains the root node . ]the procedure of the network generation is as follows . as a starting pointwe consider an undirected balanced tree graph where is the number of levels and is the branching number at each level .therefore , the total number of nodes is given by and the total number of edges is . due to the fact that the total number of edges in a complete undirected graph is ,the remaining unconnected number of pairs for the tree structure is given by .notice that our starting network does not have any cycles .having a highly structured tree - like network with known statistical characteristics allows us to control the network topology in two different ways .one is targeting the network topology directly by introducing cycles of a chosen length . the other way lets us to evaluate what happens if we control the sparseness of the network by randomly introducing a certain amount of additional edges .we are going to refer to the introduction of cycles as the _ single link addition_. here the length of the cycle is the control parameter .the other method will be referred to as _ random link addition _ and , in this case , the control parameter is the probability multiplying the number of unconnected pairs .we want to point out that this network generation method is very similar to the ones used to generate _ small - world _structures .however , the substantial difference is that our initial network is a balanced tree structure instead of a regular lattice .what we aim to evaluate is how the different lengths of cycles respectively the choice of - values change the synchronization features of the networks .we consider the resulting network structure of our two methods as the adjacency matrix , with if nodes and are connected , and otherwise .the number of connections , the degree , of node is given by .further , we define the laplacian , .its eigenvalues ( ) play an important role in characterizing the synchronizability of the system and therefore measure comprehensively what we want to determine  , i.e. changes in the eigenvalue spectrum of our designed networks .in particular , we are interested in the minimal changes or best cycle lengths that have maximum impact on the value of or ( if the network is connected ) .the magnitude of the first non - trivial eigenvalue is related to the onset of synchronization while the magnitude of the maximum eigenvalue is connected to the end of the synchronization interval  .    of the largest laplacian eigenvalue for various wiring choices .for reference , the integers denominate the length of the ( a ) shortest and ( b ) longest cycle possible for each configuration . for the link classification `` in '' vs. `` out '' see fig .[ fig : network ] . ]  while we are going to study the random link addition later when focusing on network dynamics , we start by studying just the eigenvalue pair and as a function of the cycle length after single link addition . avoiding self - loops ,the first cycle is a , i.e. a node s neighbours in the network are themselves connected . using the illustration in fig .[ fig : network ] of a balanced tree network ( only levels are shown ) we see that introducing a in this network means connecting two nodes and that lie on one of the circles highlighting the level number ( see the dashed blue highlighted line in fig .[ fig : network ] ) .in addition , note that the branching on level of the original network is highlighted as the shading of the three areas . ignoring the trivial level , all of the 3cycles that we can create lie within the same shaded area and do not break the symmetry of the network .this also means that they are linking two nodes within the same branch of the tree network ( shaded areas with the same color in fig .[ fig : network ] ) and we call these links _ in - links_. on the other hand , if a link connects nodes belonging to different branches ( e.g. the dashdotted red line in fig .[ fig : network ] ) , we denominate it as an _ out - link_. in this case , the shortest cycle created by an out - link is a .  in fig .[ fig : msf_part1]a we can see how distinguishing between in- and out - links can be used as a tool to generate cycles of different lengths ( shown for ) .starting from the centre of the network , we number all nodes and their levels based on their distance . in this matrix - like plotwe find that , for example , linking two nodes in the first level ( -out ) creates a .similarly , a can be created by linking from the second level ( -out ) back to the centre ( -out ) .as we can see in , are the longest possible cycles in this network .note also that the average path length is changed by the introduction of different cycles .especially short cycles will impact the average shortest path length while long cycles , e.g. out - links connecting leaf nodes , are likely to only have a small effect .   for connections ( a ) on the same level creating cycles of length and ( b ) between different levels creating cycles of length .]  since we are focusing on changes of the laplacian s eigenvalues , it is worth pointing out that adding links in any undirected network always positively increases the magnitude of these eigenvalues . from a synchronization point of viewwe are , for instance , interested in the maximum possible change of the maximum laplacian eigenvalue . in fig .[ fig : msf_part1 ] we show how much the different cycles change the magnitude of and .again we give these changes in terms of _ in _ and _ out _ level connections .we want to highlight some of these changes .firstly , we note that of the possible the one with the strongest impact on the eigenvalues is on , while has less impact .in particular , connecting to neighbouring leaf nodes in level five does not change the upper limit of the spectrum while the largest sensitivity to topological changes is observed close to the root node .this tendency is repeated again in the where the strongest impact is found when is connected to a note on while already connections from to do not lead to such a large change .for the and we summarize this topological effect in fig .[ fig : sum_top ] .  if we ignore the additional information on where the cycle is within the network , it is a valid question to ask for the general impact of the cycle s length on changes in the eigenvalues .this information can be found in fig .[ fig : cycle ] for the distribution of changes in the largest laplacian eigenvalue of .it is not surprising that long cycles play almost no role in small - sized networks , while 4cycles are dominating the change .furthermore , 3cycles induce changes in the spectrum comparable to long cycles , suggesting that such topological patterns have weak impact on network dynamics as well .this is somewhat remarkable , given how pervasive 3cycles are in real - world networks  .noteworthy , it was verified ( not shown here ) that changes in due to the share of long cycles increase with the system size , but still are dominating . were consistently found to be responsible for lower changes in the spectrum than .this has also been addressed by lodato et al .comparing the synchronizability of -node subgraphs , quantified by the ratio of the laplacian eigenvalues  .their analysis showed that -node motifs containing a single -cycle have lower synchronizability than motifs with the same number of nodes and links containing a -cycle .interestingly , a single link addition that creates two -cycles in a -node motif does not increase its synchronizability .therefore , their result suggests that networks with a higher number of -cycles are more prone to exhibit a synchronized state .    concerning the random link addition , it is of great interest to estimate the probability for cycles of a particular length being created . as we can see from the relative occurrence of cycle lengths plotted in fig .[ fig : hist ] it is more likely that we choose longer cycles , i.e. on average a randomly chosen link would create a .therefore we can expect that large changes associated , for example , with some of the will be less likely to dominate the eigenvalue spectrum , while the sum of changes resulting from several longer cycles will make up the main effect that changes the eigenvalue spectrum .balanced tree . ]by means of the above mentioned methods , we obtain the adjacency matrices of the networks containing our prototypical dynamic systems .we investigate two paragons of dynamical systems theory , namely rssler oscillators , as an example for an autonomous continuous system , and the well - established logistic map as a discrete one .our main tool allowing us to find relationships between the network spectral properties and the stability of synchronous regimes of the considered dynamical systems is the master stability function ( msf ) approach .  to illustrate the method, we first assume a continuous ( autonomous ) dynamical system of , for instance , coupled oscillators described by a state vector at site in a network :    where is the individual oscillator s dynamics , the coupling function between coupled elements , an element of the network s adjacency matrix and the overall coupling strength . to assesswhether our networks allow for stable solutions of complete synchronization , i.e. , we need to obtain the msf . for identical oscillatorsthis function is given by :    where with laplacian eigenvalues and and are the jacobian of the system and of the coupling function . if we assume for a moment that the argument is a continuous ( in general complex ) variable , the real roots of the maximum lyapunov exponent of determine the boundaries and of the stability interval . note that for periodic systems , there is only one root as . in order to have a stable solution , the coupling needs to be chosen in a way that , for to be negative and hence to have an asymptotically stable synchronous regime ( cf . fig .[ fig : msf ] ) .  as discussed in the section before ,the most crucial non - trivial laplacian eigenvalues to be fit in the stability interval are the second minimum and the largest ones , since the rest of the eigenvalues are distributed between them .  in the following, we apply this approach to the rssler system as oscillatory units that are coupled through their :    computing , we find that and are given by    for our investigations we choose the parameters such that we have one set in the periodic regime ( , , ) and one in the chaotic regime ( , , , ) .notice that the msf in general only yields a valid linearization in a neighbourhood around a point from the synchronization manifold which we determine from numerical simulations of the system .we obtain and for the periodic and chaotic cases , respectively .  as a second applicationwe use the time - discrete logistic map as an oscillator given by :    where the local node dynamics is the logistic map (  ] is the coupling strength between the units and an adjacency matrix .the logistic map is one - dimensional , therefore the jacobian of eq .( [ eq : logmap ] ) is the derivative of with respect to and the associated msf is given by    where , are the eigenvalues of the laplacian as stated above .  for our researchwe choose the control parameter of the logistic map , , such that we have one periodic ( , period 3 ) and one chaotic case ( ) .for the periodic case , again has only one root which is found as ( fig .[ fig : msf ] ) . the roots for the chaotic local dynamicsare found to be and .we demonstrate the stability of the coupled oscillators representing the nodes of our networks .we study the synchronization behaviour of two different node types , either the nodes dynamics are given by logistic map or the continuous rssler system .both systems are analyzed for two different dynamical regimes , periodic and chaotic . as we can see in fig .[ fig : eig_rand ] the eigenvalues and change with a different rate depending on the parameter which controls the random link addition .while the initial growth of dominates for low -values , it seems to saturate for high values where we observe most of the change for .   for different probability of link addition ( see text for details ) .the shaded area denotes the existence of stable synchronous solutions . in all panels ,the initial topology was . ]  hence , if just a few links are added at random locations , the initial growth of leads to a spreading of the laplacian spectrum , with the potential to cross the upper limit of the synchronization interval .this can be counteracted by lowering the coupling , i.e. small changes in the network should be accommodated by reducing to safely pertain in the synchronous interval .  in summary ,[ fig : msf ] depicts the qualitative results for msfs of the two test systems , the shaded areas denote the synchronous region . in the white regions the synchronous regime is unstable .as mentioned above , increasing -values leads to an increase in edges and similarly the laplacian s eigenvalues increase as well .\i ) the lower boundary for the periodic cases is ( cf .[ fig : msf]a , c ) , so that one only needs to consider the upper limit to ensure stable synchronization .\ii ) the lower boundary of the chaotic case is positive .therefore both limits are needed to be considered ( cf .[ fig : msf]b , d ) .  as we can see in all four cases ,there is a maximum amount of links that we can add before the system reaches the upper threshold of the synchronization region .clearly if we choose , some values are no longer within the shaded synchronization regions in all panels of fig .[ fig : msf ] ( see the highlighted pentagons above each graph ) .more interesting is the situation for the oscillators being chaotic ( cf .[ fig : msf]b , d ) .we can see that without adding any links ( ) , the network does not support synchronization .increasing the number of links leads to networks that can support synchronization , but again too many links will increase the eigenvalues by too much and we can not find synchronization anymore . comparing these results with the previous fig . [fig : eig_rand ] we see that needs to be big enough to increase above the level where the lower bound is already in the synchronization region , while is still below the upper threshold of the band .   and for varying linkage probability , the shading indicates one standard deviation .the inset magnifies the regime of .note that the two eigenvalues and bound the eigenvalue spectrum of the laplacian . ]since large natural or man - made networks are locally tree - like , we have focused on small trees as the starting point of constructing our network .we have seen that even including one edge can substantially alter the synchronization behaviour of the system and that short cycles connecting different levels of the tree have the largest impact on the eigenvalues of the laplacian , especially if they are created close to the root node .we have highlighted a way to add cycles of defined length in our trees and therefore give the option to design networks having a particular synchronization behaviour .  using the master stability framework we were able to analyze how random link addition alters the synchronization phenomenon using the logistic map as well as the rssler oscillator .the most striking example , that we studied , is the synchronization behaviour of nodes having chaotic dynamics .while without additional links the networks are unable to synchronize the dynamics of the nodes , adding some links to the networks alters the global dynamics essentially and the systems can synchronize .moreover we have found that adding too many links causes desynchronization .  in conclusionour work provides a method on how to optimize networks in such a way , that they can become synchronized in an improved manner .given that synchronization is of outermost importance in many networks , e.g. power grids  , neuronal  and communication networks  , our findings may be used to increase the stability of the synchronization regime by adding short cycles .the authors wish to thank the nesin foundation for an amazing working group activity in nesin math village and wish to thank tiago pereira for fruitful discussions .ps and jk acknowledge gratefully the support of bmbf , condynet , fk . 03sf0472a .tp acknowledges fapesp ( no . 2012/22160 - 7 and no .2015/02486 - 3 ) and irtg 1740 . de acknowledge support by the leibniz association ( wgl ) under grant no .saw-2013-izw-2 .34ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] _( , ) * * ,  ( ) _ _ ( , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ( ) * * ,  ( ) * * ,  ( ) _ _ , vol .( , , ) p.  * * ,  ( ) in  _ _( , ) pp .  * * ,  ( ) * * ,  ( ) ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: natural and man made networks often possess locally tree - like sub - structures . taking such tree networks as our starting point , we show how the addition of links changes the synchronization properties of the network . we focus on two different methods of link addition . the first method adds single links that create cycles of a well - defined length . following a topological approach we introduce cycles of varying length and analyze how this feature , as well as the position in the network , alters the synchronous behaviour . we show that in particular short cycles can lead to a maximum change of the laplacian s eigenvalue spectrum , dictating the synchronization properties of such networks . the second method connects a certain proportion of the initially unconnected nodes . we simulate dynamical systems on these network topologies , with the nodes local dynamics being either a discrete or continuous . here our main result is that a certain amount of additional links , with the relative position in the network being crucial , can be beneficial to ensure stable synchronization .


communication of information requires an encoding of the information into a physical system . the laws of physics therefore govern the limits on processing and communication of information . by modelling real world noise in terms of simpler models , the maximum rate for information transfermay be obtained .the seminal work of shannon showed that a memoryless noisy channel , where the noise acts independently on each symbol sent through the channel , can be parameterized by a single quantity , the capacity of the channel .shannon defined the capacity as the maximum rate that information may be sent through a channel , and he also showed that there exist codes that asymptotically achieve this rate with a vanishing probability of error .the work of shannon has been extended to include channels with `` memory '' , where the noise is no longer independent of past channel uses , or where the channel consists of an arbitrary transformation of the input states .encoding classical information into quantum states of physical systems gives a physical implementation of the constructs of information theory .the majority of research into quantum communication channels has focused on the memoryless case , although there have been a number of important results obtained for quantum channels with correlated noise operators , or more general quantum channels .recently a model for quantum channels with memory has been proposed that can consistently define quantum channels with markovian correlated noise .the model also extends to describe channels that act on transmitted states in such a way that there is no requirement for interaction with an environment within the model .  in this paper a single upper bound on the capacity of arbitrary indecomposable quantum channels is derived .a quantum channel is defined as a completely positive , trace preserving map from the set of density operators to itself .any such map may be represented as a unitary operation between the system state and an environment with a known initial state .memoryless quantum channels act on each input state independently of the previous input or output states .for a single channel use the output state is given by , \ ] ] with the input state , the initial state of the environment , a unitary operation between the state and environment , and the output state . for a sequence of transmissions through a memoryless channel , the output state is given by , \nonumber \\ & = \big(\lambda_{n } \otimes ... \otimes \lambda_{1 } \big ) \rho_q \label{eqn : memoryless_model}\end{aligned}\ ] ] where the state now represents a ( possibly entangled ) input state across the channel uses , the unitary operations are all identical , and the environment state is a product state .one model of a quantum memory channel is where each state going through the channel acts with a unitary interaction _ on the same channel memory state _ , as well as an independent environment .the backaction of the channel state on the message state therefore gives a memory to the channel .the general model thus includes a channel memory , and the independent environments for each qubit .hence , \nonumber \\ & = \mathrm{tr}_{m } \big [ \lambda_{n , m } ... \lambda_{1,m }  \big(\rho_q \otimes \omega_m \big ) \big ] \label{eqn : memory_model}\end{aligned}\ ] ] where is the initial memory state , and are the input and the output states of the channel , respectively , and the trace over the environment is over all environment states .figure [ fig : memory_channel ] illustrates the action of the unitary operators on the input , memory and environment states representing the channel .if the unitaries factor into independent unitaries acting on the memory and the combined state and environment , that is , , then the memory traces out and we have a memoryless channel .if the unitaries reduce to , we can call it a _ perfect _ memory channel , as no information is lost to the environment .the mapping of the memory state under the unitary operation also corresponds to a quantum channel on the memory state .memory channel representations where the action on the memory state is independent of the input state are termed _point _ memory channels , as the memory state will have a fixed point under the action of the represetation \omega_m = \omega_m ] . due to the monotonicity of the von neumann mutual information , the upper bound may be expressed as , where is the set of all separable states .the maximum amount of mutual information that may be generated through a quantum channel is therefore bounded by the maximum amount of `` classical '' correlation that may be shared by states through the channel .the upper bound on the classical capacity of a finite memory channel , which includes the classically correlated channels , is given by , ) \label{eqn : mem_holevo_upperbound}\ ] ] where = \lambda^{(n)}[\omega_m]\rho_q ] and $ ] denote the channels with initial memories and acting on input states to , and the maximum difference in entropy for the first output states is bounded by .therefore , taking in ( [ eqn : ave_ent_bound ] ) , the average entropies converge asymptotically for any state .the difference between the upper bounds and must therefore also converge asymptotically .a single upper bound on the capacity for the class of indecomposable quantum memory channels has been derived .

Abstract: the maximum rates for information transmission through noisy quantum channels has primarily been developed for memoryless channels , where the noise on each transmitted state is treated as independent . many real world communication channels experience noise which is modelled better by errors that are correlated between separate channel uses . in this paper , upper bounds on the classical information capacities of a class of quantum memory channels are derived . the class of channels consists of indecomposable quantum memory channels , a generalization of classical indecomposable finite state channels .


anisotropic mesh adaptation is an established technique for ensuring computational efficiency in the context of multiscale problems , sometimes reducing the computational work with several orders of magnitude . within the field of structural optimisation fixed structuredmeshes remain popular due to ease of implementation and compatibility with mathematical optimisers .it is thus mainly in the context of methods utilising various continuous sensitivities that mesh adaptivity has been applied , and even then mainly in the context of mesh refinement , but some of the most recent advancements also relate to the use of swapping operations .parallelism is another popular technique for speeding up computations and for truly large scale problems , it indeed is the only option .the community of structural optimisation has devoted significant effort towards not only finding stiff and light structures , but also structures that do not break under load .that is the structure has to satisfy a stress constraint .recent strategies include methods with global stress constraints , either with an explicit design representation , such that computation of void elements and related stress is avoided , or an implicit representation .an essential question in this context is a global versus a local constraint as investigated in the context of level - set methods .one can also choose to let the number of constraints vary throughout the optimisation .an older , but still popular approach , approximates the local stress constraints with a single global constraint using a p - norm .the approximation is good for large values of p , but this causes numerical problems and therefore a compromise has to be made in practice . if the numerical problems are due to discretisation errors , one can hope to achieve a better compromise with more computational power or by being more efficient with the power available .it is the second option , that we choose to pursue with anisotropic mesh adaptation in this paper .the mesh adaptation introduces inconsistencies , but this drawback can be outweighed by the benefits of improved resolution of small length scales .we choose to calculate discrete sensitivities as this allows us to harness the power of libraries for automatic adjoint derivation .a continuous sensitivity is required for driving the mesh adaptation as well as when interpolating between meshes , but this is trivial to compute by dividing the discrete sensitivity with the design variable volumes , and one can invert the operation , if discrete variables are desired for the optimiser .stretched elements with small or large angles are normally discouraged in the context of the finite element method , as they give rise to not only large discretisation errors , but also cause problems with iterative solvers .this wisdom is , however , only applicable to problems without any anisotropic features in the solution .in fact , problems with strong anisotropy require anisotropic meshes for optimal use of the computational degrees of freedom . note that the problems with large angles and iterative solvers are only severe for extreme anisotropy and , furthermore , it is possible to create anisotropic meshes with few large angles .the properties of the quasi optimal mesh can be derived in a continuous sense using a metric tensor , .this is a symmetric and positive definite tensor field that maps the optimal element to the unit element , which has unit length edges .the first step is to compute the hessian , , of the variable whose discretisation error is to be minimised and convert it into a positive definite matrix .this is done by taking the absolute value in the principal frame using the operator , which corresponds to removing the sign of the eigenvalues .the optimal mesh metric can then be expressed as \right)^{-\frac{1}{2q+d } } { \underline{\underline{\mathbf{\mathrm{abs}}}}}({\underline{\underline{\mathbf{h } } } } ) \label{eqn : m } ,  \end{aligned}\ ] ] where is the number of dimensions , is the error norm to be minimised , is the determinant and is a scaling factor .it is possible to combine the metrics of several variables using the inner ellipse method illustrated in figure [ fig : ellipse ] , see for implementation details .note that the unit of is always so that the metric in equation ( [ eqn : m ] ) has dimension of squared inverse length .once a final metric has been calculated , it can be passed to an anisotropic mesh generator together with the current mesh , in fact our metric is defined on the nodes of the current mesh .we use two mesh generators , which applies four local mesh modifications : coarsening , refinement , swapping and smoothing , see figure [ fig : meshmod ] .the mesh generators work by applying the modifications to optimise a heuristic quality measure , which quantifies the difference between the discrete mesh and the optimal continuous one .we use an optimised c++ mesh generator , pragmatic to illustrate the negligible performance penalty of mesh adaptation , but presently this does not support curved geometries , so in that case we resort to a slower matlab / octave implementation .our implementation can generate a metric from a function represented on continuous 2nd order polynomials , but 1st order polynomials can also be used . in the latter case galerkin projectionis used as derivative recovery technique to compute the hessian . in any case , galerkin projection is used to convert the element wise constant metric to a node based representation , before it is passed to the mesh generator .in an effort to maximise the impact of this paper , we have tried to choose the most popular method for topology optimisation with stress constraints , but we see no reason why other methods could not benefit equally well from the use of mesh adaptation . similarly other problems with small length scales , such as many convective problems , could most definitely be solved more efficiently using anisotropic mesh adaptation .this being said , we will focus on the density method with simp penalisation and a p - norm for relaxing the local stress constraints to a single global constraint .the idea is that the local constraint is satisfied in the limit of going to infinity . in practice a finite valueis chosen , but in the following we will show that adaptive meshing allows for and that this effectively removes kinks in the design .we use a stress penalisation scheme to eliminate problems with void stress , and the method of moving asymptotes is used to update the design variables .it is likely that the perfect solution to stress constrained topology optimisation will continue to allude the scientific community , but one might hope that scholars demonstrate their various heuristic schemes in the context of mesh independence such that the uncertainties of objective and constraint functions can be estimated , allowing for techniques to be compared on an objective basis .we have found the selection of papers including quantitative information related to computational cost in the context of stress constrained topology optimisation to be rather scarce , and we have been unable to identify any papers containing a discussion of mesh independence .this might be due to it being well known that the stress does not converge for designs with kinks , but in fact it has been shown that the design curvature can be effectively controlled using projection methods .it is well known that the p - norm underestimates the maximum stress , but we have deliberately chosen not to implement a local stress fix scheme , that is a scheme that changes the maximum stress to a conservative value such that the local stress constraint is satisfied .this is due to the fact that local stress constraints constitute a much more difficult problem .the alternative is to solve the relaxed stress problem first , and address the local stress problem by employing a conservative maximum stress in combination with a large p - norm .regardless , the design has to go through a manual post - processing step as it is already typical for designs obtained with topology optimisation , but the topology is normally fixed in the post - processing and therefore it is important that the optimisation result is robust and thus mesh independent .in order to avoid a stress singularity in the problem definitions .sharp corners can also give rise to stress singularities , and we have investigated the effect of this by rounding the corner of the l - bracket .the rollers to the lower right of the portal frame are associated with zero normal displacement , ,scaledwidth=35.0% ]  we consider two 2d problems , the l - bracket and the portal frame , both with finite load and support areas , and , as illustrated in figure [ fig : lgeom ] .we model plane stress and use a helmholtz filter to compute a filtered design , , with a minimum length scale , , ^t \right ) , \quad \partial_z u_z = -\frac{\nu}{1-\nu } { \boldsymbol{\nabla}}\cdot{\mathbf{u } } , { \nonumber}\\ g & = & e\frac{1}{2(1+\nu ) } , \quad \lambda = e\frac{\nu}{(1+\nu)(1 - 2\nu ) } , { \nonumber}\\ \tilde{\rho } & = & \rho + { \boldsymbol{\nabla}}\cdot l_\mathrm{min}^2 \cdot { \boldsymbol{\nabla}}\tilde{\rho } , \label{eqn : filter}\\ e & = & e_\mathrm{min}+(e_\mathrm{max}-e_\mathrm{min } ) \rho^{p_e } \label{eqn : e}\end{aligned}\ ] ] where is the lagrangian displacement , is the stress , is the strain , is the out of plane deformation , is the identity tensor , is the trace , is the shear modulus , is lam s first parameter , is young s modulus , is the poisson ratio , is the design variable , and is the simp penalisation exponent . note that we intend to use a sensitivity filter for the compliance .we thus avoid using the filtered design variable in equation ( [ eqn : e ] ) , it is only calculated for the purpose of driving the mesh adaptation .  the displacement as well as the design variables are discretised with continuous 1st order polynomials , while we use 2nd order when applying the pde filter ( [ eqn : filter ] ) .the use of discontinuous constant design variables gives rise to excessive stress concentrations in the context of a sensitivity filter , and thus we have found significantly better results ( objective functions ) using nodal densities .element wise constant densities might be used with a density filter , but then the issue of dealing with negative filtered design variables in a robust way arises , hence we use a sensitivity filter .  the forward problem defined in equations ( [ eqn : stress]-[eqn : e ] )is solved using fenics , an open source finite element engine with a high degree of automation .we use a direct solver for the forward , adjoint and filter problems , and an iterative solver for galerkin projections between different element types .we use the current design and displacement to calculate objective and constraint functions , ^{1/p } - 1 { \nonumber } , \ ] ] where is the maximum compliance , is the stress penalisation exponent , is the von mises stress and is its maximum value .the interpolations of the young s modulus in equations ( [ eqn : e ] ) and ( [ eqn : es ] ) agree in the limits of and corresponding to void and material , respectively , but one has to penalise the intermediate values differently for stress constrained optimisations .the discrete gradient of can be calculated explicitly , and we use dolfint - adjoint to calculate the gradients of and . these two discrete gradients are converted to continuous ones by division with the gradient of and anisotropic helmholtz smoothing is applied to the stress sensitivity .this smoothing is defined on the continuous level using an equation identical to ( [ eqn : filter ] ) with a tensor version of , which is based on the steiner ellipse of the elements .we use the helmholtz filter ( [ eqn : filter ] ) for the compliance sensitivity , that is we do not use a sensitivity filter related to non - local elasticity as discussed in .these continuous and smooth sensitivities are then used to calculate metrics associated with the compliance and stress constraints .the filtered design variable is used to calculate a metric associated with the volume constraint and all three metrics are combined using the inner ellipse method , see figure [ fig : ellipse ] .after the metric has been used to adapt the mesh , the optimiser variables ( the asymptotes , continuous sensitivities , previous , and previous previous design variables ) are interpolated on to the new mesh .due to extrapolation at curved boundaries it can become necessary to enforce the box constraints on the asymptotes and design variables after this interpolation step .once this interpolation step is complete , the continuous sensitivities are converted to discrete ones , and the optimiser is called to update the design variables .this completes the optimisation loop as illustrated in figure [ fig : flowchart ] .this methodology is inconsistent from a mathematical programming point of view , but except for the compliance sensitivity filter , we expect the errors to decrease with mesh refinement .we consider volume minimisation under stress and compliance constraints , the formal problem statement being the optimisation is initialised with a completely solid design , .we make the problem dimensionless using and as characteristic length scale and stress , respectively .this is reflected in the set of parameters used in the optimisations , the maximum compliance and two of the length scales differ between the two benchmarks as shown in table [ tab : tab1 ] .ideally the values for the maximum stress and compliance should be chosen according to the application , but for the academic benchmarks considered here , we just choose a maximum compliance that results in a non - extreme volume fraction and a maximum stress that results in the stress constraint being active ..the and length scale differs between the portal and l - bracket problems , and the maximum compliance is significantly higher for the l - bracket problem . [cols="^,^,^",options="header " , ]   [ [ computational - cost ] ] computational cost + + + + + + + + + + + + + + + + + +  all computations are single threaded , but it is possible to perform mesh adaptation in parallel , see .we terminate the optimisations using an iteration limit only , and the total computation time before this triggers is shown in hours at the top of each plot in figure [ fig : designs ] .we use a direct solver for the filter operations as well as for the forward and adjoint problems .the coarse optimisation of the l - bracket with a rounded corner probably only took around an hour to find the best design , but others have found similar designs with an order of magnitude less effort and comparing figures [ fig : static ] and [ fig : designs](a - c ) reveals that it is possible to achieve a similar volume fraction using a fixed mesh and a similar amount of computational time .there is thus reason to believe that the presented method needs improvement , if it is to be competitive .it is , however , worth noting that the resolution of the design near the corner is significantly worse for the static mesh , even compared to the most coarse adaptive optimisation , so the fixed mesh might be benefiting from under resolving the stress singularity .the optimisations with sharp corners are performed with pragmatic as mesh adaptation library , which is an optimised c++ implementation , so the actual mesh adaptation only takes 1 - 2% of the total computation time .the optimisations with a rounded corner use an almost identical octave / matlab implementation , which ( although fully vectorised ) is significantly slower , and therefore the mesh generation takes 20 - 30% of the total computation time. note that although the meshes conform well to the designs in figure [ fig : designs ] , this is not true for the intermediate iterations .this is due to the fact that we optimise with , so small imperfections in the designs can cause concentrations of stress away from the corner , which lead to many small elements in areas that are close to becoming void .we calculate the steiner ellipse for each element , and use this to calculate the element aspect ratio by diving the radii product with the square of the smaller radius .this element aspect ratio is between 4 and 5 on average throughout the optimisations , which is a indication of speedup due to the anisotropic elements .one can expect the square of this in 3d corresponding to an order of magnitude reduction in computational cost compared to isotropic adaptation .we have performed stress constrained topology optimisations using a combination of anisotropic mesh adaptation and the method of moving asymptotes with interpolation of the asymptotes between iterations .we find that it is necessary to use continuous linear design variables and a sensitivity filter .the computational cost of introducing the mesh adaptation is negligible for an optimised implementation .we argue that it might be beneficial to relax the problem statement to a rounded corner , as a radius of just 1% of the characteristic length scale helps the optimiser find better designs .at least , in this case we are able to demonstrate mesh independence .it would be interesting to see the method applied to design of compliant mechanisms or meta materials due to the fact that the stress concentration results from the nature of the problem rather than the geometry .another interesting aspect is the post - processing of the result , i.e. to which extend is it beneficial to have many small elements aligned with the design contour , when the design is to be extracted ?furthermore , the fact that anisotropic elements can give rise to large angles which cause problems for iterative solvers should be quantified in three dimensions and possibly also addressed by means of a meshing technique relying on advancing fronts .another point for three dimensions is , that due to the trend of increasing number of cores in workstations , the fastest methods generally rely on shared memory parallelism .therefore one ought to investigate the benefit of parallelism , when extending the method to three dimensions .finally , the meshes that occur during the optimisation loop have only small variations during the later stages of the procedure , meaning that the degrees of freedom are chosen efficiently in the spatial dimension only , not in the optimisation dimension .we suggest to address this issue using time - space elements and an optimiser defined at the continuous level .this work is supported by the villum foundation .wagdi g habashi , julien dompierre , yves bourgault , djaffar ait - ali - yahia , michel fortin , and marie - gabrielle vallet .anisotropic mesh adaptation : towards user - independent , mesh - independent and solver - independent cfd .part i : general principles . , 32(6):725744 , 2000 .pierre duysinx and ole sigmund .new developments in handling stress constraints in optimal material distribution . in _ proc of the 7th aiaa / usaf / nasaissmo symp on multidisciplinary analysis and optimization _ , volume 1 , pages 15011509 , 1998 .

Abstract: the p - norm often used in stress constrained topology optimisation supposedly mimics a delta function and it is thus characterised by a small length scale and ideally one would also prefer to have the solid - void transition occur over a small length scale , since the material in this transition does not have a clear physical interpretation . we propose to resolve these small length scales using anisotropic mesh adaptation . we use the method of moving asymptotes with interpolation of sensitivities , asymptotes and design variables between iterations . we demonstrate this combination for the portal and l - bracket problems with p=10 , and we are able to investigate mesh dependence . finally , we suggest relaxing the l - bracket problem statement by introducing a rounded corner .


in a segmented calorimeter , where the energy deposition is sampled in layers along the shower , the fluctuations in a layer at a given depth arise from fluctuations in starting point and development of the shower . already in 1937bhabha and heitler  , in their work on electromagnetic cascades , studied fluctuations and proposed that these follow a poisson distribution .furry  used a multiplicative model for the shower development and arrived at a wider distribution , often called the furry distribution .mitra  used the negative binomial or plya distribution to describe air showers , with the poisson and furry distributions as limiting cases .the plya distribution is mentioned in conjunction with cosmic rays for the first time in 1943  .it has been used e.g. to describe photomultiplier electron statistics  and in particle physics to describe hadronic inelastic non - diffractive multiplicity distributions of produced particles  .fluctuations in electromagnetic cascades have been considered as an application of markov processes  and analytic solutions have been found  .only few experimental or simulation based studies of fluctuations at a given depth can be found in the literature .longo and sestili  have used simulations to study fluctuations in the last radiation length of lead glass of different lengths for photons with energies up to 1 gev .they used the plya distribution for comparison .gaisser  , in discussing cosmic air showers , points out that these fluctuations are smallest near shower maximum and that they approximately follow a log - normal distribution , reflecting the multiplicative nature of the shower development . in a recent paper ,souza et al . studied the fluctuations of maximum number of particles in air showers in order to estimate how well the shower size at maximum depth can determine the shower energy .general properties of calorimeter fluctuations are reported e.g. by amaldi  . in this contributionwe report on a simulation study of fluctuations as function of depth in a segmented csi calorimeter .we also use these simulations to compare different methods for energy reconstruction especially for non - contained electro - magnetic showers .in the simulation studies the calorimeter consisted of csi ( radiation length 1.85 cm ) crystal bars arranged in layers perpendicular to the incident particles . the layout , shown in fig .[ fig : simcal ] , resembles a glast electromagnetic calorimeter tower  , which is built up by 8 orthogonally oriented layers of 12 csi(tl ) crystal bars of size 32.60.67.99  , giving a total thickness of about 8.6 radiation lengths for perpendicular incidence . in order to study the basic fluctuation phenomena, the simulated calorimeter does not contain any supporting material or gaps , the layers were not placed orthogonal , and in total 20 layers were implemented .each layer consisted of 6 crystals .furthermore , in the simulations the energy deposited in the crystals was taken as the measured energy , i.e. no readout system has been applied .  .the incident particle enters the calorimeter from left , width=240 ]  the simulations were performed using geant4 ( v7.0p01 ) , a monte carlo particle transport toolkit which is well supported and used by several experiments . only standard electromagnetic processeswere considered , all photo - nuclear interactions have been neglected .ajimura et al . have measured the photo - nuclear interactions to contribute to less than in csi for energies above 1 gev .electromagnetic processes are well simulated by geant4 in the energy range relevant here , for a validation see e.g.  .geant4 uses the particle range to set the energy cut - off below which secondary particles are no longer tracked and the energy is simply dumped at that location .we used a range cut of 1 mm , corresponding to an energy cut - off of 38 kev , 692 kev and 658 kev for respectively photons , electrons and positrons in csi .simulations were performed with incident electrons and photons in the energy range 1100 gev with a minimum of 100000 events in each simulation .most studies were done with perpendicular input , but the case with non - perpendicular input having a incident angle of was also examined .[ fig : meanedep ] shows the mean longitudinal profiles for showers induced by electrons and photons having initial energies of 1 gev , 10 gev and 100 gev .the showers induced by photons reach their maxima later than those induced by electrons .this is due to the differences in electron and photon interaction mechanisms .an electron starts losing energy immediately as it enters the calorimeter media , while photons may travel a distance before interacting .see further the work of wigmans and zeyrek  for a study on differences between photon- and electron - induced showers .the profiles are well fitted with the gamma distribution : where is the depth in radiation lengths , an energy - dependent parameter and a -dependent parameter varying slowly with energy . is the initial energy of the particle , is the gamma function .we see that the shower maxima for initial energies of 1 gev , 10 gev and 100 gev occur at layer 4 , 6 and 8 ( 4.3 , 6.5 and 8.6 radiation lengths ) for electrons and at layer 5 , 7 and 9 ( 5.4 , 7.5 and 9.7 radiation lengths ) for photons .  ).,width=240 ]   the fluctuations in energy deposition in each layer were studied both for showers induced by electrons and photons of initial energy of 1 gev , 10 gev and 100 gev . in each case 100000incident particles ( electrons or photons ) were simulated . figs .[ fig : fluct_lay ] shows how the fluctuations change from layer to layer in the first 15 layers ( 16 radiation lengths ) for a particle energy of 10 gev.the energy fluctuations in layer 1620 have the same exponential - like shape as layer 15 , only shifted towards lower energy as the depth increases .all of the 10 gev electrons have deposited energy in the first layer ( 1.1 radiation lengths ) whereas about 5% of the 10 gev photons have not interacted at this depth .the shape of the energy fluctuations changes systematically with depth for the three different energies .both electron- and photon - induced showers show fluctuations with a high - energy tail at small depths , which is reduced as the depth increases and the distribution becomes more symmetric and gaussian - like .moving deeper into the calorimeter , a low - energy tail develops . eventually the fluctuations become more symmetric again andfinally a high - energy tail emerges .the non - asymmetry of the energy distributions can be readily understood from fluctuations in the shower developments .indeed , in the first few radiation lengths early shower developments can give rise to large total energy deposits giving a high - energy tail .at the location of the mean maximum energy deposition , fluctuations to higher energies are less frequent than those to lower energies , giving distributions with a slight low - energy tail .the most symmetric gaussian - like distributions occur just before and after the maximum energy deposition where there are both high- and low - energy contributions from early respectively late initiated showers .these tendencies become more pronounced with increasing energy as the longitudinal shower profile becomes more stretched out .  the energy fluctuations in each layerwere fitted with negative binomial , log - normal and gaussian distributions for incident electrons and photons of energy 1 gev , 10 gev and 100 gev .the fits were performed over a range in energy corresponding to 1% of the peak value of the distribution . as expected from the variations as function of depth , quality of fits varies considerably and in most cases there are deviations from the fitted form in the tails of the distributions .an important conclusion from the fits is that for each layer one but not the same distribution gives the best fit . with 100000 simulated eventsonly a few layers give a fully acceptable fit as judged by the .  as an example we show in fig .[ fig : fluctfit ] the 10 gev electron and photon energy distributions in layers 5 , 8 and 12 ( 5.4 , 8.6 and 12.8 radiation lengths ) .these three layers are before , close to and after the shower maximum .the fitting range covers 97.1 , 99.5 and 98.9% ( 88.8 , 97.8 and 98.4% ) of the 100000 events in the simulation for electrons ( photons ) .the low - energy tails in layers 5 and 8 are not well fitted .the high - energy tails are quite well described by the log - normal or negative binomial distributions .[ fig : parchi ] shows as function of layer number the reduced of the fits , giving a relative evaluation of how well the three distributions fit the energy fluctuations .the systematic change of the fluctuations with depth can be seen .we include in fig .[ fig : parchi ] the reduced for layers 1620 although the number of events there is small . for 1 gev incident electrons and photons, the gaussian distribution gives the best fit in the first 5 layers which is expected since the the mean shower maximum occurs in layer 4 . at larger depths the energy fluctuationsare best fitted with the negative binomial .note that the energy fluctuations become close to exponential at layer 14 and beyond , giving a large uncertainty in the fitted parameters . for 10 gevthe gaussian distribution gives the best fit a few layers before and after the mean shower maximum , as expected .when the high - energy tail emerges , the best fit is given by the negative binomial , and as it becomes more prominent , the log - normal .for 100 gev the gaussian again gives the best fit before and after the location of the shower maximum , and when the high - energy tail emerges the log - normal distribution gives the best fit .the negative binomial gives quite a good fit up to layer 4 , but after that it completely fails .there are no larger differences in the between electron- and photon - induced showers since the width changes but not the shape of the fluctuations .     of the negative binomial ( solid line ) ,log - normal ( dashed line ) and the gaussian ( dotted line ) fit for 1 gev , 10 gev and 100 gev electrons and photons .the lines have been drawn to guide the eye .the arrows mark the mean shower maximum.,width=240 ]  in this way parameterizations of the deposited energy distributions can be obtained as function of energy and incident angle .more detailed results of this simulation study are presented in  .the simplest method to estimate the energy in a segmented calorimeter is to sum up the energies deposited in each segment .this estimate is only accurate if the energies of the incident particles are small as compared to the average energy lost of the particles in the calorimeter . in the usual case ,corrections have to be applied for the energy lost . in this contextit is worth noting that the energy lost due to leakage is correlated with the energy deposited in the last layer ( see figure [ fig : last_layer ] ) .a corrected energy can thus be calculated from : where is the energy estimated corrected for leakage , is the total energy deposited in the calorimeter and is the energy deposited in the last layer .the parameter is to be determined from monte - carlo simulations or from measurements under controlled conditions ( beam test ) .another method to estimate energy is to fit individual showers with the - distribution ( see eqn .[ eqn : gamma ] ) . considering a mono - energetic beam of particles in which all shower maxima are contained within the calorimeter, fitting individual showers will result in gaussian distributions of , and , centered on their true values .however , as can bee seen in figure [ fig : badindfit ] , if the shower maximum is not contained within the calorimeter the fit fails badly .thus , if the energy of the incident particles is high enough such that a significant fraction of showers have maxima outside the calorimeter volume , the distribution of will have significant high energy tails .figure [ fig:8and20 ] shows in the upper panel the distribution of for a  calorimeter as compared to one which is 22 in depth . for this figurea 10 gev electron beam was assumed . in the shallower calorimeterabout 1 % of the electrons are estimated to have an energy above 20 gev .for the deeper calorimeter the situation improves signficantly : the fraction of electrons where the estimated energy is above 11 gev is only 0.1 % .-function . in this case , 10 gev photons have been simulated at normal incidence .the upper panel shows a longitudinal profile with its maximum outside the sensitive calorimeter volume .the energy estimate obtained in this case is 692 gev .the lower panel shows an example of a profile where the shower maximum is contained .the estimated energy in this case is 9.4 gev.,width=240 ]  -distribution ( see eqn .[ eqn : gamma ] ) to indvidual longitudinal shower profiles .the upper panel shows the case of an 8 layer calorimeter , the lower panel the case of a 20 layer calorimeter .also shown is the correlation between the and the parameter.,title="fig:",width=240 ] -distribution ( see eqn .[ eqn : gamma ] ) to indvidual longitudinal shower profiles .the upper panel shows the case of an 8 layer calorimeter , the lower panel the case of a 20 layer calorimeter .also shown is the correlation between the and the parameter.,title="fig:",width=240 ]  one possible method to account for the high energy tails in the estimation , is to perform a constrained fit , where the parameters a , b and are constrained to ranges determined by monte carlo simulations or from data obtained under controlled conditions ( beam test ) .+ in table [ tab : simres ] we compare the two reconstruction methods presented above with respect to their mean estimated energy and resolution for 1 gev and 10 gev photons and electrons .the last layer correction gives a between 40 % and 50 % better energy resolution for both particle types .the improvement is larger for the lower energy ..comparsion of the two energy estimation methods [ cols= " < , < , < , < , < , < , < , < " , ]   using monte carlo simulations , in principle , it could be possible to construct the likelihood function : where denotes the reconstructed energy and a vector containing all the observations in the calorimeter .in particular , these observations could be the energy deposited in each layer .the parameterizations found in the previous sections could then be used to construct likelihood functions , since they represent probability density functions ,i.e. the probability of observing an energy given the true energy .in the simplest case , if all observations are uncorrelated , the multi - dimensional likelihood function will reduce to a multiplication of one - dimensional likelihood functions .+ in practice , this method is complicated by several factors : firstly there will be an additional parameter and observation ( the angle of the incoming photon ) and secondly we know that the assumption that the observables are uncorrelated is not correct .still , one possibility is to test the method under the assumption that correlations can be neglected and see how well it works . in case , correlations can not be neglected the only technically feasible solution ( considering limited computing resources ) is the application of learning machines , such as artificial neural networks .distributions of energy depositions in an segmented csi calorimeter have been studied using geant4 simulations .three probability distributions have been fitted to the data : negative binomial , log - normal and gaussian distributions , none of which gives a good fit over the full shower length .however , negative binomial and log normal distributions give best fits if the shower has a pronounced high energy tail . +the performed simulations have also been used to compare two different methods of energy reconstructions . the simplest method is to to add up the energies deposited in each layer of the calorimeter .this method works well for calorimeters whose size is enough such that leakage can be neglected . for finite size calorimeters , a correction for the leakage can be introduced noting that the energy lost due to leakage is correlated with the energy deposited in the last layer . +another possible method is to use a fit of the function to individual longitudinal shower profiles .this method works well , if the calorimeter is large enough so that only a non - significant fraction of the showers have their maximum outside the calorimeters sensitive volume . otherwise , the fit overestimates the energy .our simulation studies indicate that for 1 gev and 10 gev photons and electrons the method of last layer correction gives an improvement in resolution of about 50 % when compared to the fitting method .the improvement is larger for the 1 gev case .+ finally , we give a short discussion on how the parameterizations of the layer - wise energy depositions presented in this note could be used to improve maximum likelihood estimation methods .the authors would like to thank the k a wallenberg foundation and the swedish space board for financial support .computing resources made available by a grant from the gran gustafsson foundation are also acknowledged .j.c . acknowledges support from vetenskapsrdet , grant nr : 40219601 .99 bhabha h j and heitler w 1937 _ proc .london _ a * 159 * 432 furry w h 1937 _ phys . rev . _* 52 * 569 mitra a n 1957 _ nucl . phys . _ * 3 * 262 arley n 1943 _ on the theory of stochastic processes and their application to the theory of cosmic radiation _ , thesis ,copenhagen university prescott j r 1966 _ nucl .methods _ * 39 * 173 giovannini a et al . 1974 _nuovo cimento _ a * 24 * 421 alner g j et al .1985 _ phys ._ b * 160 * 193 bharucha reid a t 1960 _ elements of the theory of markov processes and their applications _, mcgraw - hill , new york messel h and potts r b 1952 _ phys .rev . _ * 87 * 759 longo e & sestili i 1975 _ nucl . instrum .methods _ * 128 * 283  gaisser t 1990 _ cosmic rays and particle physics _ , cambridge univ . press de souza v et al .2005 ( preprint astro - ph/0511677 ) amaldi u 1981 _ phys ._ * 23 * 409 agostinelli s et al . 2003 _ nucl .methods phys ._ a * 506 * 250 michelson p e 1996 _ proc ._ * 2806 * 31 johnson w n et al .1997 _ ieee nucl .symp . _ * 1 * 27 ajimura s et al .2005 _ nucl .methods phys .res . _ a * a552 * 263 amako k et al .2005 _ ieee trans ._ , * 52 * 910  wigmans r and zeyrek m t 2002 _ nucl .methods phys ._ a * 485 * 385 bergenius gavler s 2006 _ counting calories studies of energy loss in a segmented calorimeter _ , phd thesis , royal institute of technology

Abstract: in order to better understand energy estimation of electromagnetic showers in segmented calorimeters , detailed geant4 simulation studies of electromagnetic showers in the energy range 1100 gev in csi have been performed . when sampled in layers of 1.99 cm thickness , corresponding to 1.08 radiation lengths , the energy fluctuations in the samples show distributions that vary significantly with depth . the energy distributions are similar for incident electrons and photons and were found to change systematically along the shower , varying little with the initial energy . + three probability distributions have been fitted to the data : negative binomial , log - normal and gaussian distributions , none of which gives a good fit over the full shower length . the obtained parameterizations might be useful in the use of the maximum likelihood method for estimating the energy . + two methods for estimating the energy have also been studied . one method is based on fitting individual longitudinal shower profiles with a -function , the other one corrects the measured energy for leakage by utilizing the energy deposited in the last layer . our simulations indicate that the last - layer correction method applied to photons and electrons of 1 and 10 gev gives about a factor of 2 improvement in the energy resolution .   gamma - ray detectors , segmented calorimeters , energy fluctuations , energy estimation methods