the water wave equations , also known as euler s equations for inviscid , irrotational waves , describe the motion of the free surface of an ideal fluid . for periodic waves in two dimensions over a flat surface in the absence of surface tension , they are given as follows : here is the velocity potential ( related to the fluid velocity by ) , is the graph of the free surface of the water , is the acceleration of gravity , is the depth of the undisturbed fluid , and subscripts denote partial derivatives .since we are interested in periodic waves , we consider periodic boundary conditions in the horizontal direction .we note that the above equations represent a free - boundary value problem for laplace s equation , _i.e. _ both and are unknowns .these equations readily generalize to three - dimensional fluids ( with two - dimensional surfaces ) ; however , in the present work , we limit ourselves to one - dimensional surfaces .  the water - wave equations may be reformulated in terms of only the surface variables and as  where represents the dirichlet - neumann operator ( dno ) defined as where is the solution to the following boundary - value problem thus maps the given dirichlet data to the associated neumann data at the free surface .consequently , to evolve the surface variables in time using some numerical scheme , we require the solution to laplace s equation at every time step . numerically solving laplace s equation at every time step is expensive , particularly in three dimensions .  a characterization of that avoids the expensive numerical solution of laplace s equation is appealing . in the current workwe discuss four such characterizations developed in the context of water waves .these four methods are the operator expansion method of craig & sulem ( cs ) , the transformed field expansion method ( tfe ) of bruno & reitich and nicholls & reitich , the nonlocal implicit formulation of the dno given by ablowitz , fokas & musslimani ( afm ) , and a dual version to the afm method , derived by ablowitz & haut , which we denote by afm .each of these methods has had remarkable theoretical utility in deriving reduced models for water waves in various physical regimes , in deriving conserved quantities , and also in providing the theoretical framework to pose some inverse problems .additionally , each of these methods readily generalizes to the case of both varying bottom boundaries and three dimensional fluids .  of course, many traditional methods to numerically solve laplace s equation exist , including the boundary integral method , conformal mapping techniques , and the finite element method . in the present work we chose to compare the cs and tfe operator expansions and the afm / afm nonlocal formulations with the boundary integral method .we do not consider conformal mapping , as it does not extend to three dimensional problems , nor traditional finite elements , as the trade - off between high - order elements and sparsity of the stiffness matrix makes them expensive when high accuracy is desired .though it is not usually described in this way , we regard the tfe method as a spectrally accurate variant of the finite element method .the boundary integral method is particularly efficient in two dimensions for periodic problems as the lattice sums involved have an explicit analytical representation .however , in three dimensions , boundary integral methods are considerably more difficult to implement .the overall goal of the present work is to understand the relative effectiveness and accuracy of each method for two - dimensional fluids .the cs , afm and afm methods involve highly ill - conditioned intermediate calculations , so much of the paper focuses on whether accurate results can be obtained if multiple - precision arithmetic is employed in these intermediate calculations .in particular , for timestepping the water wave , it is important that the methods work if the input data is only known with limited accuracy additional precision in intermediate calculations is permissible as long as the output is roughly as accurate as the input .given the care required to obtain high accuracy in the present work , one must be cautious about using these methods in double - precision without carefully monitoring condition numbers and cancellation of digits .an outline of the paper is as follows . in  [ dno_reps ] , we briefly introduce the cs operator expansion , the afm and afm nonlocal formulations , and the boundary integral and tfe representations of the dno . in [ assessment ] we present comparisons of the first four of these methods for specific choices of the free surface .here we seek to quantify how accurate the cs , afm and afm methods are .we assume the exact dno is obtained from the boundary integral method , and , where suitable , perform computations using higher precision .section [ converge ] discusses the subtle cancellation properties associated with the series representation of craig & sulem . in [ examples ] , we present a specific example where the afm method ( in its usual interpretation as a system of equations for the coefficients of a certain series ) is guaranteed to fail , and other instances where afm and afm successfully converge to the correct neumann data .the subtle cancellation of the cs expansion is mirrored by the rapidly decaying singular values associated with the afm / afm methods .this leads us to consider regularized versions of afm / afm that involve oversampling the afm basis functions to accurately approximate a gram - schmidt orthogonalization procedure via factorization or the singular value decomposition . in the svd approach ,we also investigate the use of a pseudo - inverse cutoff threshold .we find that the afm basis functions can be more efficient at representing solutions than a fourier basis , but with the drawback of poor conditioning .finally , in [ waterwave ] , we discuss the behavior of all the methods on examples relevant to water waves .as mentioned in the introduction , the dirichlet neumann operator plays an important role in the mathematical formulation of the motion of surface gravity waves . to efficiently compute the time - dependent motion , we require a fast and efficient means to solve laplace s equation , or alternatively , a direct method to compute the dirichlet neumann operator . in this section ,we outline five commonly used approaches .the first , due to craig & sulem , involves expanding the dno in a taylor series .the second , due to ablowitz , fokas and musslimani , involves deriving a global relation between the dirichlet and neumann data that can be used as an integral equation to solve for the neumann data .the third is a dual variant of the second , formulated more directly .the fourth is a boundary integral collocation method .and the fifth is the transformed field expansion method of bruno & reitich and nicholls & reitich .consider laplace s equation posed on the domain , where is a smooth periodic function with period .further assume that is periodic in the horizontal variable with period , and thus we restrict ourselves to a flat bottom boundary at .let and be the dirichlet and neumann values of the function at . if either or is given ( in appropriate function spaces ) , the problem of determining in is well - posed in the hadamard sense . as we require the map from the dirichlet to the neumann data , assume we are given a dirichlet condition at the boundary .the associated neumann condition at the boundary is given in terms of the solution to the following boundary - value problem : in abstract terms , the dirichlet neumann operator is given by where satisfies the above boundary - value problem .note that a function of the form satisfies laplace s equation , periodicity and the boundary condition at for  .hence it is well - known ( see and references therein ) that for a lipschitz domain , the dno is an analytic function of the domain shape .thus , has a power series expansion in .writing where for , we obtain an explicit representation for from ( [ eq : g : hyp ] ) by expanding the hyperbolic terms in their respective taylor series and identifying terms of the same degree in .to lowest order we obtain by decomposing the given dirichlet condition in a fourier series , we obtain the following representation of the lowest order term of the dno = k \tanh(kh ) \mathcal{f}\left[\mathscr{d}\right].\ ] ] similarly , proceeding to higher order , we obtain further terms in the expansion of the dno . for instance , where and .terms such as are understood as pseudo differential operators , _i.e. _ they are defined through associated fourier multipliers .we remark that higher order terms of the taylor expansion of involve increasingly higher order derivatives .although the taylor series of exists for with a limited ( even finite ) degree of smoothness , the formulas for are not valid in such cases , or must be interpreted very carefully , as a whole , rather than as a sum of individual operators .even when is real analytic , there is a delicate balance existing among the terms that leads to a high degree of cancellation .this causes numerical difficulties in finite precision arithmetic .we explore the extent of these cancellations in arbitrary precision arithmetic in [ assessment ] . of course , many of these problems can be avoided by flattening the domain through a change of variables , or using boundary integral methods to compute the dno .however , the original craig sulem expansion remains of theoretical interest , and is the one most closely related to the afm approach , discussed next .following , we now derive the global relation of the ablowitz fokas musslimani reformulation of the water - wave problem .the distinguishing feature of this reformulation is the implicit nonlocal characterization of the dirichlet neumann operator . solving the resulting integral equation gives the full dno , effectively summing all the terms in the expansion of craig & sulem without having to compute them order by order .as above , the functions play a role , but now as dual functions rather than basis functions . from green s second identity, we have - \int_0^l\phi(x,\eta)\left[\psi_y(x,\eta)-\eta_x\psi_x(x,\eta)\right]dx,\\ & = \int_0^l\psi(x,\eta){\mathscr{n}}(x ) dx - \int_0^l{\mathscr{d}}(x ) \left[\psi_y(x,\eta)-\eta_x\psi_x(x,\eta)\right]dx,\label{green}\end{aligned}\ ] ] where is the normal derivative to the surface . using the definition of and noting that we obtain which is the ablowitz - fokas - musslimani ( afm ) global relation for laplace s equation .note that the global relation thus obtained is but a rephrasing of green s identity . in , the authors obtain this expression in a different , but equivalent , manner . by a spectral collocation technique , this yields an algorithm in which approximate values are obtained on a grid by solving a linear system . in that case ,( [ afm ] ) is enforced for wave numbers .a third approach , due to ablowitz & haut , can be derived from ( [ afm ] ) through a type of inverse fourier transform .the resulting dno algorithm boils down to solving for the expansion coefficients , and then computing for those functions that permit an expansion of the form ( [ dirichlet ] ) , the normal derivative is expected ( from term by term differentiation ) to be of the form ( [ normal ] ) .the operation of solving the integral equation ( [ afm ] ) is the formal adjoint of solving the system ( [ dirichlet ] ) , ( [ normal ] ) .indeed , if the ( conjugate of the ) former is written , then the latter becomes , which are consistent since is self - adjoint . here and the weights are chosen to make and bounded from to .formally , these weights cancel internally in the products and . indeed , the results are formally unchanged if both sides of ( [ afm ] ) are multiplied by , or if the weights are absorbed into in ( [ dirichlet ] ) , ( [ normal ] ) .these statements are only formal since and are not invertible .   whereas the afm method represents in the fluid as a superposition of basic solutions of the laplace equation of the form , the boundary integral method represents as a superposition of dipoles distributed along the surface , here is a parametrization of the free surface and is a field point in the fluid .we then use to reduce the integral to a period cell , and obtain using the plemelj formula , we take the limit as the field point approaches the boundary from below to obtain a second kind fredholm integral equation for : including in the formula has no effect on , but shows that is in fact a smooth function when is smooth . indeed , as , the singularities of the terms in braces cancel , yielding \big\}$ ] .once is known , the neumann data is readily shown to satisfy (\alpha )  + \frac{1}{2\pi}\int_0^{2\pi}b(\alpha,\beta)\mu'(\beta)\,d\beta , \\  \notag  & b(\alpha,\beta ) = { \operatorname{re}}\left\ {   \frac{\zeta'(\alpha)}{2 }\cot\left(\frac{\zeta(\alpha)-\zeta(\beta)}{2}\right ) -   \frac{1}{2}\cot\left(\frac{\alpha - \beta}{2}\right )  \right\},\end{aligned}\ ] ] where is the hilbert transform , with symbol . to carry this out numerically, collocation points are used to turn the integral equation ( [ bim1 ] ) into an matrix equation , where integrals are approximated by the trapezoidal rule .the derivative and hilbert transform in ( [ bim2 ] ) are easily computed using the fft .the work involved in setting up and solving these integral equations is very similar to that of the afm and afm methods .however , the condition number is much better in the bim approach since the underlying infinite dimensional system is a second - kind fredholm integral equation .this makes a big difference in practice since intermediate calculations need only be done in double - precision to achieve double - precision results , and iterative methods such as gmres can be employed to reduce the work of solving the equations from to .see for similar boundary integral methods , including formulations that incorporate a bottom boundary and allow the interface to overturn .the aim of this approach is to compute successive terms in the craig sulem expansion via formulas that do not suffer from catastrophic cancellation of digits in floating point arithmetic .the price we pay for this improvement is that the bulk fluid must be discretized . for simplicity, we consider only the finite depth case in two dimensions .the three - dimensional case is considered in , while infinite depth is treated in by introducing a fictitious interface coupling the unbounded problem on a half - space to the finite - depth problem with a curved upper boundary and a flat lower boundary .  instead of deriving the perturbation expansion for using the ill - conditioned basis functions , as was done in ( [ eq : g : hyp ] ) above, we perform a boundary - flattening change of variables : a straightforward calculation reveals that + \partial_x [ \circledast ] , \\u_{yy } & = \phi_{yy } + \big[1 - \big(1 + h^{-1}\eta \big)^{-2}\big ] u_{yy } , \ ] ] where . since , we find that where next , we write and expand and in powers of to conclude that the terms of the series satisfy as well as and . here finally , we use {y=0 } =  \big[-\eta_xu_x + \big(1 + \eta_x^2)(1 + h^{-1}\eta)^{-1}u_y\big]_{y=0 } , \\  & \mathbf{n } = ( -\eta_x,1 ) , \qquad  \nabla\phi = \big ( u_x - ( 1+h^{-1}y)(1+h^{-1}\eta)^{-1}\eta_x u_y\,,\ ,  ( 1+h^{-1}\eta)^{-1}u_y \big)\end{aligned}\ ] ] to conclude that with where empty sums ( with upper index smaller than lower index ) are zero .  in our code , and stored on a rectilinear grid with uniformly spaced mesh points in the -direction and chebyshev - lobatto nodes in the -direction , mapped by an affine transformation to obtain and .functions on the grid are stored as matrices with entries in a column indexed by , holding fixed .the formulas for are evaluated pointwise on the grid from the known values of , , and for .the fft of the zeroth order term is computed from ( [ lap : u0 ] ) by expanding in a fourier series and evaluating at the grid points .to obtain , we multiply by and take the inverse fft .similarly , is obtained by taking the ifft of .this latter function is computed by transforming the rows of to their chebyshev coefficients ( also using an fft ) , differentiating the chebyshev polynomials , and evaluating the result on the grid using the clenshaw recurrence formula .the differentiation procedure amounts to determining the coefficients of from those of : where is the chebyshev polynomial .finally , ( [ lap : un ] ) is solved using an fft in the -direction to convert the pde into a collection of uncoupled boundary value problems in .the bvp is subject to , . instead of using the chebyshev tau method to solve this bvp, we multiply by a test function , integrate the first term by parts , and proceed as if implementing a finite element method using chebyshev polynomials as the basis functions .our implementation is similar to what was done in to study the projected dynamics of kinetic diffusion equations in spaces of orthogonal polynomials .once is known for each , we compute and as described above for and .the key observation is that the derivatives on and in ( [ lap : un ] ) , and on at the end of the procedure ( to obtain and ) , are balanced by the inverse laplacian in ( [ lap : un ] ) . in other words , at the point in the algorithm where large numbers enter the computation due to applying derivatives in fourier space , we divide by even larger numbers by applying the inverse laplacian .by contrast , in the cs expansion , is expressed as a sum of several terms , each involving derivatives of products of with powers of .each derivative amplifies roundoff error , making it difficult to extract the desired sum , which is often many orders of magnitude smaller than the individual terms , as shown below .in this section we assess the merits and shortcomings of the cs method and the afm / afm methods . in [ converge ] , we explore the convergence of the cs expansion and quantify the delicate cancellation of terms mentioned in . in [ examples ] , we present an example illustrating a circumstance in which the afm method ( in its usual interpretation ) is guaranteed to fail . by oversampling the columns of the linear operators involved in the afm / afm methods , we are once again able to compute the true normal derivative with spectral accuracy. we also report on the performance of the cs expansion method for the examples considered for the afm method . finally in [ waterwave ] we consider the performance of these three methods , as well as the tfe and bim methods , on two examples in which the dirichlet data and free surface come from solutions of the water - wave equations .the first is a standing water wave and the second involves two interacting traveling water waves .   for simplicity , until [ waterwave ] , we restrict to the case of two - dimensional fluids of infinite depth . in that case , the dno expansion takes the form  where the symbols of and are and , respectively .note that when .computationally , it is convenient to absorb the term into the first term : where for . in fourier space, is an infinite matrix with the quadrants containing the main diagonal zeroed out : this already accomplishes a fair amount of cancellation since when , so the rapid growth of is balanced by decay of when is smooth . indeed ,if is real analytic , one may show that there exist and such that , which is enough to guarantee that maps sequences to exponentially decaying sequences : here is the set of positive integers when is negative and the set of negative integers when is positive .thus , if is real analytic , ( [ recur ] ) implies that is real analytic for when is merely .also , aside from , is bounded on when is real analytic .numerical experiments reveal that significant additional cancellations occur in ( [ recur ] ) , beyond combining with . indeed , without these cancellations ,even when is real analytic , one would expect , which is one of the terms in ( [ recur ] ) when the recursion is unrolled , to grow super - exponentially with . in figure[ fig : growth ] , we plot the frobenius norm of the operators and versus for the functions which are band - limited , real - analytic , and , respectively .the frobenius norm of a matrix ( in this case doubly - infinite ) is the root sum of squares of the matrix entries .we computed it in arbitrary precision arithmetic using ( [ eq : an : formula ] ) to evaluate and ( [ recur ] ) to evaluate .the computations were done column by column , in parallel .we worked in fourier space except when computing , which was done by applying the inverse fft to a column of , then multiplying by in real space , and finally applying the fft again .the frobenius norm of was computed from the indices in the range where was chosen large enough that , for the range of s considered , the terms with are small compared to the largest of those satisfying , and may be set to zero . as explained above ,this is possible since the exponential decay of dominates the polynomial growth of in ( [ eq : an : formula ] ) .this was also used as the number of grid points in the fft .we include a factor of 2 when summing the squares of the matrix entries to account for the entries in the opposite quadrant , , .the other consideration for choosing is that errors near the boundary propagate inward when computing .thus , we choose a smaller integer and compute the frobenius norm of from the entries with indices the remaining columns of ( with ) are not computed , although the rows are computed out to .we always zero out the nyquist frequency , . as before , is accounted for in figure [ fig : growth ] with a factor of in the root sum of squares .the parameters used in these computations were here `` bits '' refers to the binary precision of the mantissa , where 53 would correspond to double - precision .we used mpfr for the floating - point arithmetic , which provides ieee - like arbitrary precision rounding behavior .we also used double - precision and quadruple - precision arithmetic ( using the qd package ) in some cases .the following table gives running times for multiplying two matrices on a 3.33 ghz intel xeon x5680 system with 12 cores : the double - precision calculation is particularly fast due to the use of intel s math kernel library .the higher - precision examples were parallelized using openmp , but do not employ block - matrix algorithms to re - use data that has been pulled from main memory to cache .our general experience ( excluding level 3 blas routines such as matrix - matrix multiplication ) is that switching from double to quadruple to arbitrary precision slows down the calculation by a factor of 10 each . in figure[ fig : cosa ] , we plot the non - zero matrix entries of and for example 1 . because , the fourier modes are zero for or odd . as a result , and have only finitely many nonzero terms in this example : in figure [ fig : cosa ] , we explicitly filtered the data to zero out matrix entries of and with indices in these ranges .if this is not done , roundoff error from the fft is rapidly amplified by the recurrence ( [ recur ] ) , and requires additional precision to maintain accuracy .this is demonstrated in figure [ fig : cosb ] , where we did not filter the data .increasing the precision from 300 bits to 600 causes the correct values of to emerge from the roundoff noise .this is less of an issue for , which involves errors from taking the fft of , amplified by , but no recurrence .  in figures [ fig : alpha ] and[ fig : two3rds ] , we plot selected columns of and for examples 2 and 3 .the main change from the band - limited case of example 1 is that the matrices are no longer of finite rank , and do not have compactly supported columns . for small ,the decay rate of each column ( with respect to row index ) is still very fast , and the cancellations in obtaining from is fairly mild .however , as increases , the cancellations become quite severe . the cancellations can be seen in the figures as the vertical difference from one curve to the other . recall that is the first term in the formula ( [ recur ] ) for .the functions ( [ ex2 ] ) and ( [ ex3 ] ) have fourier modes of the form where in example 2 and in example 3 .example 2 is intended to represent a typical real - analytic function , while example 3 was designed to check if super - exponential growth in the norms of the operators might cause to also grow super - exponentially with .to see why grows super - exponentially , note that iterated convolution of with itself yields for functions of the form ( [ eq : fab ] ) .we then consider the column of ( [ eq : an : formula ] ) and maximize the maximum on the right occurs near , so we set and obtain , via sterling s formula , we tried and .the decay in the former case was too slow for the problem to be computationally tractable beyond .so we present the results in figure [ fig : two3rds ] with , where we were able to compute terms out to .returning to figure [ fig : growth ] , the right panel shows that the frobenius norm of does indeed grow super - exponentially , but the cancellations are strong enough that remains fairly flat . as a result, decays like for large even though eventually stops decaying for any positive .in addition to visual confirmation that roundoff error has not corrupted for small , as demonstrated in figures [ fig : cosb][fig : two3rds ] , we also validate the results by checking self - adjointness of each .this is done by measuring in figure [ fig : sym ] , we plot versus for each of the three examples .note that 300 bits of precision was not sufficient in the unfiltered case of example 1 to avoid errors in the symmetry of , whereas 600 bits gives at least 90 correct digits .this is consistent with the results of figure [ fig : cosb ] , where the signal is barely distinguishable from the noise in the left panel , but is many orders of magnitude larger in the center panel . in hindsight, 1500 bits was overkill for example 3 since the symmetry errors in are still below when .further validation of the correctness of the dno expansion will be given in the following section , where it will be used to compute in a case where the solution is known .in this section we present examples which show that even for band - limited wave profiles of arbitrarily small amplitude , the system ( [ dirichlet]-[normal ] ) , with the sums interpreted as infinite series , may fail to produce the neumann data for dirichlet data of a certain class , which includes real analytic functions , regardless of how many digits of arithmetic are used in the computation .both afm and afm methods involve linear operators acting on the unknown neumann data . in this section ,we study the singular value decomposition ( svd ) of truncations of these linear operators . by over - sampling the columns, we obtain better approximations of the svd of the underlying quasi - matrices and in ( [ wts ] ) , whose columns are continuous functions .this leads to regularized versions of the afm / afm methods , the simplest version ( with no pseudo - inverse cutoff ) being equivalent to performing a gram - schmidt orthogonalization of the basis functions before attempting to represent or using these functions .the regularized approach enables these methods to be used even if no series of the form ( [ dirichlet ] ) exists for .we now construct a function for which no such series exists .consider the function which is -periodic in and harmonic for all outside of the set .further , and hence for all .evaluating at , we obtain for values of in the range , we may also write in the form ( [ wilken_ex ] ) , is well - behaved except at the poles . however , the afm formulation is based on representing via the series ( [ phi : ex1 ] ) , which is divergent for .thus , we expect trouble for wave profiles that extend above .the simplest example illustrating these difficulties is the infinite depth case with introducing the weights in ( [ wts ] ) and taking the limit as , the system ( [ afm ] ) becomes where .similarly , ( [ dirichlet ] ) , ( [ normal ] ) become in our case , using in ( [ wilken_ex ] ) , we have since dips below the poles at , is harmonic on .moreover , is real analytic and -periodic .nevertheless , there is no solution of ( [ dirichlet : h : inf ] ) valid over the whole interval . from ( [ phi : ex1 ] ) , we see that the coefficients will work over , but not elsewhere ; see figure [ fig : wilken ] .comparison of the series representation ( [ dirichlet : h : inf ] ) , ( [ normal : h : inf ] ) , ( [ eq : ck : exact ] ) of and , ( blue curves ) , to the exact formulas ( [ dn : exact ] ) , ( red markers ) , for , .the series was truncated at , and the results were plotted at 128 grid points . a centered finite difference was used to compute the derivative in ( [ normal : h : inf ] ) . as expected, the series diverges for , where . ]there remains the possibility that for any prescribed tolerance , a choice of the can be made such that ( [ dirichlet : h : inf ] ) is satisfied approximately , to the specified tolerance . in other words ,the left - hand side is not treated as a series , but instead as a finite linear combination of basis functions that can approximate to arbitrary accuracy .rather than just add more terms to improve accuracy , it may be necessary to change all the coefficients . to explore this possibility, we construct the by matrices and with entries \big )  e^{ikx_j } , \qquad  b_{jk } = i{\operatorname{sgn}}(k)a_{jk},\ ] ] where , , and .we then compute the singular value decomposition , with and square and of dimension , and evaluate here is computed spectrally via the fft ( with no filter ) .the idea here is to sample each column of and in ( [ wts ] ) with enough points that computing the svd of the resulting matrix in ( [ eq : ab ] ) is equivalent ( up to scaling by in various places ) to computing the svd of the quasi - matrix in ( [ wts ] ) , whose columns are continuous , functions : here has been truncated to have columns , and are unitary , is diagonal with positive decreasing entries , and is equipped with the inner product to avoid factors of elsewhere .the columns of form an orthonormal basis for the column - span of .once sufficient grid resolution is reached , approximating and at the collocation points leads to accurate approximation throughout , using trigonometric polynomials to interpolate between grid points .the columns of the matrix version of can be thought of as sampled versions of the columns of the quasi - matrix from ( [ eq : quasi ] ) , up to a factor of .for smaller values of , trigonometric interpolation becomes less accurate , and the errors can be amplified significantly on division by small singular values .we remark that if the pseudo - inverse in ( [ n : afm ] ) and ( [ n : afm1 ] ) is replaced by an inverse , then one can use a factorization instead of the svd to obtain an orthonormal basis for the column span of .this gives up some flexibility in regularizing the afm / afm methods , but is cheaper and has the advantage that the leading basis functions do not change if is increased .this approach is equivalent to gram - schmidt orthogonalization when the columns are sampled sufficiently .the results of computing from in ( [ dn : exact ] ) are shown in figure [ fig : off0 ] .the first panel shows the singular values of with and . in all four cases ,the singular values decay very rapidly , with slightly slower decay when is larger .by contrast , the singular values of in the boundary integral approach ( bim ) remain nearly constant .panel b shows the error in from ( [ n : afm ] ) , ,\ ] ] where refers to the exact solution ( [ dn : exact ] ) , plotted in panel d. the error depends on the pseudo - inverse cutoff index , , defined by it consists of two parts , one due to how well is approximated by the leading columns of , and one by how well the coefficients are approximated by for all values of , the error in panel b decreases initially as increases .this suggests that the leading entries of are a good approximation of those of .when is close to ( or in the figure ) , the error reaches a minimum at an optimal , and then increases rapidly .this occurs because the singular values become so small that the high - index entries of become large and no longer approximate the corresponding entries of . for larger ,the singular values decay more slowly and the error curve decreases monotonically all the way to .panel e shows the vector version of the error , namely in ( [ eq : error ] ) , corresponding to the minima of the error curves in panel b. for smaller values of , the error is largest where is smallest .this is not surprising since the corresponding rows of are smaller in this region due to the exponential growth of the basis functions in the -direction .what is surprising is the extreme accuracy that is achieved by the afm method in the region where for smaller values of .we do not know why this occurs .this additional accuracy disappears as the errors are reduced in the region where by increasing .once reaches 384 , the columns of are well - resolved in as discussed above , and the error is roughly uniform throughout the domain .panels c and f show the same results for the afm method .the results are very poor when , presumably a consequence of aliasing errors in sampling the columns of being amplified on division by small singular values .as increases , the aliasing errors become smaller and the singular values become larger . by the time reaches 384 , the error of the afm method is similar to that of the afm method , of order . by contrast , the boundary integral method has errors of order with .there is little amplification of roundoff error in the bim approach since is so well - conditioned .figure [ fig : off1 ] shows a similar computation to the above , but with offset vertically so that the series ( [ phi : ex1 ] ) converges at all points on the curve .specifically , we set the formulas ( [ dn : exact ] ) remain nearly the same , with replaced by .the singular values in panel a are the same as those in figure [ fig : off0 ] since is also shifted downward by 1 in ( [ eq : ab ] ) .the afm method behaves similarly to before , achieving exceptional accuracy in the region where is largest when is close to , and achieving nearly uniform accuracy once is large enough to fully resolve the columns of the continuous version of . because and are smoother ( with faster decay of fourier modes ) ,all three methods ( afm , afm and bim ) yield smaller errors in figure [ fig : off1 ] than in figure [ fig : off0 ] .the afm method turns out to be superior to the afm method on this example for all four choices of .by contrast , in figure [ fig : off0 ] , the afm method was better for and , and the two methods were equal when .a partial explanation is that the left - hand side of ( [ dirichlet : h : inf ] ) is a genuinely convergent series in this second example , as opposed to a finite linear combination of vectors from a dense set .thus , the afm method has an easier time selecting the coefficients that best represent .by contrast , for the afm method , there is little difference between the two examples . in either case, the exact solutions and satisfy the afm global relation , and the question of convergence comes down to how well the trapezoidal rule approximates the integrals , and how much the errors are amplified by the poorly - conditioned matrix .we do not know why the afm method turns out to be 25 orders of magnitude more accurate once reaches 384 .as before , the bim method is superior to both afm methods .  in figure[ fig : craigerr ] , we check convergence of the craig - sulem expansion for this example . as in figure[ fig : off0 ] , we consider , , with dirichlet data as in ( [ dn : exact ] ) .the computations were done on a 256-point grid with 360 bits of precision .panel a shows the errors in the partial sums , defined as (x).\ ] ] the errors decrease steadily until , where reaches , the level of aliasing errors in the fourier representation of and on a 256-point grid .recall that the error in the boundary integral method with 256 points was also around .panel b shows the fourier spectrum of ( from the exact solution ) and .the rapid decay of the fourier modes of show that the zeroth order approximation does an excellent job of predicting the high - frequency components of , but not the low - frequency ones .all further corrections will only be made to the first 50 fourier modes .in other words , we actually use ^\wedge_k , & |k|<50 , \\\hat{\mathscr{n}}_k - [ g_0(\eta){\mathscr{d}}]^\wedge_k , & |k|\ge50   \end{cases}\ ] ] when reconstructing in panels e h . as a result ,the modes marked `` unrecoverable error '' in panel b are frozen , and will not be altered by successive corrections .this is reasonable since these modes are dominated by aliasing errors in sampling on the 256-point grid . on a larger grid , if more accuracy were desired , the cutoff would need to be increased . the error plateau in panel a is due to these aliasing errors .panel c shows the magnitude of versus for and .each successive correction reduces the error in the leading fourier modes of the neumann data .note that the bulge near in is below , and therefore additional corrections do not improve the global error , which is dominated by high - frequency modes at that point .the rapid growth of above is due to truncating the fourier series to when computing the craig - sulem expansion .analogous behavior was seen in figures [ fig : cosb][fig : sym ] , where we observed that errors in the columns of propagate inward from high to low wave numbers . due to the cutoff, these large errors do not affect the reconstruction of .panel d shows the exact solution while panels e h show the errors in various partial sum reconstructions . because the low - order modes are the least accurate ( as seen in panel c ) , the corrections are remarkably smooth , non - oscillatory curves .the highly oscillatory error in is due to the `` unrecoverable error '' in panel b ; if were increased , would also be very smooth . in summary ,the craig - sulem expansion performs well on this example , converging to the exact solution , up to aliasing errors of order , in 95 iterations .we conclude with two examples in which the wave profile and velocity potential come from solutions of the water wave problem .the first is a large - amplitude standing water wave shortly before reaching maximum height .the second consists of two stokes waves of different amplitudes traveling to the right .both examples can be evolved efficiently using the boundary integral method .our interest here is whether the dno expansion method and the two afm methods can take data ( and ) that are only known to double- or quadruple - precision accuracy and return neumann data ( ) with the same accuracy . we allow ourselves to use additional precision for intermediate calculations .figures [ fig : stand2][fig : afm : stand ] show the calculation of the dno operator for a large - amplitude standing water wave .we selected the infinite depth wave corresponding to the first local maximum of wave height ( half the vertical crest - to - trough distance ) for this example . when wavelength is set to and the acceleration of gravityis , this wave has period , wave height , crest acceleration , and fifth fourier mode of at ( a good bifurcation parameter in this regime ) of .see for details on how the wave was computed . since the wave comes to rest at , at that time .thus , to avoid a trivial dno calculation while keeping close to its maximum - amplitude state , we selected the wave profile at for this example .figure [ fig : stand2 ] shows snapshots of at equal intervals of size over a quarter - period , along with the dirichlet and neumann data corresponding to .figure [ fig : craig : stand ] shows the results of the dno calculation using the craig - sulem method .the left panel shows the error as a function of order for .the middle panel shows the fourier decomposition of the error at orders , and , together with the fourier modes of the `` exact '' solution . unlike the results of figure [ fig : craigerr ], the zeroth order approximation does not lead to a large improvement in the high - frequency modes ( seen as a faster decay rate in than in figure [ fig : craigerr ] ) .in fact , is difficult to distinguish from , so we plotted instead , which has a similar decay rate but is shifted down slightly .the data for this problem , and , are specified via their leading 750 fourier modes ( recorded in quadruple - precision , i.e. 32 digits ) .this gives an approximation of the standing wave to 25 digits of accuracy , but is regarded here as specifying the dno problem with infinite precision .the `` exact '' solution was computed using the boundary integral method with 2304 collocation points and 212 bits of precision , leading to approximately 40 digits of accuracy .the resulting 1152 fourier modes of this `` exact '' solution are labeled in the middle panel .the craig - sulem expansion was performed using grid points for the fft and 900 bits of precision in intermediate calculations in order to achieve accurate results up to order for modes , .this cutoff was chosen to agree with the last computed mode of the `` exact '' solution .the right panel shows the error in the 120th order approximation as a function of .we see that the error is a smooth function of order that is largest near the crest tip at . in summary , using 900 bits ( 271 digits ) and grid points , the 150th order craig - sulem method is able to achieve errors around , which is comparable to the original standing wave calculation , which was done in quadruple - precision ( 32 digits ) with 2048 grid points using the boundary integral method .though the craig - sulem method is not competitive , it is still remarkable that such a large - amplitude wave would be inside the radius of convergence of the dno expansion .figure [ fig : afm : stand ] shows the same calculation using the afm and afm methods .the same `` exact '' solution as in figure [ fig : craig : stand ] was used to measure errors . the large ( crest - to - trough ) wave height leads to very rapidly - decaying singular values . nevertheless , using modes and collocation points with 360 bits ( 108 digits ) of precision , we are able to achieve 25 digits of accuracy in the solution .the afm and afm methods are comparable for cutoffs up to about .after that , the error in the afm method grows rapidly while the afm method flattens out .thus , the afm method is somewhat more robust in this example .the final example of this section consists of two stokes waves traveling right on a -periodic domain .initially , one wave is centered at and the other is centered at .the fluid depth is and the parameters of the waves are where is the wave speed .plots of , and are given in figure [ fig : stokes2 ] .figure [ fig : craigstokes ] shows the results of the dno calculation using the craig - sulem method .the recursion ( [ recur ] ) is modified as follows to account for finite depth :     ( n!)^{-1}|d|^{n-1}\big ( d f^n d - g_0 f^n g_0 \big ) , & n\text { odd }   \end{array}\right\ } , \\a_n(f)^\wedge_{kj } & = \frac{jk^n}{n ! } a_{nkj } ( f^n)^\wedge_{k - j } , \qquad  a_{nkj } = \left\{\begin{array}{ll }    \tanh kh - \tanh jh , & n\text { even } \\    1 - ( \tanh kh)(\tanh jh ) , & n\text { odd }    \end{array}\right\}.\end{aligned}\ ] ] we used grid points for the fft and computed the expansion through order .the `` exact '' solution was computed via the boundary - integral method in quadruple - precision , which is correct to about 28 digits of accuracy .the craig - sulem expansion reaches this level of accuracy at order .the errors then increase ( left panel ) due to the error growth region crossing the cutoff mode at 92nd order ( center panel ) .increasing would delay this crossing , but is unnecessary since order is sufficient to reach the accuracy of the underlying `` exact '' solution .the right panel shows the error as a function of at order .the error is concentrated near , where is largest . in the regionnot shown , the error is uniformly less than , even near , the location of the second stokes wave. thus , most of the work goes into resolving the solution near the larger peak .since this last example has finite - depth , we can also use the variant of the transformed field expansion method described in [ sec : tfe ] to compute the neumann data from the dirichlet data .the left panel of figure [ fig : tfestokes ] shows the error from ( [ eq : en : def ] ) in the neumann data as a function of the order , as well as the difference between the order terms from the tfe and cs expansions , the error reaches a plateau of for in double - precision , and for in quadruple - precision . as before ,the `` exact '' solution was computed in quadruple - precision using the boundary integral method .the orange markers show that the terms in the tfe expansion agree with the corresponding terms in the cs expansion to roundoff error accuracy .in particular , a plot of for the tfe method ( not shown ) looks identical to that of the cs expansion in the right panel of figure [ fig : craigstokes ] .we also note in figure [ fig : tfestokes ] that exhibits a downward trend as increases , indicating that the terms in the tfe expansion maintain several correct digits of relative accuracy beyond the point that reaches the plateau region , i.e. the point where successive terms are smaller in magnitude than the absolute errors of the leading terms . in the double - precision case ,most of the error in the plateau region is due to the error in the zeroth order term  is larger than the sum of the other . in quadruple - precision , is still largest , although the with are of comparable size. this may be partly due to the cutoff at used in the cs expansion in figure [ fig : craigstokes ] to eliminate high - frequency noise .certainly , the rapid growth in for in figure [ fig : tfestokes ] is due to errors in the cs expansion rather than the tfe expansion .indeed , in figure [ fig : craigstokes ] grows rapidly for while it remains flat for in figure [ fig : tfestokes ] .recall that a third method ( the bim method ) was used for the exact solution in both plots .the middle and right panels of figure [ fig : tfestokes ] were used to decide how many grid points to use in the tfe method .we used , in double - precision and , in quadruple - precision .these plots show the partial norms where are the chebyshev coefficients of the fourier mode of : recall that is defined on a rectangle and expanded in powers of , , where .the partial norms and are the norms of the rows and columns of the matrix of fourier - chebyshev coefficients of the function .to resolve the solution spectrally , the mesh needs to be large enough that the partial norms decay to the desired tolerance as or .as shown in figure [ fig : tfestokes ] , the meshes we selected are sufficient to reach roundoff level tolerances in these limits .it is worth noting that many fewer grid points are needed in the -direction than in the -direction , so the price of discretizing the bulk fluid is not as severe as one might imagine .moreover , as with the bim method , intermediate calculations can be done in double or quadruple - precision arithmetic to achieve similar levels of accuracy in the solution .the overall running times ( in seconds ) of the various methods on this example are given in the following table : the code was run on a 3.33 ghz intel xeon x5680 system with 12 cores . here( d ) and ( q ) stand for double and quadruple - precision , and the other methods were run with 360 bits ( 108 digits ) of precision .the running time of afm is nearly identical to that of afm , and the afm - qr variant will be described below .clearly , the bim and tfe approaches are superior to the cs and afm - based methods since arbitrary - precision arithmetic is not required .we conclude with the results of the afm and afm methods on this example , which we selected initially as being relevant to water waves and likely to cause difficulties for the afm and afm methods .our reasoning was that the second wave is large enough to require many fourier modes to resolve its shape , but small enough that is many orders of magnitude larger at the crest of the first wave than at the crest of the second ( for large ) .thus , substantial cancellation must occur near in order to resolve the behavior of near .however , the singular values in the left panel of figure [ fig : afmstokes ] decay slowly in comparison to the infinite - depth standing wave case of figure [ fig : afm : stand ] .this is because the waves in these two examples have similar fourier decay rates for and , but the deep - water standing wave has a much larger vertical crest - to - trough height .to the extent that this is generally the case for waves in deep versus shallow water , the afm and afm methods appear to be better suited for finite depth problems .the errors in the afm and afm methods are nearly identical for every choice of pseudo - inverse cutoff in the center panel of figure [ fig : afmstokes ] .thus , the methods are equally effective at computing the dno for this example . beyond ,the error in the center panel reaches a plateau of .this is due to errors in the `` exact '' solution .since this plateau is reached already at , it would appear that and possibly can be reduced while still achieving the target accuracy of .however , reducing leads to worse results ( not shown ) .the reason is that the high - index columns of in }{\cosh[k\eta_\text{max}+h ] }  \begin{pmatrix } \cos kx \\[2pt ] \sin kx \end{pmatrix } , \qquad  ( 1\le k < k/2)\ ] ] comprise a small but important part of the low - index singular vectors of the singular value decomposition , .a better way to understand how large needs to be in order to achieve a prescribed accuracy is to perform a qr factorization , , which is a numerically robust way to perform a gram - schmidt orthogonalization of the columns of .we then define the `` discrete afm transform '' and , in figure [ fig : afmdecay ] , compare it to the discrete fourier transform the red curves give the fourier modes of the input dirichlet data , which are taken to specify the problem with infinite precision even though they only describe the standing wave and traveling waves to around 25 and 30 digits , respectively .the black curves give the fourier modes of the boundary integral solutions , which were computed with 212 and 106 bits of precision , respectively .the orange and blue curves give the `` afm transform '' of the dirichlet and neumann data .the observation that can not be reduced significantly below 2048 for the traveling wave problem is seen clearly in the right panel , which shows that does not reach roundoff error until just below .  the fact that the afm coefficients and decay faster than the fourier coefficients and suggests that the gram - schmidt version of the afm basis is more efficient at representing the dirichlet and neumann data of many problems of physical interest ( such as standing and traveling waves ) than the fourier basis .the abrupt change in slope of the orange and blue curves in the left panel occurs when the afm method begins to resolve errors in the given dirichlet data , which is taken to be exact even though it only agrees with physical standing waves to 25 digits .while it is interesting that the gram - schmidt version of the afm basis is more efficient than a fourier basis for these problems , computing the orthogonal afm basis is expensive since it requires a great deal of additional precision to avoid losing all significant digits during the qr factorization process .comparison of the coefficients in an expansion of the dirichlet and neumann data in an orthogonalized afm basis versus a fourier basis . ]we have shown that the craig - sulem expansion of the dno operator , the implicit formulation due to ablowitz , fokas and musslimani , and its dual , due to ablowitz and haut , can all be used to compute spectrally accurate solutions of the dirichlet - neumann problem .all three methods involve ill - conditioned intermediate calculations when the vertical crest - to - trough distance becomes large relative to the inverse of the highest - frequency wave numbers involved in a fourier description of the wave profile and the dirichlet data . however , this ill - conditioning can be tamed using extended precision arithmetic .most importantly , and need only be specified in double - precision to obtain double - precision results for ; extended precision is only required in intermediate calculations . while it is undesirable to work in extended precision arithmetic , a spectrally accurate method requiring 64 digits of accuracy in intermediate calculations may require less work than a more traditional 16-digit calculation using a 2nd or 4th order method if high accuracy is desired .  nevertheless , our goal in writing the paper was not to advocate the use of these methods , but to explore their limits of applicability .we originally thought that the afm and afm would break down if the potential can not be extended analytically to a strip containing the peak of the wave profile .this was the motivation for studying a potential with poles on the real axis and a wave profile that dips below the poles but also extends above them .it is surprising that the afm basis can still be used to approximate in this case to arbitrary accuracy , and that the term - by - term neumann data turns out to be a good approximation of the correct solution .we also did not expect the cs expansion to converge for realistic large - amplitude standing waves , but it was able to achieve 25 digits of accuracy at 150th order .for the same problem , the afm and afm methods achieved 25 digits of accuracy in spite of singular values dropping below . for waves in shallow water ,the condition numbers are much more reasonable , making the afm methods more appealing .as noted in , the cs expansion suffers from significant cancellations requirements .our numerical investigations confirm this behavior .we note that a suitable rearrangement of the series permits one to account for some of these cancellations , though not all .one of the goals of the present work was to understand whether this behavior is present in the afm formulation and to what extent as the afm method may be interpreted as a certain summation of the cs expansion .our investigations reveal the afm method is similarly ill - conditioned .indeed the rapid decay of singular values in the afm ( and afm * ) method mirrors the cancellation properties observed in the cs expansion .given the delicate nature of cs , afm and afm * we recommend the bim , especially for two - dimensional flows .the tfe version of the cs expansion also performs remarkably well without the need for extended precision arithmetic in intermediate calculations .  regarding extension to three dimensions ,the cs and tfe expansions work with little change .since the latter can be run effectively in double - precision , it is a viable method .although the bulk fluid must be discretized , the system of equations that must be solved decouples into many one - dimensional boundary value problems rather than a large system of equations such as arise in finite element methods .the afm and afm methods will lead to large , highly ill - conditioned linear systems that are easy to set up but nearly impossible to solve . performing the svd or qr factorization of such a large matrix in extended precision arithmeticwould be extremely costly , and iterative methods such as gmres will not converge when the condition number grows to or higher .by contrast , the boundary integral method yields condition numbers close to 1 ; hence , gmres converges in just a few iterations .although it is difficult to implement boundary integral methods in three dimensions due to the slowly decaying lattice sums involved , techniques such as ewald summation are available to accelerate convergence .jw was supported in part by the director , office of science , computational and technology research , u.s .department of energy under contract no .de - ac02 - 05ch11231 , and by the national science foundation through grant dms-0955078 .any opinions , findings , and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding sources .dyachenko , e.a .kuznetsov , m.d .spector , and v.e .analytical description of the free surface dynamics of an ideal fluid ( canonical formalism and conformal mapping ) .letters a _ , 221:0 73 79 , 1996 .laurent fousse , guillaume hanrot , vincent lefvre , patrick plissier , and paul zimmermann . : a multiple - precision binary floating - point library with correct rounding ._ acm transactions on mathematical software _ , 330 ( 2):0 13:113:15 , june 2007url http://doi.acm.org/10.1145/1236463.1236468 .

Abstract: we compare the effectiveness of solving dirichlet neumann problems via the craig  sulem ( cs ) expansion , the ablowitz - fokas - musslimani ( afm ) implicit formulation , the dual afm formulation ( afm ) , a boundary integral collocation method ( bim ) , and the transformed field expansion ( tfe ) method . the first three methods involve highly ill - conditioned intermediate calculations that we show can be overcome using multiple - precision arithmetic . the latter two methods avoid catastrophic cancellation of digits in intermediate results , and are much better suited to numerical computation .  for the craig  sulem expansion , we explore the cancellation of terms at each order ( up to 150th ) for three types of wave profiles , namely band - limited , real - analytic , or smooth . for the afm and afm methods , we present an example in which representing the dirichlet or neumann data as a series using the afm basis functions is impossible , causing the methods to fail . the example involves band - limited wave profiles of arbitrarily small amplitude , with analytic dirichlet data . we then show how to regularize the afm and afm methods by over - sampling the basis functions and using the singular value decomposition or qr - factorization to orthogonalize them . two additional examples are used to compare all five methods in the context of water waves , namely a large - amplitude standing wave in deep water , and a pair of interacting traveling waves in finite depth .   = 1


space exploration is a difficult task and the search for life is no different .the equipement size a probe can bring severely limit the scope of the search .every sample can not be analyzed and selecting those that can be is not trivial .this project investigates the possibilities of remote detection using infrared spectroscopy in order to select those few samples to pick and analyse further .the reason for using ir spectroscopy is mainly that almost all biological marker molecules ( i.e. biomarkers ) will show some spectral features in the near to far ir ( ) .furthermore , the spectroscopy will not destroy the sample ( e.g. there is no contact with the sample , no contamination ) .previous techniques of detection tended to destroy , or damage , the endolithes or the environment in which they lived ( e.g. by using electron - microscope , chemical and biological or analysis ) .biomarkers are an important source of information in the search for evidence of life in geological samples .even if the organisms are dead or dormant , it is still possible to detect their presence , byproducts or even their chemical alterations of the environment .  in searching for extraterrestrial life ,it is important to have a minimum of preconception about them in order to find it .endolithes are organisms that live inside rocks or in the pores between mineral grains. there are thousands of known species of endolithes , including members from bacteria , archaea , and fungi .they represent near half the earth s biomass and also present an ideal model of life for mars ( , ) .this study used two groups of samples .one from the guelph region ( west of toronto , ontario ) and the other near eureka on the ellesmere island in the nunavut ( ) .both regions have different geology and climate .samples were scanned using a technique called diffuse scattering .the ir beam was directed toward the sample and a series of mirrors redirected the scattered beam to the ir sensor .the spectra were obtained using an ir nicolet spectrometer .the apparatus used to collect the scattered ir beam could not receive large rock samples .it was necessary to break them into smaller pieces .the same device was very sensible to the angle of incidence and reflectance .it was important to position both mirrors near the vertical above the sample .the adjustment of each mirrors , in order to optimize the reception of light , was very time consuming .those adjustments were performed using an aluminum plate as the sample to maximize the received flux to the sensor . when the rock sample was placed in the light path , the total flux drops a lot but the signal - to - noise ratio was high enough .spectra in middle ir ( 132 ) and near ir ( 49 ) on 15 samples from the two groups were taken .the best results were from the middle ir .most of the spectra were taken at a resolution of 4 or 8 .after analysis , it appears that a spectral resolution of 8 is enough for our purpose .the middle ir spectra ( from 4000 to 650 ) were then processed using principal component analysis in order to classify them .principal component analysis ( ) is a technique of factorial analysis ( e.g. multivariate statistics ) .it is often used to find a new coordinate system in which the original data will be better aligned on some axes ( e.g. principal components ) .the technique can be summarized by the equation where a , u , w and v are matrices .  _a _ is a matrix containing the spectra on each row .the value of each absorption band is then regrouped in the columns which are the variables on which the pca works ._ u _ , also a matrix , contains the spectra in the new coordinate system .the matrix _ v _ , where is an identity matrix , contains the eigenvectors ._ w _ is a diagonal matrix containing the squared root of the eigenvalues and provides a clue about how many principal components ( pc ) can be used to describe the data .it is important to have a matrix _ a _ with more columns than rows ( ie . ) else _ w _ becomes a singular matrix and the whole pca fails .  to facilitate data manipulation and visualization for this paper , we have taken only the first three principal components .the result of the pca process is illustrated in figure 1 .the process can be extended to as many components as needed .based on the eigenvalues from our data , the first five components are relevant , the others being buried in the noise .we have identified several clusters . cluster _e _ groups the spectra containing features showing the presence of organic compounds . cluster _is very close to another cluster of points calculated from organic spectra used as a reference ( i.e. cluster o ) .further , most of the spectra of cluster _ e _ have been taken directly from green regions visible on the samples .our results show that it is possible to detect biological signatures using a spectrometer operating in the middle infrared range . however , it is not possible to highlight a particular region , or regions , in a spectrum to be used to identify biomarkers .the interdependance of the absorption bands related to the living are too complex to simply isolate a few .the proposed technique calls for a more subtle approach by comparing witness spectra and an unknown spectrum by plotting them in the pca s space .if the test spectrum , once projected into the pca s space , is close to the reference group then the probability of it is containing biomarkers is high . adding a non - organic spectra to the pca space as references may improve the idenfitication schemethis technique could be used to pinpoint potential life harboring rock for more detailed analysis .it could be possible to extend the method to better idenfity the unknown spectrum using a more precise reference database .hand , k.p . ,carlson , r.w . ,sun , h . , anderson , m , wadsworth , w. , levy , r. , 2005 , utilizing active mid - infrared microspectrometry for in - situ analysis of cryptoendolithic microbial communities of battleship promontory , dry valleys , antartica , astrobiology and planetary missions , vol 5906 , p.302

Abstract: on earth , the dry valleys of antarctica provide the closest martian - like environment for the study of extremophiles . colonies of bacteries are protected from the freezing temperatures , the drought and uv light . they represent almost half of the biomass of those regions . due to there resilience , endolithes are one possible model of martian biota .   we propose to use infrared spectroscopy to remotely detect those colonies even if there is no obvious sign of their presence . this remote sensing approach reduces the risk of contamination or damage to the samples .


cellular automata ( cas ) are discrete dynamical systems that have been studied for years due to their architectural simplicity and the wide spectrum of behaviors they are capable of . herewe study cas that can be said to perform a simple `` computational '' task .one such task is the so - called _ majority _ or _ density _task in which a two - state ca is to decide whether the initial state contains more zeros than ones or vice versa . in spite of its apparent simplicity ,it is a difficult problem for a ca as it requires a coordination among the automata . as such, it is a perfect paradigm of the phenomenon of _ emergence _ in complex systems .that is , the task solution is an emergent global property of a system of locally interacting agents .indeed , it has been proved that no ca can perform the task perfectly i.e. , for any possible initial binary configuration of states .however , several efficient cas for the density task have been found either by hand or by using heuristic methods , especially evolutionary computation .for a recent review see .all previous investigations have empirically shown that finding good cas for the majority task is very hard .however , there have been no investigations , to our knowledge , of the reasons that make this particular fitness landscape a difficult one . in this paperwe statistically quantify in various ways the degree of difficulty of searching the majority ca landscape .the paper proceeds as follows .the next section summarizes some known facts about cas for the density task .a description of its fitness landscape follows , focusing on the hardness and neutrality aspects .next we identify and analyze a particular subspace of the problem search space called the olympus .finally , we present our conclusions and hints to further works and open questions .the density task is a prototypical distributed computational problem for cas . for a finite ca of size it is defined as follows .let be the fraction of 1s in the _ initial configuration _( ic ) .the task is to determine whether is greater than or less than . in this version ,the problem is also known as the _ majority _ problem .if then the ca must relax to a fixed - point configuration of all 1 s that we indicate as ; otherwise it must relax to a fixed - point configuration of all 0 s , noted , after a number of time steps of the order of the grid size . here is set to 149 , the value that has been customarily used in research on the density task ( if is odd one avoids the case for which the problem is undefined ) .this computation is trivial for a computer having a central control .indeed , just scanning the array and adding up the number of , say , 1 bits will provide the answer in time .however , it is nontrivial for a small radius one - dimensional ca since such a ca can only transfer information at finite speed relying on local information exclusively , while density is a global property of the configuration of states .it has been shown that the density task can not be solved perfectly by a uniform , two - state ca with finite radius .the lack of a perfect solution does not prevent one from searching for imperfect solutions of as good a quality as possible . in general , given a desired global behavior for a ca ( e.g. , the density task ) , it is extremely difficult to infer the local ca rule that will give rise to the emergence of the computation sought .this is because of the possible nonlinearities and large - scale collective effects that can not in general be predicted from the sole local ca updating rule , even if it is deterministic . since exhaustive evaluation of all possible rules is out of the question except for elementary ( ) and perhaps radius - two automata , one possible solution consists in using evolutionary algorithms , as first proposed by packard in and further developed by mitchell et al .+ the _ standard performance _ of the best rules ( with ) found at the end of the evolution is defined as the fraction of correct classifications over randomly chosen ics .the ics are sampled according to a binomial distribution ( i.e. , each bit is independently drawn with probability of being 0 ) .+ mitchell and coworkers performed a number of studies on the emergence of synchronous ca strategies for the density task ( with ) during evolution .their results are significant since they represent one of the few instances where the dynamics of emergent computation in complex , spatially extended systems can be understood . as for the evolved cas , it was noted that , in most runs , the ga found unsophisticated strategies that consisted in expanding sufficiently large blocks of adjacent 1s or 0s .this `` block - expanding '' strategy is unsophisticated in that it mainly uses local information to reach a conclusion . as a consequence ,only those ic that have low or high density are classified correctly since they are more likely to have extended blocks of 1s or 0s .these cas have a performance around .a few runs yielded more sophisticated cas with performance ( around ) on a wide distribution of ics .however , high - performance automata have evolved only nine times out of runs of the genetic algorithm .this clearly shows that the search space is a very difficult one , even there exists some recent works on coevolutionary algorithm which able to find a number of `` block expanding '' strategies .these sophisticated strategies rely on traveling signals ( `` particles '' ) that transfer spatial and temporal information about the density in local regions through the lattice , and have been quantitatively described with a framework known as `` computational mechanics '' .the gkl rule is hand - coded but its behavior is similar to that of the best solutions found by evolution .das and davis solutions are two other good solutions that have been found by hand .other researchers have been able to artificially evolve a better ca ( ) by using genetic programming .finally , juill _ et al _ obtained still better cas ( and ) by using a coevolutionary algorithm .their coevolved ca has performance about 0.86 , which is the best result known to date .we call the six best local optima known , with a standard performance over , the _ blok _ ( tab .[ tab - opt6 ] ) .  in the next section we present a study of the overall fitness landscape ,while section [ olympus ] concentrates on the structure of the landscape around the _blok_.  .description in hexadecimal and standard performance of the 6 previously known best rules ( _ blok _ ) computed on sample size of .[tab - opt6 ] [ cols="^,^ " , ]   [ [ neutral - degree . ] ] neutral degree .+ + + + + + + + + + + + + + +  the figure [ fig - neutr]-b gives the neutral degree of solutions from olympus as a function of their performance .the solutions below performance are randomly chosen in olympus .the solutions over performance are sampled with 2 runs of a ga during generations .this ga is based on ga defined by mitchell where the operators are restricted to olympus subspace and the selection is a tournament selection of size 2 taking into account the neutrality .this ga allows to discover a lot of solutions between and and justified the useful of olympus runs , average performances are with standard deviation which is higher than of coevolutionary algorithm of pagie ] .two important are located around fitnesses and where the neutral degree is over . for solutions over ,the average of neutral degree is 37.6 which is a high neutral degree .the landscape has a considerable number of points with performance or which means that investigations based on sampling techniques on the whole landscape are unlikely to give good results .the neutrality of the landscape is high , and the neutral network topology is not completely random . exploiting similarities between the six best rules and symmetries in the landscape , we have defined the _ olympus _ landscape as a subspace of the majority problem landscape .this subspace have less solutions with performance and it is easy to find solutions over with a simple ga . we have shown that the neutrality of landscape is high even for solution over .andre , d. , bennett iii , f.h . ,koza , j.r . :discovery by genetic programming of a cellular automata rule that is better than any known rule for the majority classification problem . in koza , j.r . ,goldberg , d.e . ,fogel , d.b ., riolo , r.l ., eds . : genetic programming 1996 : proceedings of the first annual conference , cambridge , ma , the mit press ( 1996 ) 311  juill , h. , pollack , j.b .: coevolutionary learning : a case study . in : icml98 proceedings of the fifteenth international conference on machine learning , san francisco , ca , morgan kaufmann ( 1998 ) 251259  crutchfield , j.p ., mitchell , m. , das , r. : evolutionary design of collective computation in cellular automata . in crutchfield , j.p . , schuster , p. , eds . :evolutionary dynamics : exploring the interplay of selection , accident , neutrality , and function .oxford university press , oxford , uk ( 2003 ) 361411   pagie , l. , mitchell , m. : a comparison of evolutionary and coevolutionary search . in belew , r.k . ,juill , h. , eds .: coevolution : turning adaptive algorithms upon themselves , san francisco , california , usa ( 2001 ) 2025  das , r. , mitchell , m. , crutchfield , j.p . :a genetic algorithm discovers particle - based computation in cellular automata . in davidor , y. , schwefel , h.p ., mnner , r. , eds . : ppsn iii .volume 866 of lncs ., springer - verlag ( 1994 ) 344353          barnett , l. : netcrawling - optimal evolutionary search with neutral networks . in : proceedings of the 2001 congress on evolutionary computation cec2001 , coex , world trade center , 159 samseong - dong , gangnam - gu , seoul , korea , ieee press ( 2001 ) 3037

Abstract: we study in detail the fitness landscape of a difficult cellular automata computational task : the majority problem . our results show why this problem landscape is so hard to search , and we quantify the large degree of neutrality found in various ways . we show that a particular subspace of the solution space , called the `` olympus '' , is where good solutions concentrate , and give measures to quantitatively characterize this subspace .


quantum phase measurements can not be described by means of a proper observable and different operational approaches have been introduced over the years . on the other hand , from a practical point of view , phase detection of quantum fields is generally associated with interferometric devices , _i.e. _ , detection schemes aimed at the estimation of phase by measuring field- or intensity - based quantities with phase - dependent statistics . the art of interferomety , in turn , consists in answering to two question : a ) how can the unknown phase be effectively retrieved from the data sample ? andb ) which is the resulting precision ?the first point amounts to the choice of an estimator , _i.e. _ , a function from the data sample to the set of possible values of the phase - shift . among possible estimatorsbayes and maximum likelihood ones play a special role due to their asympotic ( _ i.e. _ , for large number of measurements ) properties .the second point may be properly addressed in the framework of quantum estimation theory , which addresses the inference of a physical quantity which is not directly accessible by means of the measurement of a different observable , or a set of observables , somehow related to the quantity of interest .quantum estimation is a powerful tool to infer a single parameter , as well to a set of parameters , up to the full reconstruction of the density matrix of an unknown quantum state , with or without the use of prior information .precision of any unbiased estimator is bounded by the inverse fisher information of the probability distribution of the measurements outcomes , whereas the ultimate limit is written in term of the inverse quantum fisher information ( qfi ) .  in quantum optical systems , homodyne measurements of field quadratures andgaussian signals play a leading role .indeed , measurement of quadratures has been shown to achieve phase estimation for coherent states with precision bounded by the ( classical ) fisher information .this result have been further improved by looking for the optimal state achieving the ultimate bound related to the qfi . among the pure gaussian states ,squeezed vacuum has been found to be the most sensitive state at fixed energy and homodyne detection .furthermore , it has been shown that the same signal allows optimal estimation of loss in bosonic channels and of interaction parameters of single- and two - mode bilinear bosonic hamiltonians .motivated by these results , in this paper we address optimal phase estimation by using gaussian states , homodyne measurements and bayesian estimation .we analyze the behavior for increasing number of measurements and show that optimality may be approached also with a limited number of runs upon using two - step methods acting on the squeezed vacuum probe and/or on the homodyne reference .moreover , we prove that , in principle , the performances of double homodyne detection can not beat the homodyne measurement ones , thus validating the conclusions of .the paper is structured as follows . in section [s : estimation ] we briefly review local quantum estimation theory and the ultimate bounds to precision in the phase - shift estimation by gaussian states . in section [ s : hdbi ]homodyne and double homodyne statistics are explicitly calculated for the phase shifted squeezed vacuum as input : this leads us to conclude that performances of the double homodyne detection can not reach the limit imposed by qfi , while single homodyne does .then , after describing our inference scheme , based on homodyne detection and bayesian inference , the asymptotic limit for large number of collected data is studied in details , as well as the validity of the gaussian approximation . since the performances of this kind of inference protocol depend on the actual value of the ( unknown ) phase shift , we suggest two feasible two - step adaptive methods , the first acting on the squeezing parameter , the other on the squeezing and local oscillator phases , that allow always to reach the optimal estimation .the results of simulated monte carlo experiments are reported in order to check convergence also for small data sample and give a quantitative meaning to the notion of asymptotic regime .section [ s : remarks ] summarizes our results and draws some concluding remarks .let us now consider a field mode undergoing a phase shift described by the unitary operator , with , and being the annihilation and creation field operators , respectively .usually itself can not be measured and a phase estimation problem appears . in order to infer the value of phase - dependent observable is measured and an estimator for , _i.e. _ , a function of the data sample is used .the aim of interferometry is to optimize the inference strategy by minimizing the uncertainty . in general , the lower bound to the variance  ] becomes : = ( 4\delta g^2)^{-1}.\ ] ] it is worth noticing that besides the number operator the above considerations hold for a general hamiltonian generator  .let us now come back to the problem of estimating by measurements on .our aim is to effectively estimate the phase shift at fixed energy upon optimizing the measurement over detection strategies and probe states .of course , the ultimate precision is bounded by the quantum cramr - rao relation ( [ minvar ] ) , which depends on the probe state we employ . in turn , the first stage of the optimization procedure is to find the best probe , which maximizes the qfi at fixed energy .we focus our attention onto the set of _ pure _ states and , more precisely , on gaussian pure states , whose generic element is a squeezed - displaced vacuum state given by :  and , , being the displacement and squeezing operators , respectively . in order to maximize the qfi we look for the state maximizing the energy fluctuations at fixed probe energy = \sinh^2 r + |\alpha|^2 ] achievable by a suitable processing of homodyne data mat saturate to the ultimate bound ( [ minvar : opt ] ) .  before going to the bayesian inference from of homodyne data, we notice that if we use double - homodyne detection we have no improvement in phase - shift estimation .double homodyne statistics is described by the coherent state povm , ; the probability distribution is thus given by : \right\ } } { \pi \cosh r},\end{aligned}\ ] ] where we already set .the corresponding fisher information reads as follows : ^ 2 = 4\sinh^2 r,\end{aligned}\ ] ] that is , : the use of double homodyne detection does not bring any improvement of the phase - shift measurement .this result agrees with the conclusions of , where the author considered double homodyne detection with squeezed vacuum as probe and an auxiliary squeezed state in the other input port .we stress that allows us to infer the probability of the homodyne outcome once the value of is assigned . in our case ,the value of is just the quantity we want to estimate and , in turn , we are interested in the conditional a posteriori probability distribution of _ given _ the the sample of homodyne data .this can be obtained by means of bayesian inference , as we will see in the following .if is the random variable associated with the outcome of the homodyne detection , then the bayes theorem states that : where are the conditional probabilities , is the prior assuming no a priori information , and the overall probability to observe . in turn , upon inverting eq .( [ bayes ] ) we obtain the conditional a posteriori probability of given the outcome .after independent homodyne measurement the a posteriori probability is given by  being the normalization factor : if , then ( [ bayes:1 ] ) rewrites as : where stands for the actual ( unknown ) value of the phase shift . in order to write eq .( [ bayes:2 ] ) we have used the law of large numbers and written the number of occurrences of the outcome as . in this limit probability ( [ bayes:2 ] ) can be explicitly calculated as follows : where we used .we note that the quantity in ( [ bayes : exp ] ) may be regarded as the relative entropy between the two distributions .   a posteriori distribution for different values the number of data ad squeezing parameter .the vertical line is the actual value of the phase shift .,title="fig:",scaledwidth=41.0% ]  a posteriori distribution for different values the number of data ad squeezing parameter .the vertical line is the actual value of the phase shift .,title="fig:",scaledwidth=41.0% ]  in fig .[ f : cond ] the a posteriori distribution is plotted for different values of the involved parameters as a function of .it is worth noting that because of the asymmetric form of the distribution , a suitable estimator for the actual value of the phase shift is given by the maximum of the distribution ( its _ mode _ ,  ] saturate the cramr - rao bound of eq .( [ crb ] ) : this is a consequence of the asymptotic normality of the a posteriori distribution ( laplace - bernstein - von mises theorem ) . however , two questions arises .the first concerns the range of validity of the gaussian approximation , which depends on both and the squeezing parameter .this aspect is illustrated in fig .[ f : gammasigma ] where we plot the ratio , being the variance of the asymptotic distribution . for fixed , one finds that the less is the difference between and the optimal phase given [ see eq .( [ phi : h ] ) ] , the larger is the range of validity of this approximation . on this observationis also based the two - step adaptive method we will describe below .the second question is whether the bayes estimator may saturate also the quantum cramr - rao bound , _i.e. _ , whether the fisher information of may be equal to the qfi , thus leading to phase - shift estimation with precision at the ultimate quantum limit .as what concerns this point we notice that , being the variance of bayes estimator dependent on the true value of the phase shift , some kind of feedback should be unavoidably involved . in the followingwe will describe two possible adaptive mechanisms , acting on the squeezing parameter of the probe or on the homodyne local oscillator and squeezing phase , respectively .adaptive methods for bayesian estimation allow to always attain the ultimate bound on precision and have been investigated in the case of large ensembles and qubit systems . herewe propose two realistic and feasible setups exploiting the interferometric features of homodyne detection .the first scheme is based on the fact that the variance may achieve the optimal value  ] ) and then modify the probe state , tuning its squeezing to . in fig .[ f : ratio ] we show the ratio  ] ( see text ) as a function of and for .the vertical dashed line indicates .,scaledwidth=60.0% ]  tuning the squeezing parameter , however , could be a challenging task . on the other hand , also when and , thus , the energy are fixed , it is possible to achieve the optimal variance by tuning the squeezing phase of the probe state or the phase of the homodyne quadrature .in fact , previously we set and ; if , on the contrary , we assign to these phases the generic values and , then we should simply apply the following change of variable in all the previous equations : , that is a translation of by the amount . since the optimal angle at fixed is given by eq .( [ phi : h ] ) , optimality is always achieved by choosing : as described above , we may obtain a rough estimate of by taking the maximum (  ] ( right ) .the solid lines denote results obtained with the adaptive method acting on the homodyne local oscillator and squeezing phases ; the dashed lines are obtained without the adaptive method .we set and . in the case of the adaptive method, we used of the data to argue the phase - shift rough estimate ( dotted line ) , then the left homodyne data are processed to assess and  ] ( right ) .the solid lines denote results obtained with the adaptive method acting on the homodyne local oscillator and squeezing phases ; the dashed lines are obtained without the adaptive method .we set and . in the case of the adaptive method, we used of the data to argue the phase - shift rough estimate ( dotted line ) , then the left homodyne data are processed to assess and  ] ( right ) .the solid lines denote results obtained with the adaptive method acting on the homodyne local oscillator and squeezing phases ; the dashed lines are obtained without the adaptive method .we set and . in the case of the adaptive method, we used of the data to argue the phase - shift rough estimate ( dotted line ) , then the left homodyne data are processed to assess and  ] ( right ) .the solid lines denote results obtained with the adaptive method acting on the homodyne local oscillator and squeezing phases ; the dashed lines are obtained without the adaptive method .we set and . in the case of the adaptive method, we used of the data to argue the phase - shift rough estimate ( dotted line ) , then the left homodyne data are processed to assess and  ] ( dashed lines in fig .[ f : mc : exp ] ) . with the adaptive scheme ( solid lines ) , of the data sample are used to argue the phase - shift rough estimate , then the phase difference is tuned according to eq .( [ adp ] ) and the left homodyne data are processed to assess and  ] fluctuations , as one may verify , for example , by using a fixed value for .it is worth to note that in our simulations the rough estimate is obtained as  ] ) .in this paper we have shown how bayesian inference techniques represent useful tools for phase estimation .our analysis is based on homodyne detection with squeezed vacuum as a probe state , and bayesian post - processing to infer the phase shift . in the asymptotic limit of a large number of measurements , our scheme saturates the cramr - rao bound to precision , _i.e. _ , the variance of the phase shift achieves the lower bound imposed by the inverse fisher information .moreover , we have shown that optimality may be approached also with a limited number of measurements by means of two - 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Abstract: we address phase - shift estimation by means of squeezed vacuum probe and homodyne detection . we analyze bayesian estimator , which is known to asymptotically saturate the classical cramr - rao bound to the variance , and discuss convergence looking at the a posteriori distribution as the number of measurements increases . we also suggest two feasible adaptive methods , acting on the squeezing parameter and/or the homodyne local oscillator phase , which allow to optimize homodyne detection and approach the ultimate bound to precision imposed by the quantum cramr - rao theorem . the performances of our two - step methods are investigated by means of monte carlo simulated experiments with a small number of homodyne data , thus giving a quantitative meaning to the notion of asymptotic optimality .