conjugate points in formation constrained optimal multi agent coordination a case study in this paper an optimal coordinated motion planning problem for multiple agents subject to constraints on the admissible formation patterns is formulated solutions to the problem are reinterpreted as distance minimizing geodesics on a certain manifold with boundary a geodesic on this manifold may fail to be a solution for different reasons in particular if a geodesic possesses conjugate points then it will no longer be distance minimizing beyond its first conjugate point we study a particular instance of the formation constrained optimal coordinated motion problem where a number of initially aligned agents tries to switch positions by rotating around their common centroid the complete set of conjugate points of a geodesic naturally associated with this problem is characterized analytically this allows us to prove that the geodesic will not correspond to an optimal coordinated motion when the angle of rotation exceeds a threshold that decreases to zero as the number of agents increases moreover infinitesimal perturbations that improve the performance of the geodesic after it passes the conjugate points are also determined which interestingly are characterized by a certain family of orthogonal polynomials

newton method for stationary and quasi stationary problems for smoluchowski type equations abstract an efficient implementation of the newton krylov iterative method finding numerical solutions of aggregation fragmentation equations without spontaneous fragmentation is reported it is based on application of fast numerical schemes for evaluation of the smoluchowski operator associated with these equations and structure of jacobi matrix allowing implementation of fast multiplication by a vector we also propose an efficient preconditioner based on lower triangular band matrix extracted from a jacobi matrix resulting in a dramatic speedup of internal gmres based iterations between the newton steps we compare the efficiency of the novel method with the existing ones and demonstrate a noticeable superiority of our approach moreover we show that the new method may be utilized for modeling of a wider class of aggregation fragmentation phenomena and derive a new class of quasi stationary solutions for aggregation fragmentation equations source of monomers

tensorisation of vectors and their efficient convolution in recent papers the tensorisation of vectors has been discussed in principle this is the isomorphic representation of an mathbb r n vector as a tensor black box tensor approximation methods can be used to reduce the data size of the tensor representation in particular if the vector corresponds to a grid function the resulting data size can become much smaller than n e g o log n ll n in this article we discuss the convolution of two vectors which are given via a sparse tensor representation we want to obtain the result again in the tensor representation furthermore the cost of the convolution algorithm should be related to the operands data sizes while mathbb r n vectors can be considered as grid values of function we also apply the corresponding procedure to univariate functions

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

tensor train versus monte carlo for the multicomponent smoluchowski coagulation equation in this paper we present a novel numerical algorithm for the space homogeneous multicomponent multidimensional smoluchowski coagulation equation the number of components is considered as dimensionality the new methodology is based on the classical finite difference predictor corrector scheme in a straightforward implementation of this scheme however one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid we propose to use special low parametric representations for the grid functions and as well for the coagulation kernel the corresponding multidimensional arrays are approximated by low rank tensor train decompositions reducing them to combinations of small low dimensional arrays eventually to matrices for which we can use fast algorithms of linear algebra instead of o n 2 d operations in the classical scheme we propose a new method that requires only o d 2 n log n operations where n is the number of nodes per axis in the space grid and d is the number of components in this work we accelerate the predictor corrector time scheme and use the trapezoidal rule for the computation of multidimensional integral operators thus the accuracy of the new method is o h 2 2 where h is the space grid step and is the time step

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

on the cauchy problem for higher order nonlinear dispersive equations abstract we study the higher order nonlinear dispersive equation t u x 2 j 1 u 0 j 1 j 2 2 j a j 1 j 2 x j 1 u x j 2 u x t r where u is a real or complex valued function we show that the associated initial value problem is well posed in weighted besov and sobolev spaces for small initial data we also prove ill posedness results when a 0 k 0 for some k j in the sense that this equation cannot have its flow map c 2 at the origin in h s r for any s r the same technique leads to similar ill posedness results for other higher order nonlinear dispersive equation as higher order benjamin ono and intermediate long wave equations

fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension in the present paper we present the tensor product approximation of a multidimensional convolution transform discretized via a collocation projection scheme on uniform or composite refined grids examples of convolving kernels are provided by the classical newton slater exponential and yukawa potentials 1 email protected e l x and e l x email protected with email protected r d for piecewise constant elements on the uniform grid of size n d we prove quadratic convergence o h 2 in the mesh parameter h 1 n and then justify the richardson extrapolation method on a sequence of grids that improves the order of approximation up to o h 3 a fast algorithm of complexity o dr 1r 2nlogn is described for tensor product convolution on uniform composite grids of size n d where r 1 r 2 are tensor ranks of convolving functions we also present the tensor product convolution scheme in the two level tucker canonical format and discuss the consequent rank reduction strategy finally we give numerical illustrations confirming a the approximation theory for convolution schemes of order o h 2 and o h 3 b linear logarithmic scaling of 1d discrete convolution on composite grids c linear logarithmic scaling in n of our tensor product convolution method on an nxnxn grid in the range email protected 16384

maximal function estimates of solutions to general dispersive partial differential equations let u x t sqf x t be the solution of the general dispersive initial value problem ot uiq d u o u x o f x x t ee rn x r and s f the global maximal operator of sqf sharp weighted lp esimates for sq f with f e h rtn are given for general phase functions q

optimal coordinated motions of multiple agents moving on a plane we address the problem of optimal coordinated motions of multiple agents moving in the same planar region the agents motions must satisfy a separation constraint throughout the encounter to be conflict free the objective is to determine the conflict free maneuvers motions with the least combined energy while taking into account the fact that agents may have different priorities a formal classification of conflict free maneuvers into homotopy types is introduced by using their braid representation various local and global optimality conditions are derived through variational analysis in the presence of the separation constraint in the case of two agents these optimality conditions allow us to construct the optimal maneuvers geometrically for the general multi agent case a convex optimization algorithm is proposed to compute within each homotopy type a solution to the optimization problem restricted to the class of multilegged maneuvers since the number of types grows explosively with the number of agents a stochastic algorithm is suggested as the type chooser thus leading to a randomized optimization algorithm

a hybrid particle number and particle model for efficient solution of population balance equations abstract this work presents a hybrid particle number and particle model to improve efficiency in solving population balance equations for type spaces spanning spherical and aggregate particles the particle number model tracks simpler spherical particles cheaply by storing only the number of particles with a given one dimensional internal coordinate while the particle model allows resolution of the detailed aggregate structure that occurs due to collision and coagulation between particles by storing distinct computational entries for each particle this approach is exact if primary particles are defined by their monomer count and the particle number model increments in single monomers a stochastic method is used to solve the population balance equations for the combined type space the hybrid method works well for large ensembles 2 12 particles with a detailed particle model where performing a finite number of particle number updates is demonstrated to be 40 50 cheaper than updating an equivalent ensemble of discrete particles these savings can be traded for a larger sample volume to increase the resolution in the particle size distribution or more repeat runs to reduce the total error run time improvements are curtailed at very high surface growth and coagulation rates due to the fixed cost of growth updates on the large aggregates formed however the hybrid method is still attractive in this case as its primary purpose is to reduce error by preventing saturation of the ensemble with simple particles at high inception rates

optimal small data scattering for the generalized derivative nonlinear schrodinger equations abstract in this work we consider the following generalized derivative nonlinear schrodinger equation i t u x x u i u 2 x u 0 t x r r we prove that when 2 the solution is global and scattering when the initial data is small in h s r 1 2 s 1 moreover we show that when 0 2 there exist a class of solitary wave solutions c satisfying c h 1 r 0 when c tends to some endpoint which is against the small data scattering statement therefore in this model the exponent 2 is optimal for small data scattering we remark that this exponent is larger than the short range exponent and the strauss exponent

efficient convolution with the newton potential in d dimensions the paper is concerned with the evaluation of the convolution integral int mathbb r d frac 1 left vert x y right vert f y rm d y in d dimensions usually d 3 when f is given as piecewise polynomial of possibly large degree i e f may be considered as an hp finite element function the underlying grid is locally refined using various levels of dyadically organised grids the result of the convolution is approximated in the same kind of mesh if f is given in tensor product form the d dimensional convolution can be reduced to one dimensional convolutions although the details are given for the kernel 1 left vert x right vert the basis techniques can be generalised to homogeneous kernels e g the fundamental solution const cdot left vert x right vert 2 d of the d dimensional poisson equation

tt cross approximation for multidimensional arrays abstract as is well known a rank r matrix can be recovered from a cross of r linearly independent columns and rows and an arbitrary matrix can be interpolated on the cross entries other entries by this cross or pseudo skeleton approximation are given with errors depending on the closeness of the matrix to a rank r matrix and as well on the choice of cross in this paper we extend this construction to d dimensional arrays tensors and suggest a new interpolation formula in which a d dimensional array is interpolated on the entries of some tt cross tensor train cross the total number of entries and the complexity of our interpolation algorithm depend on d linearly so the approach does not suffer from the curse of dimensionality we also propose a tt cross method for computation of d dimensional integrals and apply it to some examples with dimensionality in the range from d 100 up to d 4000 and the relative accuracy of order 10 10 in all constructions we capitalize on the new tensor decomposition in the form of tensor trains tt decomposition

pointwise convergence of solutions to schrodinger type equations abstract in this paper we study the pointwise convergence of the function e i t 2 f x whenever f belongs to some sobolev space when 2 e i t f x is the solution to the standard schrodinger equation and this problem has been widely studied here we focus on the case 0 1 the same convergence problem for the nonelliptic schrodinger equation is also discussed the proof here follows the idea of carleson s first paper on this subject but we do reprove some known theorems by this straightforward method and get some new results as well

brief paper finite time multi agent deployment a nonlinear pde motion planning approach the systematic flatness based motion planning using formal power series and suitable summability methods is considered for the finite time deployment of multi agent systems into planar formation profiles along predefined spatial temporal paths thereby a distributed parameter setting is proposed where the collective leader follower agent dynamics is modeled by two boundary controlled nonlinear time varying pdes governing the motion of an agent continuum in the plane the discretization of the pde model directly induces a decentralized communication and interconnection structure for the multi agent system which is required to achieve the desired spatial temporal paths and deployment formations

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

tensor decomposition in electronic structure calculations on 3d cartesian grids in this paper we investigate a novel approach based on the combination of tucker type and canonical tensor decomposition techniques for the efficient numerical approximation of functions and operators in electronic structure calculations in particular we study applicability of tensor approximations for the numerical solution of hartree fock and kohn sham equations on 3d cartesian grids we show that the orthogonal tucker type tensor approximation of electron density and hartree potential of simple molecules leads to low tensor rank representations this enables an efficient tensor product convolution scheme for the computation of the hartree potential using a collocation type approximation via piecewise constant basis functions on a uniform nxnxn grid combined with the richardson extrapolation our approach exhibits o h 3 convergence in the grid size h o n 1 moreover this requires o 3rn r 3 storage where r denotes the tucker rank of the electron density with r o logn almost uniformly in n for example calculations of the coulomb matrix and the hartree fock energy for the ch 4 molecule with a pseudopotential on the c atom achieved accuracies of the order of 10 6 hartree with a grid size n of several hundreds since the tensor product convolution in 3d is performed via 1d convolution transforms our scheme markedly outperforms the 3d fft in both the computing time and storage requirements

nonlocal schrodinger equations in metric measure spaces abstract in this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic schrodinger equations on spaces of homogeneous type we prove the a e convergence when the initial data belongs to a dyadic version of an l 2 based besov space

low rank method for fast solution of generalized smoluchowski equations abstract an efficient method for the fast solution of a new class of generalized smoluchowski equations is developed in contrast to conventional smoluchowski equations with constant rate coefficients that describe the aggregation kinetics the rate coefficients of the generalized smoluchowski equations depend on time the evolution of these coefficients is governed by another set of equations coupled with the aggregation equations the most prominent example of such processes is the ballistic aggregation when aggregates of different sizes have different partial temperatures the conventional approaches fail to deal efficiently with these systems when the number of equations n is large the new method overcomes this difficulty it exploits the low rank approach and drastically decreases computation time instead of o n 2 operations per time step the new method requires only o n r r log n operations where r is the system rank general techniques for an efficient solution of very large systems of ode are discussed in the context of the new approach

optimal multi agent coordination under tree formation constraints this paper presents a framework for studying the centralized optimal multi agent coordination problem under tree formation constraints the geodesic equations characterizing the optimal coordinated motions are derived in a suitably chosen coordinate system for general tree formation constraints the solutions to these equations however may fail to be optimal once extended beyond certain points called the conjugate points due to the failure of the second order optimality condition for the particular class of star formations two methods for computing the conjugate points along a natural candidate solution are introduced using these methods we derive analytically the conjugate points as well as the better solutions once the candidate solution is extended beyond its first conjugate point the optimal centralized coordinated motions derived in this paper will yield a performance lower bound for those generated by decentralized algorithms

smoothing and dispersive estimates for 1d schrodinger equations with bv coefficients and applications we prove smoothing estimates for schrodinger equations i t x a x x 0 with a x bv real and bounded from below we then bootstrap these estimates to obtain optimal strichartz and maximal function estimates all of which turn out to be identical to the constant coefficient case we also provide counterexamples showing a bv to be in a sense a minimal requirement finally we provide an application to sharp well posedness for a generalized benjamin ono equation

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

multilevel toeplitz matrices generated by tensor structured vectors and convolution with logarithmic complexity we study the tensor structure of two operations the transformation of a given multidimensional vector into a multilevel toeplitz matrix and the convolution of two given multidimensional vectors we show that the low rank tensor structure of the input is preserved in the output and propose efficient algorithms for these operations in the newly introduced quantized tensor train qtt format consider a d dimensional 2n times cdots times 2n vector boldsymbol x if it is represented elementwise the number of parameters is 2n d however if we assume that boldsymbol x is given in a qtt representation with ranks bounded by p the number of parameters is reduced to mathcal o left dp 2 log n right under this assumption we show how the multilevel toeplitz matrix generated by boldsymbol x can be obtained in the qtt format with ranks bounded by 2p in mathcal o left dp 2 log n right operations we also describe how the convolution boldsymbol x star boldsymbol y of

breaking the curse of dimensionality or how to use svd in many dimensions for d dimensional tensors with possibly large d 3 an hierarchical data structure called the tree tucker format is presented as an alternative to the canonical decomposition it has asymptotically the same and often even smaller number of representation parameters and viable stability properties the approach involves a recursive construction described by a tree with the leafs corresponding to the tucker decompositions of three dimensional tensors and is based on a sequence of svds for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial spatial dimensions it is shown how this format can be applied to the problem of multidimensional convolution convincing numerical examples are given

the schrodinger equation along curves and the quantum harmonic oscillator abstract we consider the schrodinger equation associated to the harmonic oscillator i t u h u where h x 2 with initial data in the sobolev space h s r d with d 2 and s 3 8 we prove almost everywhere convergence of the solution to its initial data as time tends to zero which improves a result of yajima 1990 30 to this end we consider almost everywhere convergence for the free schrodinger along curves as it turns out these problems are more or less equivalent to that of the free schodinger equation along vertical lines our results are obtained by showing the equivalence of related maximal estimates

on the generalized benjamin ono equation we study well posedness of the initial value problem for the generalized benjamin ono equation tu ukdxu 0 k e in sobolev spaces hs r for small data and higher nonlinearities k 2 new local and global including scattering results are established our method of proof is quite general it combines several estimates concerning the associated linear problem with the contraction principle hence it applies to other dispersive models in particular it allows us to extend the results for the generalized benjamin ono to nonlinear schrodinger equations or systems of the form atu i2u p u oxu ui axi 0

stability analysis of swarms in this note we specify an individual based continuous time model for swarm aggregation in n dimensional space and study its stability properties we show that the individuals autonomous agents or biological creatures will form a cohesive swarm in a finite time moreover we obtain an explicit bound on the swarm size which depends only on the parameters of the swarm model

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

on dyadic nonlocal schrodinger equations with besov initial data abstract in this paper we consider the pointwise convergence to the initial data for the schrodinger dirac equation i u t d u with u x 0 u 0 in a dyadic besov space here d denotes the fractional derivative of order associated to the dyadic distance on r the main tools are a summability formula for the kernel of d and pointwise estimates of the corresponding maximal operator in terms of the dyadic hardy littlewood function and the calderon sharp maximal operator

a fast numerical method for the cauchy problem for the smoluchowski equation a new solution technique is proposed for one dimensional smoluchowski equations it is based on the finite difference predictor corrector scheme and is faster than other methods using this kind of scheme the new technique capitalizes on low rank approximations of matrices arising after discretization of the coagulation kernel and includes a new fast convolution algorithm with the trapezoidal quadrature rule for the grids with n nodes the complexity of the new method is o n log n for each step with time instead of o n 2 operations required by the standard implementation of the predictor corrector scheme

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

a model system for strong interaction between internal solitary waves a mathematical theory is mounted for a complex system of equations derived by gear and grimshaw that models the strong interaction of two dimen sional long internal gravity waves propagating on neighboring pycnoclines in a stratified fluid for the model in question the cauchy problem is of interest and is shown to be globally well posed in suitably strong function spaces our results make use of kato s theory for abstract evolution equations together with somewhat delicate estimates obtained using techniques from harmonic analysis in weak function classes a local existence theory is developed the system is shown to be susceptible to the dispersive blow up phenomenon investigated recently by bona and saut for korteweg de vries type equations

well posedness of the initial value problem for the korteweg de vries equation 1 1 itu axu u1xu o x t e r u x 0 uo x the kdv equation which was first derived as a model for unidirectional propagation of nonlinear dispersive long waves 21 has been considered in different contexts namely in its relation with the inverse scattering method in plasma physics and in algebraic geometry see 24 and references therein our purpose is to study local and global well posedness of the ivp 1 1 in classical sobolev spaces hs r we shall say that the ivp 1 1 is locally resp globally well posed in the function space x if it induces a dynamical system on x by generating a continuous local resp global flow it was established in the works of bona and smith 3 bona and scott 2 saut and temam 30 and kato 1 5 that the ivp 1 1 is locally resp globally well posed in hs with s 3 2 resp s 2 roughly speaking global well posedness in hs depends on the available local theory and on the conservation laws satisfied by solutions of 1 1 namely

existence and uniqueness of solution for a generalized nonlinear derivative schrodinger equation in this work we study the well posedness for the initial value problem associated to a generalized derivative schrodinger equation for small size initial data in weighted sobolev space the techniques used include parabolic regularization method combined with sharp linear estimates an important point in our work is that the contraction principle is likely to fail but gives us inspiration to obtain certain uniform estimates that are crucial to obtain the main result to prove such uniform estimates we assume smallness on the initial data in weighted sobolev spaces

coherence on fractals versus pointwise convergence for the schrodinger equation we consider carleson s problem regarding convergence for the schrodinger equation in dimensions d ge 2 we show that if the solution converges almost everywhere with respect to alpha hausdorff measure to its initial datum as time tends to zero for all data h s mathbb r d then s ge frac d 2 d 2 d 1 alpha this strengthens and generalises results of bourgain and dahlberg kenig

numerical solution of the hartree fock equation in multilevel tensor structured format in this paper we describe a novel method for a robust and accurate iterative solution of the self consistent hartree fock equation in mathbb r 3 based on the idea of tensor structured computation of the electron density and the nonlinear hartree and nonlocal exchange operators at all steps of the iterative process we apply the self consistent field scf iteration to the galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three dimensional cartesian grid the approximation error is estimated by o h 3 where h o n 1 is the mesh size of an n times n times n tensor grid while the numerical complexity to compute the galerkin matrices scales linearly in n log n we propose the tensor truncated version of the scf iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid dependent termination criteria at each discretization level this implies that the overall computational cost scales almost linearly in the univariate problem size n numerical illustrations are presented for the all electron case of h 2 o and the pseudopotential case of ch 4 and ch 3 oh molecules the proposed scheme is not restricted to a priori given rank 1 basis sets allowing analytically integrable convolution transform with the newton kernel that opens further perspectives for promotion of the tensor structured methods in computational quantum chemistry