an a posteriori parameter choice for tikhonov regularization in hilbert scales leading to optimal convergence rates an a posteriori parameter choice for tikhonov regularization in hilbert scales is proposed that leads to optimal convergence rates the advantage of this method is that knowledge of the smoothness parameters of both the operator and the exact solution is not needed the finite dimensional realization of this method is also discussed numerical examples confirm the theoretical results

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

a quasi reversibility regularization method for an inverse heat conduction problem without initial data in this paper we study an inverse heat conduction problem without initial data in a bounded domain in which the cauchy data at x 0 are given and the solution in 0

a fast multiple scale polynomial solution for the inverse cauchy problem of elasticity in an arbitrary plane domain the polynomial expansion method together with the collocation technique is a cheap yet simple method to solve the navier equations of elasticity which is easily arranged to satisfy the governing equations and boundary conditions pointwise in this paper we propose a novel numerical algorithm for the solution of an overspecified underspecified cauchy problem of linear elasticity in an arbitrary plane domain by using the multiple scale pascal polynomial expansion method mspem of which the scales are determined a priori by the collocation points according to the idea of equilibrated matrix in the numerical tests of a direct problem as well as the overspecified underspecified cauchy problems the mspem is very accurate and stable against large relative noise up to 20 for the unknown displacements recovery problem and up to 100 absolute noise for the recovery of unknown loading force the present method is convergent very fast for most cases within 100 iteration steps

on converse and saturation results for tikhonov regularization of linear ill posed problems in this paper we prove some new converse and saturation results for tikhonov regularization of linear ill posed problems tx y where t is a linear operator between two hilbert spaces

a multiple scale pascal polynomial for 2d stokes and inverse cauchy stokes problems abstract the polynomial expansion method is a useful tool for solving both the direct and inverse stokes problems which together with the pointwise collocation technique is easy to derive the algebraic equations for satisfying the stokes differential equations and the specified boundary conditions in this paper we propose two novel numerical algorithms based on a third first order system and a third third order system to solve the direct and the inverse cauchy problems in stokes flows by developing a multiple scale pascal polynomial method of which the scales are determined a priori by the collocation points to assess the performance through numerical experiments we find that the multiple scale pascal polynomial expansion method mspem is accurate and stable against large noise

a proportional derivative control strategy for restarting the gmres m algorithm abstract restarted gmres or gmres m is normally used for solving large linear systems a x b with a general possibly nonsymmetric matrix a although the restarted gmres consumes less computational time than its counterpart full gmres if the restarting parameter is not correctly chosen its convergence cannot be guaranteed and the method may converge slowly unfortunately it is difficult to know how to choose this parameter a priori in this article we regard the gmres m method as a control problem in which the parameter m is the controlled variable and propose a new control inspired strategy for choosing the parameter m adaptively at each iteration the advantage of this control strategy method is that only a few additional vectors need to be stored and the controller has the capacity to modify the dimension of the krylov subspace whenever any convergence problem is detected numerical experiments based on benchmark problems show that the proposed control strategy accelerates the convergence of gmres m

quantum tomography under prior information we provide a detailed analysis of the question how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information we show that if the prior information restricts the possible states to a set of lower dimensionality then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states conversely we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the minkowski dimension of the set we apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints in particular we show that the minimal number of measurement outcomes povm elements necessary to identify all pure states in a d dimensional hilbert space is 4d 3 c d d for some c d in 1 2 and d being the number of ones appearing in the binary expansion of d 1

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

on local behaviour of the phase separation line in the 2d ising model the aim of this note is to discuss some statistical properties of the phase separation line in the 2d low temperature ising model we prove the functional central limit theorem for the probability distributions describing fluctuations of the phase boundary in the direction orthogonal to its orientation the limiting gaussian measure corresponds to a scaled brownian bridge with direction dependent parameters up to the temperature factor the variances of local increments of this limiting process are inversely proportional to the stiffness

a family of rules for parameter choice in tikhonov regularization of ill posed problems with inexact noise level we consider tikhonov regularization of linear ill posed problems with noisy data the choice of the regularization parameter by classical rules such as discrepancy principle needs exact noise level information these rules fail in the case of an underestimated noise level and give large error of the regularized solution in the case of very moderate overestimation of the noise level we propose a general family of parameter choice rules which includes many known rules and guarantees convergence of approximations quasi optimality is proved for a sub family of rules many rules from this family work well also in the case of many times under or overestimated noise level in the case of exact or overestimated noise level we propose to take the regularization parameter as the minimum of parameters from the post estimated monotone error rule and a certain new rule from the proposed family the advantages of the new rules are demonstrated in extensive numerical experiments

a simple strategy for varying the restart parameter in gmres m when solving a system of linear equations with the restarted gmres method a fixed restart parameter is typically chosen we present numerical experiments that demonstrate the beneficial effects of changing the value of the restart parameter in each restart cycle on the total time to solution we propose a simple strategy for varying the restart parameter and provide some heuristic explanations for its effectiveness based on analysis of the symmetric case

an a posteriori parameter choice for ordinary and iterated tikhonov regularization of ill posed problems leading to optimal convergence rates on propose un choix a posteriori du parametre pour la regularisation de tikhonov ordinaire et iteree des problemes mal poses conduisant des vitesses de convergence optimales

the tortoise and the hare restart gmres when solving large nonsymmetric systems of linear equations with the restarted gmres algorithm one is inclined to select a relatively large restart parameter in the hope of mimicking the full gmres process surprisingly cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values here two simple examples are presented where gmres 1 converges exactly in three iterations while gmres 2 stagnates one of these examples reveals that gmres 1 convergence can be extremely sensitive to small changes in the initial residual

on the regularization of projection methods for solving iii posed problems in this paper we consider a class of regularization methods for a discretized version of an operator equation which includes the case that the problem is ill posed with approximately given right hand side we propose an a priori as well as an a posteriori parameter choice method which is similar to the discrepancy principle of ivanov morozov from results on fractional powers of selfadjoint operators we obtain convergence rates which are in many cases the same for both parameter choices

applications of the modified discrepancy principle to tikhonov regularization of nonlinear ill posed problems in this paper we consider the finite dimensional approximations of tikhonov regularization for nonlinear ill posed problems with approximately given right hand sides we propose an a posteriori parameter choice strategy which is a modified form of morozov s discrepancy principle to choose the regularization parameter under certain assumptions on the nonlinear operator we obtain the convergence and rates of convergence for tikhonov regularized solutions this paper extends the results which were developed by plato and vainikko in 1990 for solving linear ill posed equations to nonlinear problems

the polynomial trefftz method for solving backward and inverse source wave problems the trefftz method is a truly meshless boundary type method because the trial solutions automatically satisfy the governing equation in order to stably solve the high dimensional backward wave problems and the one dimensional inverse source problems we develop a multiple scale polynomial trefftz method msptm of which the scales are determined a priori by the collocation points the msptm can retrieve the missing initial data and unknown time varying wave source the present method can also be extended to solve the higher dimensional wave equations long term through the introduction of a director in the two dimensional polynomial trefftz bases several numerical examples reveal that the msptm is efficient and stable for solving severely ill posed inverse problems of wave equations under large noises

on the iteratively regularized gauss newton method for solving nonlinear ill posed problems the iteratively regularized gauss newton method is applied to compute the stable solutions to nonlinear ill posed problems f x y when the data y is given approximately by y with y y in this method the iterative sequence x k is defined successively by x k 1 x k k i f x k f x k 1 f x k f x k y k x k x 0 where x 0 x 0 is an initial guess of the exact solution x and k is a given decreasing sequence of positive numbers admitting suitable properties when x k is used to approximate x the stopping index should be designated properly in this paper an a posteriori stopping rule is suggested to choose the stopping index of iteration and with the integer k determined by this rule it is proved that x k x c inf x k x k k 0 1 with a constant c independent of where x k denotes the iterative solution corresponding to the noise free case as a consequence of this result the convergence of x k is obtained and moreover the rate of convergence is derived when x 0 x satisfies a suitable source wise representation the results of this paper suggest that the iteratively regularized gauss newton method combined with our stopping rule defines a regularization method of optimal order for each 0 v 1 numerical examples for parameter estimation of a differential equation are given to test the theoretical results

multilevel algorithms for ill posed problems in this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations convergence estimates are established for multilevel algorithms applied to tikhonov type regularization methods our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter the algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations

on the microscopic validity of the wulff construction and of the generalized young equation for a large class of 1 1 dimensional interfaces of the solid on solid type we prove on a microscopic basis the validity of the wulff construction and of the generalized young equation which gives the contact angle of a sessile drop on a wall our proof relies on a new method to treat random walks with a finite number of global constraints

approximation accuracy of the krylov subspaces for linear discrete ill posed problems abstract for the large scale linear discrete ill posed problem min a x b or a x b with b contaminated by gaussian white noise the lanczos bidiagonalization based krylov solver lsqr and its mathematically equivalent cgls the conjugate gradient cg method implicitly applied to a t a x a t b are most commonly used and cgme the cg method applied to min a a t y b or a a t y b with x a t y and lsmr which is equivalent to the minimal residual minres method applied to a t a x a t b have also been choices these methods exhibit typical semi convergence feature and the iteration number k plays the role of the regularization parameter however there has been no definitive answer to the long standing fundamental question can lsqr and cgls find 2 norm filtering best possible regularized solutions the same question is for cgme and lsmr too at iteration k lsqr cgme and lsmr compute different iterates from the same k dimensional krylov subspace a first and fundamental step towards answering the above question is to accurately estimate the accuracy of the underlying k dimensional krylov subspace approximating the k dimensional dominant right singular subspace of a assuming that the singular values of a are simple we present a general sin theorem for the 2 norm distances between these two subspaces and derive accurate estimates on them for severely moderately and mildly ill posed problems we also establish some relationships between the smallest ritz values and these distances numerical experiments justify the sharpness of our results

comparison of a posteriori parameter choice rules for linear discrete ill posed problems abstract tikhonov regularization is one of the most popular methods for computing approximate solutions of linear discrete ill posed problems with error contaminated data a regularization parameter 0 balances the influence of a fidelity term which measures how well the data is approximated and of a regularization term which dampens the propagation of the data error into the computed approximate solution the quality of the computed solution is affected by the value of the regularization parameter the discrepancy principle is a popular a posteriori rule for determining a suitable value of it performs quite well when a fairly accurate estimate of the norm of the error in the data is known a modification of the discrepancy principle proposed independently by gfrerer and raus also can be used to determine analysis of this modification in an infinite dimensional hilbert space setting suggests that it will determine a value of that yields an approximate solution of higher quality than the approximate solution obtained when using the standard discrepancy principle to compute this paper compares these a posteriori rules for determining when applied to the solution of many linear discrete ill posed problems with different amounts of error in the data our comparison shows that in a discrete setting the discrepancy principle generally gives a value of that yields a computed solution of higher quality than the value of furnished by the modified discrepancy principle

phaseequal convex phase retrieval via alternating direction method of multipliers the problem of estimating an unknown complex signal from its magnitude only measurements is a classical problem known as phase retrieval this problem arises naturally because in some applications it is difficult or costly to measure the phase of the measurements in this paper we propose a new convex formulation referred to as phaseequal for solving the phase retrieval problem similar to phasemax t goldstein and c studer 2018 by goldstein and studer iee tit 2018 phaseequal works in the natural signal space and thus is computationally efficient we then extend phaseequal to the sparse signal case with the proposed convex formulation termed as compressed phaseequal different from existing convex compressed phase retrieval methods the proposed compressed phaseequal formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice order wise recovery conditions for phaseequal and its sparse version i e compressed phaseequal are analyzed our theoretical results show that phaseequal resp compressed phaseequal achieves perfect recovery with mathcal o n resp mathcal o k text log frac n k magnitude measurements provided that a well correlated reference vector is available where k and n denote the number of nonzero entries in the complex sparse signal and the dimension of the signal respectively simulation results are provided to illustrate the effectiveness of the proposed methods

near optimal parameters for tikhonov and other regularization methods choosing the regularization parameter for an ill posed problem is an art based on good heuristics and prior knowledge of the noise in the observations in this work we propose choosing the parameter without a priori information by approximately minimizing the distance between the true solution to the discrete problem and the family of regularized solutions we demonstrate the usefulness of this approach for tikhonov regularization and for an alternate family of solutions further we prove convergence of the regularization parameter to zero as the standard deviation of the noise goes to zero

a control theoretic study on iterative solutions to nonlinear equations for applications in embedded systems in this paper the fixed point iteration and newton s methods for iteratively solving nonlinear equations are studied in the control theoretical framework this work is motivated by the ever increasing demands for integrating iterative solutions of nonlinear functions into embedded control systems the use of the well established control theoretical methods for our application purpose is inspired by the recent control theoretical study on numerical analysis our study consists of two parts in the first part the existing fixed point iteration and newton s methods are analysed using the stability theory for the sector bounded lure s systems the second part is devoted to the modified iteration methods and the integration of sensor signals into the iterative computations the major results achieved in our study are besides some academic examples applied to the iterative computation of the air path model embedded in the engine control systems

second order large deviation estimates for ferromagnetic systems in the phase coexistence region we consider thed dimensional ising model with ferromagnetic nearest neighbor interaction at inverse temperature let m lambda lambda 1 sum limits i in lambda sigma i be the magnetization inside ad dimensional hyper cube be the gibbs state andm be the spontaneous magnetization for such thatm 0 we find a sufficient condition easily verified to hold for large for m a b to decay exponentially with d 1 d when m b m 1 a b ford 2 this sufficient condition is the exponential decay of a connectivity function we also prove a partial converse to this result obtain a sharper result for the magnetization ond 1 dimensional cross sections of the model and prove a similar result ford 2 m a b m and large when free boundary conditions are chosen outside

adaptive choice of the tikhonov regularization parameter to solve ill posed linear algebraic equations via liapunov optimizing control numerical problems including linear algebraic equations are frequently ill posed which means that small perturbations in the observation data can have large effects on the computed solutions in these cases some kind of regularization method must be used to find a feasible solution regularization implies the modification of the original ill posed problem so that it becomes well posed and the tikhonov regularization method is the most often used this method requires the choice of a regularization parameter and this choice is a crucial issue recently several papers were presented proposing adaptive tikhonov regularization parameters instead of constant ones this paper revisits all these adaptive choices and shows that they can be interpreted in a unified and systematic manner using the liapunov optimizing control loc method which in addition opens up the possibility of new choices of the regularization parameter examples of matrices from real applications show that one of the methods designed by the loc method called the ovm method by its discoverer can even outperform the cg method for some matrices although the cg method performs better for a larger class of matrices

system theory for numerical analysis many numerical schemes can be suitably studied from a system theoretic point of view this paper studies the relationship between the two disciplines that is numerical analysis and system theory we first see that various iterative solution schemes for linear and nonlinear equations can be suitably transformed into the form of a closed loop feedback system and show the crucial role of the internal model principle in such a context this leads to new stability criteria for newton s method we then study runge kutta type methods for solving differential equations and also derive new stability criteria based on recent results on lmi a numerical example is given to illustrate the advantage of the present theory

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

dobrushin kotecký shlosman theorem up to the critical temperature we develop a non perturbative version of the dobrushin koteck shlosman theory of phase separation in the canonical 2d ising ensemble the results are valid for all temperatures below critical

phasecode fast and efficient compressive phase retrieval based on sparse graph codes we consider the problem of recovering a complex signal x in mathbb c n from m intensity measurements of the form a i mathrm h x 1 leq i leq m where a i mathrm h is the i th row of measurement matrix a in mathbb c m times n our main focus is on the case where the measurement vectors are unconstrained and where x is exactly k sparse or the so called general compressive phase retrieval problem we introduce phasecode a novel family of fast and efficient algorithms that are based on a sparse graph coding framework we show that in the noiseless case the phasecode algorithm can recover an arbitrarily close to one fraction of the k nonzero signal components using only slightly more than 4k measurements when the support of the signal is uniformly random with the order optimal time and memory complexity of theta k 1 it is known that the fundamental limit for the number of measurements in compressive phase retrieval problem is 4k o k for the more difficult problem of recovering the signal exactly and with no assumptions on its support distribution this shows that under mild relaxation of the conditions our algorithm is the first constructive capacity approaching compressive phase retrieval algorithm in fact our algorithm is also order optimal in complexity and memory furthermore we show that for any signal x phasecode can recover a random 1 p fraction of the nonzero components of x with high probability where p can be made arbitrarily close to zero with sample complexity m c p k where c p is a small constant depending on p that can be precisely calculated with optimal time and memory complexity as a result assuming that the nonzero components of x are lower bounded by theta 1 and upper bounded by theta k gamma for some positive constant gamma we are able to provide a strong ell 1 guarantee for the estimated signal hat x as follows hat x x 1 leq p x 1 1 o 1 where p can be made arbitrarily close to zero as one instance the phasecode algorithm can provably recover with high probability a random 1 10 7 fraction of the significant signal components using at most m 14k measurements next motivated by some important practical classes of optical systems we consider a fourier friendly constrained measurement setting and show that its performance matches that of the unconstrained setting when the signal is sparse in the fourier domain with uniform support in the fourier friendly setting that we consider the measurement matrix is constrained to be a cascade of fourier matrices corresponding to optical lenses and diagonal matrices corresponding to diffraction mask patterns finally we tackle the compressive phase retrieval problem in the presence of noise where measurements are in the form of y i a i mathrm h x 2 w i and w i is the additive noise to the i th measurement we assume that the signal is quantized and each nonzero component can take l m possible magnitudes and l p possible phases we consider the regime where k beta n delta delta in 0 1 we use the same architecture of phasecode for the noiseless case and robustify it using two schemes the almost linear scheme and the sublinear scheme we prove that with high probability the almost linear scheme recovers x with sample complexity theta k log n and computational complexity theta l m l p n log n and the sublinear scheme recovers x with sample complexity theta k log 3 n and computational complexity theta l m l p k log 3 n throughout we provide extensive simulation results that validate the practical power of our proposed algorithms for the sparse unconstrained and fourier friendly measurement settings for noiseless and noisy scenarios 1 here we define the notation mathcal o cdot theta cdot and omega cdot we have f mathcal o g if and only if there exists a constant c 1 0 such that left f g right f theta g if and only if there exist two constants c 1 c 2 0 such that c 1 and f omega g if and only if there exists a constant c 1 0 such that left f g right c 1

a multiple direction trefftz method for solving the multi dimensional wave equation in an arbitrary spatial domain in this paper we first express the wave equation in terms of the minkowskian polar coordinates and generate a set of complete hyperbolic type trefftz bases r k cosh k and r k sinh k which are further transformed to wave polynomials as the trial solution bases for the one dimensional wave equation in order to stably solve the wave propagation problems long term we develop a multiple scale trefftz method mstm of which the scales are determined a priori by the collocation points then we derive a very simple method of multi dimensional wave polynomials equipped with different spatial directions which being the normalized wavenumber vectors as the polynomial trefftz bases for solving the multi dimensional wave equations which is named a multiple direction trefftz method mdtm several numerical examples of two and three dimensional wave equations demonstrate that the present method is efficient and stable

uniform convergence of regularization methods for linear ill posed problems abstract after a general discussion about convergence and convergence rates for regularization methods in banach spaces we present a general method that can be used to modify regularization methods in l2 in such a way that uniform convergence which is often preferred in concrete applications to just l2 convergence is obtained we prove results about convergence rates in the uniform norm and discuss questions of parameter choice the theoretical results will be supported by numerical examples which indicate that although the results are asymptotic in character they are of some relevance also for actual computations

on the wulff crystal in the ising model we study the phase separation phenomenon in the ising model in dimensions d 3 to this end we work in a large box with plus boundary conditions and we condition the system to have an excess amount of negative spins so that the empirical magnetization is smaller than the spontaneous magnetization m we confirm the prediction of the phenomenological theory by proving that with high probability a single droplet of the minus phase emerges surrounded by the plus phase moreover the rescaled droplet is asymptotically close to a definite deterministic shape the wulff crystal which minimizes the surface free energy in the course of the proof we establish a surface order large deviation principle for the magnetization our results are valid for temperatures t below a limit of slab thresholds t c conjectured to agree with the critical point t c moreover t should be such that there exist only two extremal translation invariant gibbs states at that temperature a property which can fail for at most countably many values and which is conjectured to be true for every t the proofs are based on the fortuin kasteleyn representation of the ising model along with coarse graining techniques to handle the emerging macroscopic objects we employ tools from geometric measure theory which provide an adequate framework for the large deviation analysis finally we propose a heuristic picture that for subcritical temperatures close enough to t c the dominant minus spin cluster of the wulff droplet permeates the entire box and has a strictly positive local density everywhere

critical region for droplet formation in the two dimensional ising model we study the formation dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence specifically we consider the 2d ising model in volumes of size l2 inverse temperature c and overall magnetization conditioned to take the value m l2 2m vl where c 1 is the critical temperature m m is the spontaneous magnetization and vl is a sequence of positive numbers we find that the critical scaling for droplet formation dissolution is when vl3 2l 2 tends to a definite limit specifically we identify a dimensionless parameter proportional to this limit a non trivial critical value c and a function such that the following holds for c there is a single wulff shaped droplet containing a fraction c 2 3 of the magnetization deficit and there are no other droplets beyond the scale of log l moreover and are related via a universal equation that apparently is independent of the details of the system

a proportional differential control method for a time delay system using the taylor expansion approximation abstract in this paper a proportional differential control algorithm is studied according to the system performance indices we uses the first order taylor expansion to approximate the time delay present an algorithm to determine the controller parameters so that the desired dynamic performance can be achieved the numerical examples are provided to show the advantages of the proposed algorithm for small time delay

sublinear time algorithms for compressive phase retrieval in the problem of compressed phase retrieval the goal is to reconstruct a sparse or approximately k sparse vector x in mathbb c n given access to y phi x where v denotes the vector obtained from taking the absolute value of v in mathbb c n coordinate wise in this paper we present sublinear time algorithms for a few for each variants of the compressive phase retrieval problem which are akin to the variants considered for the classical compressive sensing problem in theoretical computer science our algorithms use pure combinatorial techniques and near optimal number of measurements

measurement matrix design for phase retrieval based on mutual information in phase retrieval problems a signal of interest soi is reconstructed based on the magnitude of a linear transformation of the soi observed with additive noise the linear transform is typically referred to as a measurement matrix many works on phase retrieval assume that the measurement matrix is a random gaussian matrix which in the noiseless scenario with sufficiently many measurements guarantees invertability of the transformation between the soi and the observations up to an inherent phase ambiguity however in many practical applications the measurement matrix corresponds to an underlying physical setup and is therefore deterministic possibly with structural constraints in this paper we study the design of deterministic measurement matrices based on maximizing the mutual information between the soi and the observations we characterize necessary conditions for the optimality of a measurement matrix and analytically obtain the optimal matrix in the low signal to noise ratio regime practical methods for designing general measurement matrices and masked fourier measurements are proposed simulation tests demonstrate the performance gain achieved by the suggested techniques compared to random gaussian measurements for various phase recovery algorithms

a globally optimal tri vector method to solve an ill posed linear system in the present paper a future cone in the minkowski space defined in terms of the square norm of the residual vector for an ill posed linear system to be solved is used to derive a nonlinear system of ordinary differential equations then the forward euler scheme is used to generate an iterative algorithm two critical values in the critical descent tri vector are derived which lead to the largest convergence rate of the resultant iterative algorithm namely the globally optimal tri vector method gotvm some numerical examples are used to reveal the superior performance of the gotvm than the famous methods of conjugate gradient cgm and generalized minimal residual gmres through the numerical tests we also set forth the rationale by assuming the tri vector as being a better descent direction

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

an improved version of marti s method for solving ill posed linear integral equations we propose an algorithm for solving linear integral equations of the first kind that can be viewed as a variant of marti s method as opposed to that method our algorithm leads to optimal convergence rates also with noisy data 1 introduction throughout this paper let x and y be real hilbert spaces t x y a bounded linear operator with nonclosed range then the problem of determining the best approximate solution of 1 1 tx y is ill posed the best approximate solution exists only for y e d tt r t r t which we assume from now on and depends discontinuously on the right hand side here tt is the moore penrose inverse of t see 12 the best approximate solution is defined as the element of minimal norm that minimizes the residual tx yll and can be written as tty a prominent example for the ill posed

optimal a posteriori parameter choice for tikhonov regularization for solving nonlinear ill posed problems the authors propose an a posteriors strategy for choosing the regularization parameter in tikhonov regularization for solving nonlinear ill posed problems and show that under certain conditions the convergence rate obtained with this strategy is optimal as a by product a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems the authors compare the parameter choice strategy with morozov s discrepancy principle finally numerical results are presented

on an a posteriori parameter choice strategy for tikhonov regularization of nonlinear ill posed problems in the study of the choice of the regularization parameter for tikhonov regularization of nonlinear ill posed problems scherzer engl and kunisch proposed an a posteriori strategy in 1993 to prove the optimality of the strategy they imposed many very restrictive conditions on the problem under consideration their results are difficult to apply to concrete problems since one can not make sure whether their assumptions are valid in this paper we give a further study on this strategy and show that tikhonov regularization is order optimal for each 0 nu le 2 with the regularization parameter chosen according to this strategy under some simple and easy checking assumptions this paper weakens the conditions needed in the existing results and provides a theoretical guidance to numerical experiments

an efficient discretization scheme for solving ill posed problems in this paper we consider a finite dimensional approximation scheme combined with tikhonov regularization for solving ill posed problems error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite dimensional approach

using the l curve for determining optimal regularization parameters the l curve is a plot in ordinary or doubly logarithmic scale of the norm of tikhonov regularized solutions of an ill posed problem versus the norm of the residuals we show that the popular criterion of choosing the parameter corresponding to the point with maximal curvature of the l curve does not yield a convergent regularization strategy to solve the ill posed problem nevertheless the l curve can be used to compute the regularization parameters produced by morozov s discrepancy principle and by an order optimal variant of the discrepancy principle proposed by engl and gfrerer in an alternate way

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

phase separation in random cluster models iii circuit regularity we study the droplet that results from conditioning the subcritical fortuin kasteleyn random cluster model on the presence of an open circuit encircling the origin and enclosing an area of at least or exactly n 2 in this paper we prove that the resulting circuit is highly regular we define a notion of a regeneration site in such a way that for any such site v the circuit cuts through the radial line segment through v only at v we show that provided that the conditioned circuit is centred at the origin in a natural sense the set of regeneration sites reaches into all parts of the circuit with maximal distance from one such site to the next being at most logarithmic in n with high probability the result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations and as such it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in hammond arxiv 1001 1527 2009 arxiv 1001 1528 2009

on the adaptive selection of the parameter in regularization of ill posed problems we study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter as a result a rather general form of a selection criterion is proposed and its relation to the heuristical quasi optimality principle of tikhonov and glasko z vychisl mat mat fiz 4 1964 pp 564 571 and to an adaptation scheme proposed in a statistical context by lepskii theory probab appl 36 1990 pp 454 466 is discussed the advantages of the proposed criterion are illustrated by using such examples as self regularization of the trapezoidal rule for noisy abel type integral equations lavrentiev regularization for nonlinear ill posed problems and an inverse problem of the two dimensional profile reconstruction

optimally scaled vector regularization method to solve ill posed linear problems instead of the tikhonov regularization method which with a scalar being the regularization parameter liu et al 1 have proposed a novel regularization method with a vector as being the regularization parameter as a continuation we further propose an optimally scaled vector regularization method osvrm to solve the ill posed linear problems which is better than the tikhonov regularization method the presently proposed vector regularization method is shown to be well conditioning to overcome the ill posedness of the linear equations system the osvrm causes a significant improvement of stability and accuracy in the numerical solution of ill posed linear problem and its convergence speed is as fast as by solving the well posed linear problem some tests of the linear inverse problems confirm the efficiency and accuracy of the osvrm

lower bounds for boundary roughness for droplets in bernoulli percolation we consider boundary roughness for the droplet created when supercritical two dimensional bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l 2 for large l the maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull we show that for large l this maximum is at least of order l 1 3 logl 2 3 this complements the upper bound of order l 1 3 logl 2 3 proved in al3 for the average local roughness the exponent 1 3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

analysis of acceleration strategies for restarted minimal residual methods we provide an overview of existing strategies which compensate for the deterioration of convergence of minimum residual mr krylov subspace methods due to restarting we evaluate the popular practice of using nearly invariant subspaces to either augment krylov subspaces or to construct preconditioners which invert on these subspaces in the case where these spaces are exactly invariant the augmentation approach is shown to be superior we further show how a strategy recently introduced by de sturler for truncating the approximation space of an mr method can be interpreted as a controlled loosening of the condition for global mr approximation based on the canonical angles between subspaces for the special case of krylov subspace methods we give a concise derivation of the role of ritz and harmonic ritz values and vectors in the polynomial description of krylov spaces as well as of the use of the implicitly updated arnoldi method for manipulating krylov spaces

interface surface tension and reentrant pinning transition in the 2d ising model we develop a new way to look at the high temperature representation of the ising model up to the critical temperature and obtain a number of interesting consequences in the two dimensional case it is possible to use these tools to prove results on phase separation lines in the whole phase coexistence regime by way of a duality transformation we illustrate the power of these techniques by studying an ising model with a boundary magnetic field in which a reentrant pinning transition takes place more precisely we show that the typical configurations of the model can be described at the macroscopic level by interfaces which are solutions of the corresponding thermodynamic variational problem this variational problem is solved explicitly there exist values of the boundary magnetic field and temperatures 0 t 2 but is pinned for t 1 t 3 it is not pinned an important property of the surface tension which is used in this paper is the sharp triangle inequality about which we report some new results the techniques used in this work are robust and can be used in a variety of different situations

application of the newton iteration algorithm to the parameter estimation for dynamical systems this paper uses the newton iteration to study new identification methods for determining the parameters of dynamical systems from step responses on the basis of the step response analysis we present the newton iterative algorithms moreover in order to test the accuracy of the estimated parameters the frequency and step response experiment are applied to the dynamical systems between the estimated and true models the simulation results show that the obtained models can capture the dynamics of the systems i e the estimated model s outputs are close to the outputs of the actual systems this confirms the effectiveness of the proposed newton iterative identification methods

fluctuations of shapes of large areas under paths of random walks we discuss statistical properties of random walks conditioned by fixing a large area under their paths we prove the functional central limit theorem invariance principle for these conditional distributions the limiting gaussian measure coincides with the conditional probability distribution of certain timenonhomogeneous gaussian random process obtained by an integral transformation of the white noise from the point of view of statistical mechanics the studied problem is the problem of describing the fluctuations of the phase boundary in the one dimensional sos model

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance

deconvolution of rutherford backscattering spectra an inverse problem we review the problems associated with resolution correction and discuss some of the most promising procedures to solve them we select four methods to deconvolute heavy particle backscattering spectra i parametrization of the theoretical function ii using histograms with variable bin width iii modified landweber iterations and iv using mollifiers to judge the quality of the methods we treat simulated backscattering spectra as obtained in rutherford backscattering measurements with solid state detectors our results are as follows when a priori information on the shape of the spectra is available parametrization of the problem is superior to all other methods when information on high frequency components of the spectrum e g on sharp edges is of primary interest the use of histograms with variable bin width might provide good results in all other cases the choice of the best procedure depends on the specific problem and our ability to optimize the adjustable parameter of the specific method textcopyright 1996 the american physical society

fluctuations of interfaces and anisotropy the mean square vertical displacement w2 is considered for 1 1 dimensional interfaces of the sos type which make an angle theta with respect to the horizontal axis it is proved that w2 per unit horizontal length is proportional to the inverse of beta gamma theta beta gamma theta where beta is the inverse temperature gamma theta is the surface tension of the interface at angle theta and gamma theta is the second derivative of gamma with respect to its argument this result establishes on a microscopic basis the validity of the corresponding formula proposed by akutsu and akutsu 1986 using thermodynamical arguments

limit theorems in financial market models abstract invariance principle states that a scaled simple random walk converges to the standard brownian motion in this article we present a discrete time stochastic process which reflects a microstructure of market dynamics and prove a convergence to a scaling limit process with a drift term and a jump term these terms are derived from a macroscopic condition on volumes traded in some time intervals the mathematical tools for obtaining our results are dobrushin hryniv theory and the method of cluster expansion developed in mathematical studies of statistical mechanics

fluctuations of the phase boundary in the 2d ising ferromagnet we discuss some statistical properties of the phase boundary in the 2d low temperature ising ferromagnet in a box with the two component boundary conditions we prove the weak convergence in c 0 1 of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them the limiting gaussian measure coincides with the conditional distribution of certain gaussian process obtained by the integral transformation of the white noise

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc

a fast multiscale galerkin method for the first kind ill posed integral equations via iterated regularization in this paper we develop a fast multiscale galerkin method to solve the fredholm integral equations of the first kind via tikhonov regularization the method leads to fast solutions of discrete regularization methods we obtain optimal convergence rates for approximate solutions with an a priori parameter choice and a kind of discrepancy principle finally numerical experiments are given to illustrate the efficiency of the method