trans dimensional geoacoustic inversion this paper develops a general trans dimensional bayesian methodology for geoacoustic inversion trans dimensional inverse problems are a generalization of fixed dimensional inversion that includes the number and type of model parameters as unknowns in the problem by extending the inversion state space to multiple subspaces of different dimensions the posterior probability density quantifies the state of knowledge regarding inversion parameters including effects due to limited knowledge about appropriate parametrization of the environment and error processes the inversion is implemented here using a reversible jump markov chain monte carlo algorithm and the seabed is parametrized using a partition model unknown data errors are addressed by including a data error model jumps between dimensions are implemented with a birth death methodology that allows transitions between dimensions by adding or removing interfaces while maintaining detailed balance in the markov chain trans dimensional inversion results in an inherently parsimonious solution while partition modeling provides a naturally self regularizing algorithm based on data information content not on subjective regularization functions together this results in environmental estimates that quantify appropriate seabed structure as supported by the data allowing sharp discontinuities while approximating smooth transitions where needed this approach applies generally to geoacoustic inversion and is illustrated here with seabed reflection coefficient data

stochastic inversion for electromagnetic geophysics practical challenges and improving convergence efficiency traditional deterministic geophysical inversion algorithms are not designed to provide a robust evaluation of uncertainty that reflects the limitations of the geophysical technique stochastic inversions which do provide a sampling based measure of uncertainty are computationally expensive and not straightforward to implement for nonexperts nonstatisticians our results include stochastic inversion for magnetotelluric and controlled source electromagnetic data two markov chain sampling algorithms metropolis hastings and slice sampler can significantly decrease the computational expense compared to using either sampler alone the statistics of the stochastic inversion allow for 1 variances that better reveal the measurement sensitivities of the two different electromagnetic techniques than traditional techniques and 2 models defined by the median and modes of parameter probability density functions which produce amplitude and phase data that are consistent with the observed data in general pa

generalized equations and the generalized newton method we give some convergence results on the generalized newton method referred to by some authors as newton s method and the chord method when applied to generalized equations the main results of the paper extend the classical kantorovich results on newton s method to nonsmooth generalized equations our results also extend earlier results on nonsmooth equations due to eaves robinson josephy pang and chan

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

effective hamiltonians and phase diagrams for tight binding models we present rigorous results for several variants of the hubbard model in the strong coupling regime we establish a mathematically controlled perturbation expansion which shows how previously proposed effective interactions are in fact leading order terms of well defined volume independent unitarily equivalent interactions in addition in the very asymmetric falicov kimball regime we are able to apply recently developed phase diagram technology quantum pirogov sinai theory to conclude that the zero temperature phase diagrams obtained for the leading classical part remain valid except for thin excluded regions and small deformations for the full fledged quantum interaction at zero or low temperature moreover the phase diagram is stable against addition of arbitrary but sufficiently small further quantum terms that do not break the ground state symmetries this generalizes and unifies a number of previous results on the subject in particular published results on the zero temperature phase diagram of the falikov kimball model with and without magnetic flux are extended to small temperatures and or small ionic hopping we give explicit expressions for the first few orders in the hopping amplitude of equivalent interactions and we describe the resulting phase diagram our approach yields algorithms to compute equivalent interactions to arbitrarily high order in the hopping amplitude

scalable uncertainty estimation for nonlinear inverse problems using parameter reduction constraint mapping and geometric sampling marine controlled source electromagnetic examples we have developed a new uncertainty estimation method that accounts for nonlinearity inherent in most geophysical problems allows for the explicit search of model posterior space is scalable and maintains computational efficiencies on the order of deterministic inverse solutions we accomplish this by combining an efficient parameter reduction technique a parameter constraint mapping routine a sparse geometric sampling scheme and an efficient forward solver in order to reduce our model domain and determine an independent basis we implement both a typical principal component analysis which factorizes the model covariance matrix and an alternative compression method based on singular value decomposition which acts on training models directly and is storage efficient once we have a reduced base we map parameter constraints from our original model domain to this reduced domain to define a bounded geometric region of feasible model space we utilize an optimal scheme to sample this reduced mo

on primal regularity estimates for set valued mappings abstract we prove several generalizations of the results in 6 for set valued mappings in some cases we improve also the statements for single valued mappings linear openness of the set valued mapping in question is deduced from the properties of its suitable approximation this approach goes back to the classical lyusternik graves theorem saying that a continuously differentiable single valued mapping between banach spaces is linearly open around an interior point of its domain provided that its derivative at this point is surjective in this paper we consider approximations given by a graphical derivative a contingent variation a strict pseudo h derivative and a bunch of linear mappings

newton s method for solving inclusions using set valued approximations results on stability of both local and global metric regularity under set valued perturbations are presented as an application we study super linear convergence of a newton type iterative process for solving generalized equations we investigate several iterative schemes such as the inexact newton s method the nonsmooth newton s method for semismooth functions the inexact proximal point algorithm etc moreover we also cover a forward backward splitting algorithm for finding a zero of the sum of two multivalued not necessarily monotone operators finally a globalization of the newton s method is discussed

on the flux phase conjecture at half filling an improved proof we present a simplification of lieb s proof of the flux phase conjecture for interacting fermion systems such as the hubbard model at half filling on a general class of graphs the main ingredient is a procedure which transforms a class of fermionic hamiltonians into reflection positive form the method can also be applied to other problems which we briefly illustrate with two examples concerning the t v model and an extended falicov kimball model

openness stability and implicit multifunction theorems applications to variational systems abstract in this paper we aim to present two general results regarding on one hand the openness stability of set valued maps and on the other hand the metric regularity behavior of the implicit multifunction related to a generalized variational system then these results are applied in order to obtain in a natural way and in a widely studied case several relations between the metric regularity moduli of the field maps defining the variational system and the solution map our approach allows us to complete and extend several very recent results from the literature

quantum lattice models at intermediate temperature we analyze the free energy and construct the gibbs kms states for a class of quantum lattice systems at low temperature and when the interactions are almost diagonal in a suitable basis the models we study may have continuous symmetries our results however apply to intermediate temperatures where discrete symmetries are broken but continuous symmetries are not our results are based on quantum pirogov sinai theory and a combination of high and low temperature expansions

low temperature expansions for the gibbs states of weakly interacting quantum ising lattice systems low temperature expansions for the gibbs states of weakly interacting transverse ising like models are developed by conditioning the states on a sub algebra of observables the conditioned states have effective classical hamiltonians which are estimated by the solution to a simple implicit equation provided the interaction is sufficiently weak but fixed independent of temperature and the temperature is sufficiently low exponential clustering of the correlation functions holds

stability of frustration free hamiltonians we prove stability of the spectral gap for gapped frustration free hamiltonians under general quasi local perturbations we present a necessary and sufficient condition for stability which we call local topological quantum order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace this result extends previous work by bravyi et al on the stability of topological quantum order for hamiltonians composed of commuting projections with a common zero energy subspace we conclude with a list of open problems relevant to spectral gaps and topological quantum order

inexact newton methods and dennis more theorems for nonsmooth generalized equations in this paper we study local convergence of inexact newton methods of the form big f x k a k x k 1 x k f x k 1 big cap r k x k neq emptyset with a k in cal h x k for solving the generalized equation f x f x ni 0 in banach spaces where the function f is continuous but not necessarily smooth and f is a set valued mapping with closed graph the mapping cal h plays the role of a generalized set valued derivative of f which in finite dimensions may be represented by clarke s generalized jacobian while in banach spaces it may be identified with ioffe s strict prederivative the set valued mappings r k represent inexactness we utilize conditions divided into three groups the first concerns the kind of nonsmoothness of the function f the second involves metric regularity properties of an approximation of the mapping f f and the third is about the sequence of mappings r k under various combinations of these conditions we show linear superlinear or

ekeland s inverse function theorem in graded frechet spaces revisited for multifunctions abstract in this paper we present some inverse function theorems and implicit function theorems for set valued mappings between frechet spaces the proof relies on lebesgue s dominated convergence theorem and on ekeland s variational principle an application to the existence of solutions of differential equations in frechet spaces with non smooth data is given

phase diagram of the two dimensional t t falicov kimball model the ground state phase diagram of the two dimensional falicov kimball model with nearest neighbour and next nearest neighbour hoppings has been studied in the perturbative regime where hoppings are small compared with the on site coulomb interaction the phase diagram at fourth order exhibits a richer structure than the one of the ordinary falicov kimball model

set valued approximations and newton s methods to rm under the assumption of semi smoothness of the mapping we prove that the approximation can be obtained through the clarke generalized jacobian ioffe ralph generalized jacobian b subdifferential and their approximations as an application we propose a generalized newton s method based on the point based set valued approximation for solving nonsmooth equations we show that the proposed method converges locally superlinearly without the assumption of semi smoothness finally we include some well known generalized newton s methods in our method and consolidate the convergence results of these methods

an inverse map result and some applications to sensitivity of generalized equations abstract this work deals with non global inverse map results for the sum of two maps we prove two theorems which shed some new light on this aspect some implications in terms of sensitivity of parametric generalized equations are investigated finally a class of well conditioned operators is identified

conditioning for optimization problems under general perturbations a b s t r a c t given a function f c 1 1 in the literature a parametrized optimization problem involves usually an objective function and constraints depending on parameters the stability results are related to the study of various continuity properties of the solution map with respect to the parameter while the sensitivity analysis refers to the quantitative investigation like for instance the study of lipschitz properties of the solution map generally speaking the concept of well posedness of an optimization problem focuses on existence uniqueness and continuity properties of the solutions both optimal solutions and optimal value with respect to data perturbations and therefore deals with the stability properties see for instance 1 2 hence well posedness is not directly related to the different issues of sensitivity analysis although there are obvious connections between the two research areas on the other hand the condition number represents a measure of the sensitivity of the solutions with respect to small changes in the problem s data and it is usually conceived as an upper bound of the ratio of the size of the solution error to the size of the data error at the same time this number says how we can perturb the initial problem in order to preserve its good properties in the literature the conditioning is essentially related to numerical problems a prototype result involving condition numbers is the well known distance theorem of eckart young see 3 via the norm of the inverse matrix it provides an upper bound on how much a given matrix may be perturbed in order to preserve its nonsingularity this remarkable result has been extended to several problems like computation of eigenvalues and eigenvectors and of zeros of polynomials and to various optimization problems it is worthwhile mentioning in the framework of linear programming the exhaustive paper by renegar 4

iterative estimation of reflectivity and image texture least squares migration with an empirical bayes approach in many geophysical inverse problems smoothness assumptions on the underlying geology are used to mitigate the effects of nonuniqueness poor data coverage and noise in the data and to improve the quality of the inferred model parameters within a bayesian inference framework a priori assumptions about the probabilistic structure of the model parameters can impose such a smoothness constraint analogous to regularization in a deterministic inverse problem we have considered anempirical bayes generalization of the kirchhoff based least squares migration lsm problem we have developed a novel methodology for estimation ofthe reflectivitymodel andregularization parameters using a bayesian statistical framework that treats both of these as random variables to be inferred from the data hence rather than fixing the regularization parameters prior to inverting for the image we allow the data to dictate where to regularize estimating these regularization parameters givesusinformation about the degree of conditional correlation or lack thereof between neighboring image parameters and subsequently incorporating this information in the final model produces more clearly visible discontinuities in the estimated image the inference framework is verified on 2d synthetic data sets in which the empirical bayes imaging results significantly outperform standard lsm images we note that although we evaluated this method within the context of seismic imaging it is in fact a general methodology that can be applied to any linear inverse problem in which there are spatially varying correlations in the model parameter space

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

lie schwinger block diagonalization and gapped quantum chains analyticity of the ground state energy abstract we consider quantum chains whose hamiltonians are perturbations by interactions of short range of a hamiltonian that does not couple the degrees of freedom located at different sites of the chain and has a strictly positive energy gap above its ground state energy for interactions that are form bounded w r t the on site hamiltonian terms we have proven that the spectral gap of the perturbed hamiltonian above its ground state energy is bounded from below by a positive constant uniformly in the length of the chain for small values of a coupling constant see 5 the main result of this paper is that under the same hypotheses the ground state energy is analytic for values of the coupling constant belonging to a sufficiently small disk in the complex plane centred at the origin uniformly in the length of the chain furthermore assuming that the interaction potentials are invariant under translations we prove that in the thermodynamic limit the energy per site is analytic for values of the coupling constant in the same disk in our proof we use a new method introduced in 7 which is based on local lie schwinger conjugations of the hamiltonians associated with connected subsets of the chain indeed to do this we prove a rather strong result concerning complex hamiltonians corresponding to complex values of the coupling constant

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

geometric aspects of quantum spin states a number of interesting features of the ground states of quantum spin chains are analyzed with the help of a functional integral representation of the system s equilibrium states methods of general applicability are introduced in the context of the su 2s 1 invariant quantum spin s chains with the interaction p o wherep o is the projection onto the singlet state of a pair of nearest neighbor spins the phenomena discussed here include the absence of neel order the possibility of dimerization conditions for the existence of a spectral gap and a dichotomy analogous to one found by affleck and lieb stating that the systems exhibit either slow decay of correlations or translation symmetry breaking our representation elucidates the relation evidence for which was found earlier of the p o spin s systems with the potts and the fortuin kasteleyn random cluster models in one more dimension the method reveals the geometric aspects of the listed phenomena and gives a precise sense to a picture of the ground state in which the spins are grouped into random clusters of zero total spin e g within such structure the dichotomy is implied by a topological argument and the alternatives correspond to whether or not the clusters are of finite mean length

on semiregularity of mappings abstract there are two basic ways of weakening the definition of the well known metric regularity property by fixing one of the points involved in the definition the first resulting property is called metric subregularity and has attracted a lot of attention during the last decades on the other hand the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature we provide a self contained material gathering and extending the existing theory on the topic we demonstrate a clear relationship with other regularity properties for example the equivalence with the so called openness with a linear rate at the reference point is shown in particular cases we derive necessary and or sufficient conditions of both primal and dual type we illustrate the importance of semiregularity in the convergence analysis of an inexact newton type scheme for generalized equations with not necessarily differentiable single valued part

segregation in the falicov kimball model the falicov kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on site repulsion alternatively it is a model of itinerant electrons and fixed nuclei it can be seen as a simplification of the hubbard model by neglecting the kinetic hopping energy of the spin up particles one gets the falicov kimball model we show that away from half filling i e if the sum of the densities of both kinds of particles differs from 1 the particles segregate at zero temperature and for large enough repulsion in the language of the hubbard model this means creating two regions with a positive and a negative magnetization our key mathematical results are lower and upper bounds for the sum of the low est eigenvalues of the discrete laplace operator in an arbitrary domain with dirichlet boundary conditions the lower bound consists of a bulk term independent of the shape of the domain and of a term proportional to the boundary therefore one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary for the falicov kimball model this corresponds to having a single compact domain that has no heavy particles

estimations of the free energy for the hubbard model a series of upper bounds and two lower bounds for the partition function of the hubbard model have been derived these bounds are expressible by certain properties of the falicov kimball model the upper bounds have been derived mainly by the use of the golden thompson inequality and its generalizations and the holder inequality the lower bounds are based on the bogoliubov peierls inequality the numerical values of bounds have also been calculated for small systems

two dimensional berezin li yau inequalities with a correction term we improve the berezin li yau inequality in dimension two by adding a positive correction term to its right hand side it is also shown that the asymptotical behaviour of the correction term is almost optimal this improves a previous result by melas 11

unitary dressing transformations and exponential decay below threshold for quantum spin systems parts i and ii this is the continuation of a series of articles concerning a class of quantum spin systems with hamiltonian operators of the form h lambda sum limits x in lambda s x sum limits yo subset lambda lambda left yo right c 1 t yo where is a graph is a small parameter andsx has a gap 1 for all x e i in the singular seti the gap ofsx can be arbitrarily small part iii is devoted to the proof of a preliminary result while in part iv we consider the case in whichi is a subset of finite density of this completes the first iteration step of the deterministic part of the proof of localization in the ground state of the random field quantumxy model

to bayes or not to bayes the goal of geophysical inversion is to make quantitative inferences about the earth from noisy finite data the limitations of noise and the inadequacy of the data mean that geophysical inversion problems are fundamentally problems of statistical inference we do not invert data to find models as much as we might like to we invert data to make inferences about models there will usually be an infinity of models that fit the data thus we must look to probability theory for help

segregation in the asymmetric hubbard model we study the asymmetric hubbard model where hoppings of electrons depend on their spin for strong interactions and sufficiently asymmetric hoppings it is proved that the ground state displays phase separation away from half filling this extends a recent result obtained with freericks and lieb for the falicov kimball model it is based on estimates for the sum of lowest eigenvalues of the discrete laplacian in arbitrary domains

lower bound for the segregation energy in the falicov kimball model in this work a lower bound for the ground state energy of the falicov kimball model for intermediate densities is derived the explicit derivation is important in the proof of the conjecture of segregation of the two kinds of fermions in the falicov kimball model for sufficiently large interactions this bound is given by a bulk term plus a boundary term of the form 1 n where is the region devoid of classical particles and n is the density of electrons a detailed proof is presented for n 1 2 where the coefficient 1 1 2 10 13 is obtained for the two dimensional case although clearly not optimal in terms of order of magnitude this is the largest explicitly calculated coefficient in the range of intermediate densities with suitable modifications the method can also be used to obtain a coefficient for all densities that is the topic of the last section where a sketch of the proof for n 1 2 is shown

some properties of correlations of quantum lattice systems in thermal equilibrium simple proofs of uniqueness of the thermodynamic limit of kms states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures new quantum correlation inequalities for general heisenberg models are described finally a simplified derivation of a general result on power law decay of correlations in 2d quantum lattice systems with continuous symmetries is given extending results of mcbryan and spencer for the 2d classical xy model

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

minimizing the ground state energy of an electron in a randomly deformed lattice we provide a characterization of the spectral minimum for a random schrodinger operator of the form h delta sum i in mathbb z d q x i omega i in l 2 mathbb r d where the single site potential q is reflection symmetric compactly supported in the unit cube centered at 0 and the displacement parameters i are restricted so that adjacent single site potentials do not overlap in particular we show that a minimizing configuration of the displacements is given by a periodic pattern of densest possible 2 d clusters of single site potentials

strong metric subregularity of mappings in variational analysis and optimization abstract although the property of strong metric subregularity of set valued mappings has been present in the literature under various names and with various equivalent definitions for more than two decades it has attracted much less attention than its older siblings the metric regularity and the strong metric regularity the purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications we show that the strong metric subregularity of a mapping f acting between metric spaces is stable under perturbations of the form f f where f is a function with a small calmness constant this result is parallel to the lyusternik graves theorem for metric regularity and to the robinson theorem for strong regularity where the perturbations are represented by a function f with a small lipschitz constant then we study perturbation stability of the same kind for mappings acting between banach spaces where f is not necessarily differentiable but admits a set valued derivative like approximation strong metric q subregularity is also considered where q is a positive real constant appearing as exponent in the definition rockafellar s criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite dimensional spaces a sufficient condition for strong metric subregularity is established in terms of surjectivity of the frechet coderivative and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property in general then various versions of newton s method for solving generalized equations are considered including inexact and semismooth methods for which superlinear convergence is shown under strong metric subregularity as applications to optimization a characterization of the strong metric subregularity of the kkt mapping is obtained as well as a radius theorem for the optimality mapping of a nonlinear programming problem finally an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the pontryagin principle

chain rules for linear openness in general banach spaces this paper is devoted to the investigation of the linear openness of a general type of set valued composition in banach spaces as an application of the main result we study an important issue recently brought into attention by a paper of arutyunov the relation between some assertions on fixed points of set valued compositions and linear openness results of lyusternik graves type then we show on one hand that the chain rule for linear openness we obtain here covers several well known and recent results in the field and on the other hand that it generates new interesting conclusions

implicit multifunction theorems with positively homogeneous maps abstract we extend robinson s and ledyaev and zhu s implicit multifunction theorems using the language of generalized derivatives with positively homogeneous maps allowing us to obtain results that more closely resemble the classical single valued implicit function theorem we highlight that using linear openness instead of metric regularity gives simpler proofs and stronger results as part of our analysis we study perturbations of generalized linear openness and metric regularity finally we discuss how our methods may also be adapted to study generalized calmness

variational bayesian inversion of synthetic 3d controlled source electromagnetic geophysical data abstractinversion of controlled source electromagnetic data is dealt with for a geophysical application the goal is to retrieve a map of conductivity of an unknown body embedded in a layered underground from measurements of the scattered electric field that results from its interaction with a known interrogating wave this constitutes an inverse scattering problem whose associated forward problem is described by means of electric field domain integral equations the inverse problem is solved in a bayesian framework in which prior information is introduced via a gauss markov potts model this model describes the body as being composed of a finite number of different materials distributed into compact homogeneous regions the posterior distribution of the unknowns is approached by means of the variational bayesian approximation as a separable distribution that minimizes the kullback leibler divergence with respect to the posterior law thus we get a parametric model for the distributions of the induced cu

robust and accelerated bayesian inversion of marine controlled source electromagnetic data using parallel tempering bayesian methods can quantify the model uncertainty that is inherent in inversion of highly nonlinear geophysical problems in this approach a model likelihood function based on knowledge of the data noise statistics is used to sample the posterior model distribution which conveys information on the resolvability of the model parameters because these distributions are multidimensional and nonlinear we used markov chain monte carlo methods for highly efficient sampling because a single markov chain can become stuck in a local probability mode we run various randomized markov chains independently to some extent this problem can be mitigated by running independent markov chains but unless a very large number of chains are run biased results may be obtained we got around these limitations by running parallel interacting markov chains with annealed or tempered likelihoods which enable the whole system of chains to effectively escape local probability maxima we tested this approach using a transdimensional algorithm where the number of model parameters as well as the parameters themselves were treated as unknowns during the inversion this gave us a measure of uncertainty that was independent of any particular parameterization we then subset the ensemble of inversion models to either reduce uncertainty based on a priori constraints or to examine the probability of various geologic scenarios we demonstrated our algorithms fast convergence to the posterior model distribution with a synthetic 1d marine controlled source electromagnetic data example the speed up gained from this new approach will facilitate the practical implementation of future 2d and 3d bayesian inversions where the cost of each forward evaluation is significantly more expensive than for the 1d case

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

influence of anisotropic next nearest neighbour hopping on diagonal charge striped phases we consider the model of a strongly correlated system of electrons on a two dimensional square lattice described by an extended falicov kimball hamiltonian where the stability of some axial and diagonal striped phases was proved introducing a next nearest neighbour hopping small enough not to destroy the striped structure we examine rigorously how the presence of the next nearest neighbour hopping anisotropy reduces the 2 rotation degeneracy of the diagonal striped phase the effect appears to be similar to that in the case of anisotropy of the nearest neighbour hopping diagonal stripes are oriented in the direction of the weaker next nearest neighbour hopping

towards variational analysis in metric spaces metric regularity and fixed points the main results of the paper include a a theorem containing estimates for the surjection modulus of a partial composition of set valued mappings between metric spaces which contains as a particlar case well known milyutin s theorem about additive perturbation of a mapping into a banach space by a lipschitz mapping b a double fixed point theorem for a couple of mappings one from x into y and another from y to x which implies a fairly general version of the set valued contraction mapping principle and also a certain different version of the first theorem

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

2 5d forward and inverse modeling for interpreting low frequency electromagnetic measurements we present 2 5d fast and rigorous forward and inversion algorithms for deep electromagnetic em applications that include crosswell and controlled source em measurements the forward algorithm is based on a finite difference approach in which a multifrontal lu decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single source experiment for each frequency of operation when the size of the linear system of equations is large the use of this noniterative solver is impractical hence we use the optimal grid technique to limit the number of unknowns in the forward problem the inversion algorithm employs a regularized gauss newton minimization approach with a multiplicative cost function by using this multiplicative cost function we do not need a priori data to determine the so called regularization parameter in the optimization process making the algorithm fully automated the algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image to increase the robustness of the algorithm we also constrain the minimization and use a line search approach to guarantee the reduction of the cost function after each iteration to demonstrate the pros and cons of the algorithm we present synthetic and field data inversion results for crosswell and controlled source em measurements

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

parallel tempering for strongly nonlinear geoacoustic inversion this paper applies parallel tempering within a bayesian formulation for strongly nonlinear geoacoustic inverse problems bayesian geoacoustic inversion consists of sampling the posterior probability density ppd of seabed parameters to estimate integral properties such as marginal probability distributions based on ocean acoustic data and prior information this sampling is usually carried out using the markov chain monte carlo method of metropolis hastings sampling however standard sampling methods can be very inefficient for strongly nonlinear problems involving multi modal ppds with the potential to miss important regions of the parameter space and to significantly underestimate parameter uncertainties parallel tempering achieves efficient effective sampling of challenging parameter spaces with the ability to transition freely between multiple ppd modes by running parallel markov chains at a series of increasing sampling temperatures with probabilistic interchanges between chains the approach is illustrated for inversion of simulated acoustic reverberation data for which the ppd is highly multi modal while metropolis hastings sampling gives poor results even with very large sample sizes parallel tempering provides efficient convergent sampling of the ppd methods to enhance the efficiency of parallel tempering are also considered

resolution and uncertainty in 1d csem inversion a bayesian approach and open source implementation we show that resolution and uncertainty in csem inversion is most naturally approached using a bayesian framework resolution can be inferred by either hierarchical models with free parameters for effective correlation lengths bayesian smoothing or model choice frameworks applied to variable resolution spatial models bayesian splitting merging we find that typical 1d csem data can be modelled with quite parsimonious models typically o 10 parameters per common midpoint efficient opt imizations for the csem problem must address the challenges of poor scaling strong nonlinearity multimodality and the necessity of bound constraints the posterior parameter uncertainties are frequently controlled by the nonlinearity and linearised approaches to uncertainty are usually very poor in markov chain monte carlo mcmc approaches the nonlinearity and poor scaling make good mixing hard to achieve a novel approximate frequentist method we call the bayesianized parametric bootstrap sometimes called randomized maximum likelihood is much more efficient than mcmc in this problem is considerably better than linearized analysis but tends to modestly overstate uncertainties the software that implements these ideas for the 1d csem problem is made available under an open source license agreement

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

prior information and uncertainty in inverse problems solving any inverse problem requires understanding the uncertainties in the data to know what it means to fit the data we also need methods to incorporate data independent prior information to eliminate unreasonable models that fit the data both of these issues involve subtle choices that may significantly influence the results of inverse calculations the specification of prior information is especially controversial how does one quantify information what does it mean to know something about a parameter a priori in this tutorial we discuss bayesian and frequentist methodologies that can be used to incorporate information into inverse calculations in particular we show that apparently conservative bayesian choices such as representing interval constraints by uniform probabilities as is commonly done when using genetic algorithms for example may lead to artificially small uncertainties we also describe tools from statistical decision theory that can be used to characterize the performance of inv

prior information sampling distributions and the curse of dimensionality this tutorial addresses geometrical issues that concern the specification of high dimensional sampling distributions in bayesian inversion we illustrate that simple low dimensional geometrical concepts that are sometimes used to construct such distributions may become completely distorted and even untrue in higher dimensional problems this has important implications for bayesian inversion if a convenient sampling distribution is constructed using low dimensional geometrical concepts which cause it to differ from the distribution representing our prior information these differences can become extremely expensive to correct in higher dimensions indeed they may make a nonlinear inversion computationally intractable when this need not be the case a crucial factor in bayesian inversion is therefore whether one firmly believes in a particular prior distribution if so this distribution may constitute the most efficient sampling distribution even in cases where it is not straight forward to draw sa

the curse of dimensionality in inverse problems abstract nonlinear inverse problems in real problems in industry have typically a very underdetermined character due to the high number of parameters that are usually needed to achieve accurate forward predictions the corresponding inverse problem is ill posed that is there exist many solutions which are compatible with the prior information fitting the observed data within the same error bounds these solutions are located in one or several flat curvilinear and disconnected valleys of the cost function topography the random sampling of these equivalent models is impossible due to the curse of dimensionality and to the high computational cost needed to provide the corresponding forward predictions this paper generalizes the curse of dimensionality to linear and nonlinear inverse problems outlining the main differences between them with a simple 2d example we show that nonlinearities allow for a reduction in size of the nonlinear equivalence region that could be embedded in a linear hyperquadric with smaller condition number than the corresponding linearized equivalence region we also analyze the effect of the regularization in the posterior sampling and that of the dimensionality reduction which is needed to perform efficient sampling of the region of uncertainty equivalence in high dimensional problems we hope that the additional theoretical knowledge provided by this research will help practitioners to design more efficient methods of sampling

berezin li yau inequalities on domains on the sphere abstract we prove berezin li yau inequalities for the dirichlet and neumann eigenvalues on domains on the sphere s d 1 a sharp explicit bound for the sums of the neumann eigenvalues is obtained for all dimensions d in the case of s 2 we also obtain sharp lower bounds with correction terms for the vector laplacian and the stokes operator

on fixed point stability for set valued contractive mappings with applications to generalized differential equations where d x c inf d x c a set valued mapping t x c x is a ci 1 and for x y e x if x is complete then every set valued contraction has a fixed point i e a point x with x e tx the set of fixed points of t will be denoted by f t stability of fixed points of set valued contractions was investigated in 12 and lo in lo a stability theorem was proved under some rather restrictive conditions including i the domain of the maps being a closed convex bounded subset of a hilbert space ii the image of each point under each map being a closed convex subset theorem 1 below removes these conditions

low temperature phase diagrams of quantum lattice systems i stability for quantum perturbations of classical systems with finitely many ground states starting from classical lattice systems ind 2 dimensions with a regular zerotemperature phase diagram involving a finite number of periodic ground states we prove that adding a small quantum perturbation and or increasing the temperature produce only smooth deformations of their phase diagrams the quantum perturbations can involve bosons or fermions and can be of infinite range but decaying exponentially fast with the size of the bonds for fermions the interactions must be given by monomials of even degree in creation and annihilation operators our methods can be applied to some anyonic systems as well our analysis is based on an extension of pirogov sinai theory to contour expansions ind 1 dimensions obtained by iteration of the duhamel formula

low temperature expansions for the gibbs states of quantum ising lattice systems by conditioning on a subalgebra of observables we develop low temperature expansions for certain quantum ising models which include transverse ising models and anisotropic xy models these expansions are majorized by the series for functions solving certain nonlinear partial differential equations convergence for the majorizing series follows from cauchy kowalevski theory applied to the partial differential equations the low temperature expansions imply long range order