nonlinear parametric inversion using interpolatory model reduction nonlinear parametric inverse problems appear in several prominent applications one such application is diffuse optical tomography dot in medical image reconstruction such inverse problems present huge computational challenges mostly due to the need for solving a sequence of large scale discretized parametrized partial differential equations in the forward model in this paper we show how interpolatory parametric model reduction can significantly reduce the cost of the inversion process in dot by drastically reducing the computational cost of solving the forward problems the key observation is that function evaluations for the underlying optimization problem may be viewed as transfer function evaluations along the imaginary axis a similar observation holds for jacobian evaluations as well this motivates the use of system theoretic model order reduction methods we discuss the construction and use of interpolatory parametric reduced models as surrogates for the full forward model within the dot s

an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

accelerating optimization of parametric linear systems by model order reduction design optimization problems are often formulated as an optimization problem whose objective is a function of the output of a large scale parametric linear system obtained from the discretization of a pde to reduce the high computational cost of the objective and its gradient model order reduction techniques can be used this paper uses interpolatory reduced models as surrogate models in an optimization procedure we replace the standard first order condition by the relaxed first order condition which is more suitable when algebraic reduced models are used as surrogate models the relaxed first order condition imposes that the approximation quality of the surrogate model at the interpolation point can be measured and refined and that the surrogate model is equipped with an error bound on the entire parameter space we propose two optimization algorithms one uses the error bound to define a trust region and the other penalizes the objective with the error bound we prove convergence of both methods un

on level set regularization for highly ill posed distributed parameter estimation problems the recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward pdes is fraught with theoretical and practical difficulties better results are obtained for problems where the solution may take on at each point only one of two values thus yielding a shape recovery problem this article considers level set regularization for such problems however rather than explicitly integrating a time embedded pde to steady state which typically requires thousands of iterations methods based on gauss newton are applied more directly one of these can be viewed as damped gauss newton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a tikhonov type regularization with a sharpening sub step at each iteration in practice this method is eclipsed however by a special finite time or levenberg marquardt type method which we call dynamic regularization applied to the output least squares formulation our stopping criterion for the iteration does not involve knowledge of the true solution the regularization functional is applied to the smooth level set function rather than to the discontinuous function to be recovered and the second focus of this article is on selecting this functional typical choices may lead to flat level sets that in turn cause ill conditioning we propose a new quartic non local regularization term that penalizes flatness and produces a smooth level set evolution and compare its performance to more usual choices two numerical test cases are considered a potential problem and the classical eit dc resistivity problem

a new selection operator for the discrete empirical interpolation method improved a priori error bound and extensions this paper introduces a new framework for constructing the discrete empirical interpolation method sf deim projection operator the interpolation node selection procedure is formulated using the qr factorization with column pivoting and it enjoys a sharper error bound for the sf deim projection error furthermore for a subspace mathcal u given as the range of an orthonormal mathsf u the sf deim projection does not change if mathsf u is replaced by mathsf u omega with arbitrary unitary matrix omega in a large scale setting the new approach allows modifications that use only randomly sampled rows of mathsf u but with the potential of producing good approximations with corresponding probabilistic error bounds another salient feature of the new framework is that robust and efficient software implementation is easily developed based on readily available high performance linear algebra packages

inexact solves in interpolatory model reduction we investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem we give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods we show that when a petrov galerkin framework is employed for the inexact solves the associated reduced order model is an exact interpolatory model for a nearby full order system thus demonstrating backward stability we also give evidence that for h2 optimal interpolation points interpolatory model reduction is robust with respect to perturbations due to inexact solves finally we demonstrate the effectiveness of direct use of inexact solves in optimal h2 approximation the result is an effective model reduction strategy that is applicable in realistically large scale settings

on the existence and uniqueness of solutions of a class of fractional differential equations abstract in this paper we investigate the existence and uniqueness of solutions for the following class of multi order fractional differential equations d 1 1 1 d n n n u t i 1 n d i i i u t d i n i i u t f t u t t 0 1 u 0 0 i 1 n i 1 i 0 i 0 1 i n where d i n i i denotes the generalized erdelyi kober operator of fractional derivative of order i moreover some properties concerning the positive maximal minimal and continuation of solutions are obtained

solution of the time domain inverse resistivity problem in the model reduction framework part i one dimensional problem with siso data many time domain problems in engineering applications can be described by means of parameter dependent time invariant dynamic systems we are interested in parameter estimation by fitting available transient measurements using the nonlinear least square method as the main application we consider the control source electromagnetic csem method of geophysical exploration governed by the diffusion maxwell system where the unknown parameters describe the spatial distribution of electrical resistivity we propose a novel model reduction approach for constructing an efficient approximation of the jacobian the reduction is based on subspace projection of the state variable and it allows us to split the time and space dependence of the derivative we examine several popular rational krylov subspaces and single out the h 2 optimal subspace that not only minimizes the approximation error but completely annuls its influence on the inversion result preliminary numerical experiments with a simplified one dim

the morley element for fourth order elliptic equations in any dimensions in this paper the well known nonconforming morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion the general n dimensional morley element consists of all quadratic polynomials defined on each n simplex with degrees of freedom given by the integral average of the normal derivative on each n 1 subsimplex and the integral average of the function value on each n 2 subsimplex explicit expressions of nodal basis functions are also obtained for this element on general n simplicial grids convergence analysis is given for this element when it is applied as a nonconforming finite element discretization for the biharmonic equation

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

h2 tf optimality conditions for a finite time horizon abstract in this paper we establish the interpolatory model reduction framework for optimal approximation of mimo dynamical systems with respect to the h 2 norm over a finite time horizon denoted as the h 2 t f norm using the underlying inner product space we derive the interpolatory first order necessary optimality conditions for approximation in the h 2 t f norm then we develop an algorithm which yields a locally optimal reduced model that satisfies the established interpolation based optimality conditions we test the algorithm on various numerical examples to illustrate its performance

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators in this paper we mainly investigate controlled fractional evolution hemivariational inequalities of degenerate type in caputo and riemann liouville fractional derivatives of order in 1 2 respect

a multiscale method for model order reduction in pde parameter estimation abstract estimating parameters of partial differential equations pdes is of interest in a number of applications such as geophysical and medical imaging parameter estimation is commonly phrased as a pde constrained optimization problem that can be solved iteratively using gradient based optimization a computational bottleneck in such approaches is that the underlying pdes need to be solved numerous times before the model is reconstructed with sufficient accuracy one way to reduce this computational burden is by using model order reduction mor techniques such as the multiscale finite volume method msfv in this paper we apply msfv for solving high dimensional parameter estimation problems given a finite volume discretization of the pde on a fine mesh the msfv method reduces the problem size by computing a parameter dependent projection onto a nested coarse mesh a novelty in our work is the integration of msfv into a pde constrained optimization framework which updates the reduced space in each iteration we present a computationally tractable way of explicitly differentiating the mor solution that acknowledges the change of basis as we demonstrate in our numerical experiments our method leads to computational savings for large scale parameter estimation problems where iterative pde solvers are necessary and offers potential for additional speed ups through parallel implementation

a regularized gauss newton trust region approach to imaging in diffuse optical tomography we present a new algorithm for the solution of nonlinear least squares problems arising from parameterized imaging problems with diffuse optical tomographic data d boas et al ieee signal process mag 18 2001 pp 57 75 the parameterization arises from the use of parametric level sets for regularization m e kilmer et al proc spie 5559 2004 pp 381 391 a aghasi m e kilmer and e l miller siam j imaging sci 4 2011 pp 618 650 such problems lead to jacobians that have relatively few columns a relatively modest number of rows and are ill conditioned moreover such problems have function and jacobian evaluations that are computationally expensive our optimization algorithm is appropriate for any inverse or imaging problem with those characteristics in fact we expect our algorithm to be effective for more general problems with ill conditioned jacobians the algorithm aims to minimize the total number of function and jacobian evaluations by analyzing which spectral components of the gauss newton direction should be discarded or damped the analysis considers for each component the reduction of the objective function and the contribution to the search direction restricting the computed search direction to be within a trust region the result is a truncated svd like approach to choosing the search direction however we do not necessarily discard components in order of decreasing singular value and some components may be scaled to maintain fidelity to the trust region model our algorithm uses the basic trust region algorithm from a r conn n i m gould and ph l toint trust region methods siam philadelphia 2000 we prove that our algorithm is globally convergent to a critical point our numerical results show that the new algorithm generally outperforms competing methods applied to the dot imaging problem with parametric level sets and regularly does so by a significant factor

dynamical analysis of fractional order modified logistic model in this paper we study a fractional differential equation model of the single species multiplicative allee effect first we study the stability of equilibrium points further we give some sufficient conditions ensuring the existence and uniqueness of integral solution in the last section we perform several numerical simulations to validate our analytical findings

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

stable finite element pair for stokes problem and discrete stokes complex on quadrilateral grids in this paper we construct a stable finite element pair for incompressible stokes problem and a discrete stokes complex associated with that pair on general quadrilateral grids the finite element spaces involved consist of piecewise polynomials only and the divergence free condition is imposed in a primal formulation these constructions can be generated onto grids that consist of both triangular and quadrilateral cells

three limit cycles in a leslie gower predator prey model with additive allee effect in this work a bidimensional continuous time differential equations system is analyzed which is derived of leslie type predator prey schemes by considering a nonmonotonic functional response and allee effect on population prey for the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant the only quadrant of interest for the sake of realism we show that under certain conditions over the parameters the system allows the existence of three limit cycles the first two cycles are infinitesimal ones generated by hopf bifurcation the third one arises from a homoclinic bifurcation furthermore we give conditions over the parameters such that the model allows long term extinction or survival of both populations in particular the presence of a weak allee effect does not imply extinction of populations necessarily for our model

asymptotically non expansive self maps and global stability with ultimate boundedness of dynamic systems this paper investigates self maps t x x which satisfy a distance constraint in a metric space with mixed point dependent non expansive properties or in particular contractive ones and potentially expansive properties related to some distance threshold the above mentioned constraint is feasible in certain real world problems of usefulness for instance when discussing ultimate boundedness in dynamic systems which guarantees lyapunov stability this fact makes the proposed analysis to be potentially useful to investigate global stability properties in dynamic systems in the potential presence of some locally unstable equilibrium points the results can be applied to stability problems of dynamics systems and circuit theory as the given examples suggest

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh