Alfred Carasso

• National Institute of Standards and Technology

This paper considers the ill-posed data assimilation problem associated with hyperbolic/parabolic systems describing 2D coupled sound and heat flow. Given hypothetical data at time T > 0, that may not correspond to an actual solution of the dissipative system at time T, initial data at time t = 0 are sought that can evolve, through the dissipative...

A non iterative direct blind deconvolution procedure, previously used successfully to sharpen Hubble Space Telescope imagery, is now found useful in sharpening nanoscale scanning electron microscope (SEM) and helium ion microscope (HIM) images. The method is restricted to images $g(x,y)$, whose Fourier transforms $\hat{g}(\xi,\eta)$ are such that $...

For the 2D incompressible Navier-Stokes equations, with given hypothetical non smooth data at time $T > 0 $that may not correspond to an actual solution at time $T$, a previously developed stabilized backward marching explicit leapfrog finite difference scheme is applied to these data, to find initial values at time $t = 0$ that can evolve into use...

An artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find in...

Publication History Approved by the NIST Editorial Review Board on 2022-07-05 How to cite this NIST Technical Series Publication: Alfred S. Carasso (2022) Data assimilation in 2D nonlinear advection diffusion equations, using an explicit O(∆t) 2 stabilized leapfrog scheme run backward in time. Abstract With an artifcial example of a 2D nonlinear ad...

The 2D viscous Burgers equation is a system of two nonlinear equations in two unknowns, u(x,y,t),v(x,y,t). This paper considers the data assimilation problem of finding initial values [u(⋅,0),v(⋅,0)], that can evolve into a close approximation to a desired target result [u∗(⋅,T),v∗(⋅,T)], at some realistic T>0. Highly nonsmooth target data are cons...

Richardson's leapfrog scheme is notoriously unconditionally unstable in well-posed, forward, linear dissipative evolution equations. Remarkably, that scheme can be stabilized, marched backward in time, and provide useful reconstructions in an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier–Stokes initial value pr...

This paper constructs an unconditionally stable explicit finite difference scheme, marching backward in time, that can solve an interesting but limited class of ill-posed, time-reversed, 2D incompressible Navier–Stokes initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instabi...

This paper constructs an unconditionally stable explicit difference scheme, marching backward in time, that can solve a limited, but important class of time-reversed 2D Burgers' initial value problems. Stability is achieved by applying a compensating smoothing operator at each time step to quench the instability. This leads to a distortion away fro...

This paper develops stabilized explicit marching difference schemes that can successfully solve a significant but limited class of multidimensional, ill-posed, backward in time problems for coupled hyperbolic/parabolic systems associated with vibrating thermoelastic plates and coupled sound and heat flow. Stabilization is achieved by applying compe...

The numerical computation of ill-posed, time-reversed, nonlinear parabolic equations presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable, and result in explosive noise amplification. This paper develops a method for stabilizing the expli...

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equ...

This paper analyses an effective technique for stabilizing pure explicit time differencing in the numerical computation of multidimensional nonlinear parabolic equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers of the negative Laplac...

This paper discusses a two step enhancement technique applicable to noisy Helium Ion Microscope images in which background structures are not easily discernible due to a weak signal. The method is based on a preliminary adaptive histogram equalization, followed by 'slow motion' low-exponent Levy fractional diffusion smoothing. This combined approac...

Photoshop processing of latent fingerprints is the preferred methodology among law enforcement forensic experts, but that appproach is not fully reproducible and may lead to questionable enhancements. Alternative, independent, fully reproducible enhancements, using IDL Histogram Equalization and IDL Adaptive Histogram Equalization, can produce bett...

Linear and nonlinear parabolic advection–dispersion–reaction equations play a significant role in the geophysical sciences, where they are widely used in modeling contaminant transport in subsurface acquifers and rivers, as well as the dispersion of air pollutants from elevated sources. Environmental forensics aims at reconstructing the contaminant...

Identifying sources of ground water pollution, and deblurring nanoscale imagery as well as astronomical galaxy images, are two important applications involving numerical computation of parabolic equations backward in time. Surprisingly, very little is known about backward continuation in nonlinear parabolic equations. In this paper, an iterative pr...

Identifying sources of ground water pollution and deblurring astronomical galaxy images are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particul...

Helium ion microscopes (HIM) are capable of acquiring images with better than 1 nm resolution, and HIM images are particularly rich in morphological surface details. However, such images are generally quite noisy. A major challenge is to denoise these images while preserving delicate surface information. This paper presents a powerful slow motion d...

Generalized Linnik processes and associated logarithmic diffusion equations can be constructed by appropriate Bochner randomization of the time variable in Brownian motion and the related heat conduction equation. Remarkably, over a large but finite frequency range, generalized Linnik characteristic functions can exhibit almost Gaussian behavior ne...

Given a blurred image g(x,y), variational blind deconvolution seeks to reconstruct both the unknown blur k(x,y) and the unknown sharp image f(x,y), by minimizing an appropriate cost functional. This paper restricts its attention to a rich and significant class of infinitely divisible isotropic blurs that includes Gaussians, Lorentzians, and other h...

Most images f(x,y) are not smoothly differentiable functions of x and y, but display edges, localized singularities, and other significant fine-scale roughness, or texture. Correct characterization and calibration of image roughness is vital in many image processing tasks. The L1 Lipschitz exponent α, where 0<α≤1, measures fine-scale image roughnes...

The APEX method is a noniterative direct blind deconvolution technique that can sharpen certain kinds of high-resolution images in quasi real time. The method is predicated on a restricted class of blurs, in the form of 2-D radially symmetric, bell-shaped, heavy-tailed, probability density functions. Not all images can be usefully enhanced with the...

Total variation (TV) image deblurring is a PDE-based technique that preserves edges, but often eliminates vital small-scale information, or {\em texture}. This phenomenon reflects the fact that most natural images are not of bounded variation. The present paper reconsiders the image deblurring problem in Lipschitz spaces $\Lambda(\alpha, p, q)$, wh...

The APEX method is an FFT-based direct blind deconvolution technique that can process complex high resolution imagery in seconds or minutes on current desktop platforms. The method is predicated on a restricted class of shift-invariant blurs that can be expressed as finite convolution products of two-dimensional radially symmetric Lévy stable proba...

Loss of resolution due to image blurring is a major concern in electron microscopy. The point spread function describing that blur is generally unknown. We discuss the use of a recently developed fast Fourier transform (FFT)-based direct (noniterative) blind deconvolution procedure, the APEX method, that can process 512×512 images in seconds of CPU...

Blind deconvolution seeks to deblur an image without knowing the cause of the blur. Iterative methods are commonly applied to that problem, but the iterative process is slow, uncertain, and often ill-behaved. This paper considers a significant but limited class of blurs that can be expressed as convolutions of two-dimensional symmetric Lévy “stable...

Nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods. This paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging, where reconstruction of high frequency detail is of prime interest, and where avoidance of a priori smoot...

This paper examines a wide class of ill-posed initial value problems for partial differential equations, and surveys logarithmic convexity results leading to Hölder-continuous dependence on data for solutions satisfying prescribed bounds. The discussion includes analytic continuation in the unit disc, time-reversed parabolic equations in Lp spaces,...

For ill-posed initial value problems, step by step marching computations are unconditionally unstable, and necessarily blow-up numerically as the mesh is refined. However, for the 1D nonlinear inverse heat conduction problem, the author shows how to construct consistent marching schemes that blow-up much more slowly than the counterpart analytical...

This paper deals with image deblurring when the unknown original image is not smooth and a priori bounds on its derivatives cannot be prescribed in the inversion algorithm. A significant class of such deblurring problems occurring in medical, industrial, military, astronomical, and environmental applications is shown to be equivalent to backwards-i...

The effects of prescribing a physically motivated supplementary constraint, the so-called slow evolution from the continuation boundary (SECB) constraint is presented. The theoretical result obtained is valid for a class of ill-posed continuation problems. It also deals with a class of ill-posed problems in partial differential equations and certai...

A new a-priori constraint sharply reduces noise contamination in ill- posed continuation problems. One application concerns a significant class of image deblurring problems that can be formulated as ill-posed time- reversed parabolic initial value problems.

A new supplementary a-priori constraint, the slow evolution from the boundary constraint, (SEB), sharply reduces noise contamination in a large class of space-invariant image deblurring problems that occur in medical, industrial, surveillance, environmental, and astronomical application. The noise suppressing properties of SEB restoration can be pr...

Recently developed 'slowly divergent' space marching difference schemes, coupled with Tikhonov regularization, can solve the one-dimensional inverse heat conduction problem at values of the nondimensional time step delta-t+ as low as delta-t+ = 0.0003. A Lax-Richtmyer analysis is used to demonstrate dramatic differences in error amplification behav...

Infinitely divisible probability density functions on the half-line t ≥ 0 form a convolution semigroup on t ≥ 0, as they describe stochastic processes with stationary, non-negative, independent increments. A subclass D of such densities are C∞ functions on the whole t-line when extended by zero for t ∞ approximations to the Dirac δ-function, with f...

If [ e − t A ] [{e^{ - tA}}] is a uniformly bounded C 0 {C_0} semigroup on a complex Banach space X X , then − A α , - {A^\alpha }, , 0 > α > 1 0 > \alpha > 1 , generates a holomorphic semigroup on X X , and [ e − t A α ] [{e^{ - t{A^\alpha }}}] is subordinated to [ e − t A ] [{e^{ - tA}}] through the Lévy stable density function. This was proved b...

Experimental identification of dynamic behavior in linear structural systems can be achieved by exciting the structure with a specifically synthesized pulse, and reconstructing the relevant dynamic Green's function by deconvolution of the measured response. The reconstruction procedure involves the solution of an ill-posed integral equation in the...

The effects of noise from interface devices on the performance of the impulse-response acquisition scheme (IRAS) proposed by Carasso (1987) for linear time-invariant systems are investigated analytically. The IRAS employs an infinitely divisible probe waveform and recovers the impulse response by solving the Cauchy problem for the corresponding gen...

This paper presents the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems. The method consists of exciting the structure with specific pulses and reconstructing the dynamic Green's functions by deconvolution of the measured response. The reconstruction procedure involve...

The paper outlines the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems. The method consists of exciting the structure with specific pulses and reconstructing the Green’s functions by deconvolution of the measured response. The reconstruction involves the solution of a...

This paper addresses the problem of determining the impulse response of a linear time invariant system, by probing the system with a causal, C** infinity approximation to the Dirac delta -function. We analyze the ill-posed deconvolution problem which results from a wide choice of possibly multimodal, infinitely divisible, probe pulses. The notion o...

When a C** infinity approximation to the Dirac delta -function, in the form of an inverse Gaussian pulse, is used as input into a linear time invariant system, the output waveform is an approximation to that system's Green's function, in which the singularities have been smoothed out. The ill-posed deconvolution problem for the output signal aims a...

The authors propose a new time domain method for the experimental determination of the 'impulse respose' of linear systems. The technique centers around the use of specifically designed probe waveforms. These waveforms are particular C** infinity approximations to the Diract delta -function and the Heaviside unit step function, and lead to a subseq...

The tethers of tension leg platforms (TLP's) undergoing surge motions are subjected to inertia and hydrodynamic loads. The purpose of this paper is to present an investigation into the effects of the tether curvature caused by these loads. The investigation was conducted by solving the coupled equations of surge motion of the tethers/platform syste...

In this paper we present some preliminary results on a new approach to the problem of characterizing flaws using ultrasonics. The approach takes advantage of the fact that we have control over the time waveform of the probing pulse in an ultrasonic test. It also takes advantage of some special properties of the inverse Gaussian function and an effe...

Estimates of tension leg platform (TLP) surge presented in the literature are generally based on the assumption that the tethers are straight at all times, i.e., that transverse deformations of tethers due to hydrodynamic and inertial forces may be disregarded. The purpose of this note is to verify the validity of this assumption in the case of a d...

We develop and analyze a marching procedure for the numerical computation of backwards parabolic equations with variable coefficients and noisy initial data. The scheme is stable (but inconsistent) and leads to error bounds of logarithmic convexity type for t bounded away from the line t = T, where the solution is only of class L2. The scheme is a...

The inverse heat transfer problem is considered for the case of the one-dimensional heat equation in a quarter plane, and a new solution algorithm is presented together with error bounds of logarithmic convexity type. The method is based on regularizing the initial value problem for the heat equation run sideways. A numerical experiment with rapidl...

We consider the image restoration problem with Gaussian-like point spread functions and reformulate it as an initial value problem for the backwards diffusion equation. This approach leads to rigorous bounds on the reliability of the restoration, as a function of the noise variance, without any assumptions on the spectral characteristics of either...

The snow-plough model is considered for describing cylindrical plasma collapse for a specified constant driving term. This coupled nonlinear system consists of five partial differential equations in two independent variables, one of which is the time variable. It was shown that by direct construction of the unique solution, explicitly in terms of t...

The authors develop a tractable, and mathematically rigorous, asymptotic theory for the development of instability in viscous fluid flow down an inclined plane, at supercritical Reynolds numbers. The theory involves consideration of an ill posed problem, and provides a new example of the way such problems come up in applied mathematics. It is shown...

We construct and analyze an algorithm for the numerical computation of Burgers' equation for preceding times, given an a priori bound for the solution and an approximation to the terminal data. The method is based on the “backward beam equation” coupled with an iterative procedure for the solution of the nonlinear problem via a sequence of linear p...

Consider the following problem. Given the positive constants $\delta $, M, T and $f(x)$ in $L^2 (\Omega )$, find all solutions of $u_t = \Delta u$ in $\Omega \times (0,T]$, $u = 0$ on $\partial \Omega \times (0,T]$, such that $\| {u( \cdot ,T) - f} \|_{L^2 } \leqq \delta $, $\| {u( \cdot ,0) - f} \|_{L^2 } \leqq M$. It is known that if $u_1 (x,t)$,...

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain $\Omega$ in $R^N$, with homogeneous Dirichlet boundary conditions. The method is based on Crank-Nicolson time differencing. To approximately solve the resulting syste...

We construct and analyze a least-squares procedure for approximately solving the initial-value problem for the linearized equations of coupled sound and heat flow, in a bounded domain 2 in A, with homogeneous Dirichlet boundary conditions.The method is based on Crank Nicolson time differencing To approximately solve the resulting system of boundary...

A survey is given of the backward beam equation approach in the numerical computation of backwards parabolic equations. The discussion includes problems with variable coefficients as well as an example of non-linear equation, Burgers' equation. Numerical experiments are presented.

The pure implicit least squares scheme of Bramble and Thomeé can be rendered “A-stable” by approximating the solution at each time step in a different norm. This new norm involves normal derivatives of the solution in the boundary integrals. These derivatives are unknown. By using suitable weights, these derivatives can be given the wrong value zer...

We develop and analyze a least squares procedure for approximating the homogeneous Dirichlet problem for the wave equation in a bounded domain $\Omega$ in $R^N$. This procedure is based on the pure implicit scheme for time differencing. Surprisingly, it is the normal derivative of $u$ rather than $u$ itself which must be included in the boundary fu...

We develop and analyze a least squares procedure for approximating the homogeneous Dirichlet problem for the wave equation in a bounded domain SI in R. This procedure is based on the pure implicit scheme for time differencing. Surprisingly, it is the normal derivative of u rather than u itself which must be included in the boundary functional. This...

We develop and analyze a general procedure for computing selfadjoint parabolic problems backwards in time, given an a priori bound on the solutions. The method is applicable to mixed problems with variable coefficients which may depend on time. We obtain error bounds which are naturally related to certain convexity inequalities in parabolic equatio...

Author discusses the methods for solving the difference equations in one dimensional case.

We analyze the convergence of a ut = [a (x, t) ux ]x + b (x, t) ux - f(x, t, u)u_t = [a (x, t) u_x ]_x + b (x, t) u_x - f(x, t, u) , wherea(x, t) a 0>0, andf u 0, and the solutionu reaches a steady state ast . Such a procedure yields an error estimate, which is uniform int. We also discuss an iterative method of solving the difference equations.

We derive a posteriori bounds for (V - |hat V) and its difference quotient $(V - \hat V)_x$, where V and |hat V are, respectively, the exact and computed solution of a difference approximation to a mildly nonlinear parabolic initial boundary problem, with a known steady-state solution. It is assumed that the computation is over a long interval of t...

Finite-difference methods for parabolic initial boundary problems are usually treated as marching procedures. However, if the solution reaches a known steady state value as t → ∞, one may provide approximate values on a line t = T for a preselected T suitably large. With this extra data, it is feasible to consider the use of elliptic boundary-value...

In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where [ unk ] [{\text {unk}}] has smooth coefficients and a ( x ) ≧ a 0 > 0 a(x) \geqq {a_0} > 0 on [0, 1]. We show that the rate of convergence is O ( Δ x 2 ) O(\Delta {x^2}) as in the selfadjoint case for...

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In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem \begin{equation*}\begin{split}\tag{1} \mathfrak{L}\lbrack u \rbrack \equiv - \lbrack a(x) u' \rbrack' - b(x)u' + c(x)u = \lambda u, & 0 $a(x) \geqq a_0 > 0$ on [ 0, 1 ]. We show that the rate of convergence...