Density control and local regularity

We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension d ≤ 3, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen–Loève expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen–Loève expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipschitz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finite-dimensional Karhunen–Loève truncation.

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

An extension of the conventional linear and nonlinear scale-space models for shape and image simplification, indexing and retrieval is presented. The linear model can be used for shape based retrieval when the main video objects have been identified. In this context an algorithm for contour simplification is introduced. Two different nonlinear filtering techniques for image simplification are also described. The first one is generated by convolution with a group of anisotropic weighted filter kernels. The second is based on a parabolic differential equation in divergence form. The linear and nonlinear form of these diffusion filters allows to integrate additional information to control the evolution in both spatial and temporal directions. This property is used to extend the conventional nonlinear scale-space in order to perform content-based segmentation of natural video. (5 pages)

We define entropy as a measure of the disorder of a system. Disorder is the lack of order. And what is order? To answer this question we must ask a second one: order with respect to what? Suppose we have several large and small balls. The large balls are put in one corner of a flat container and the small balls in another corner (Figure 9.1(a)). Such a system is ordered with respect to the size of the balls. In contrast, suppose the balls are all mixed up, so that the neighbors of a particular ball can be a large ball or a small ball with equal likelihood (Figure 9.1(b)). This system is disordered.

In [GG], Giaquinta & Giusti introduced the notion of quasiminima, a concept which is very useful in unifying the C 0,α-regularity theory of elliptic equations in divergence form. In particular, their treatment provides regularity results for minima of nondifferentiable variational integrals. They also show that weak solutions of nonlinear elliptic equations in divergence form, under natural and general structural conditions, are quasiminima as are solutions to a wide class of obstacle problems. Because of the great interplay between quasiminima and the Holder theory of partial differential equations, it is important to develop the subject fully. Indeed, significant progress has already been made. Recently, for example, Harnack inequalities have been established in [DBT] for sub and super-quasiminima which run parallel to the results obtained by Trudinger [T] for weak sub and supersolutions of quasilinear equations in divergence form. In particular, the results in [DBT] imply that quasiminima are Holder continuous, a result first established by Giaquinta & Giusti, [GG].

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

Rearrangements of functions have proved to be a fairly interesting tool in analysis. Systematically introduced by Hardy & Littlewood, they have been used by a number of authors in real and harmonic analysis, in investigations about singular integrals, function spaces and interpolation theory. See [2, 3, 5–8] for instance. Polya & Szego and their followers have demonstrated a good many isoperimetric theorems and inequalities by means of rearrangements—[9] is a source book for this matter. More recent investigations have shown that rearrangements of functions also fit well into the theory of elliptic second-order partial differential equations. See [4, 11], for example, and the references cited therein.

This paper treats the asymptotic behavior of solutions of a linear secondorder elliptic partial differential equation defined on a two-dimensional semiinfinite strip. The equation has divergence form and variable coefficients. Such equations arise in the theory of steady-state heat conduction for inhomogeneous anisotropic materials, as well as in the theory of anti-plane shear deformations for a linearized inhomogeneous anisotropic elastic solid. Solutions of such equations that vanish on the long sides of the strip are shown to satisfy a theorem of Phragmen-Lindelof type, providing estimates for the rate of growth or decay which are optimal for the case of constant coefficients. The results are illustrated by several examples. The estimates obtained in this paper can be used to assess the influence of inhomogeneity and anisotropy on the decay of end effects arising in connection with Saint-Venant's principle.

Objective To evaluate the image quality and registration accuracy of a three-dimensional (3D) dynamic phantom in four-dimensional computed tomography (4DCT) and four-dimensional cone-beam computed tomography (4DCBCT). Methods The Computerized Imaging Reference Systems Dynamic Thorax Phantom Model 008A was scanned to get 4DCT and 4DCBCT images. Two balls with different diameters (O= 1 cm and O= 2 cm) were used to simulate tumors with different sizes. The motion mode of the balls was 3D sinusoidal motion at 0.25 Hz (the amplitudes along the x, y, and z axes were ±1.0 cm, ±0.4 cm, and ±0.2 cm, respectively). Gross target volumes (GTVs) from 10-phase bins, internal gross target volumes (IGTV), and target volumes on maximum intensity projection (MIP) and mean intensity projection (MeanIP) images were contoured and calculated. Target volumes on 4DCT or 4DCBCT images were compared with the static and dynamic volumes of the balls (VS and VD). The matching index (MI) of target volumes between the 4DCT and 4DCBCT images was analyzed after rigid image registration. Results The GTV in each phase of the image was larger than VS. The difference between the average GTV derived from 10 phases of 4DCT or 4DCBCT images and Vs of the small ball was larger than that of the large ball (35.03% vs. 22.66%; 32.62% vs. 17.00%). All the IGTVs and target volumes on MIP images were slightly larger than VD, but target volumes on MeanIP images were smaller than VD. The average MI of 10-phase bins of the small ball was smaller than that of the large ball (66.76% vs. 82.21%). Moreover, MIs of IGTV, MIP, and MeanIP of the small ball were also smaller than those of the large ball (77.39% vs. 90.29%; 75.90% vs. 89.28%; 74.47% vs. 82.74%). Conclusions In the case of a relatively small tumor volume and a relatively large motion amplitude, 4DCT and 4DCBCT should be used with caution for comparison of image registration. Key words: Tomography, X-ray computed; Tomography, X-ray computed, cone-beam; Imaging registration; Phantom

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.

Optimal Holder regularity results on solutions to second order equations in divergence form with measurable coefficients are presented. Caccioppolli inequality on level sets and De Girogis method along with Moser iteration are discussed. Harnack inequality and local boundedness of solutions are also studied at the end.

Deep least-squares methods: An unsupervised learning-based numerical method for solving elliptic PDEs

The discretization of second-order elliptic partial differential equations over three-dimensional rectangular regions, in general, leads to very large sparse linear systems. Because of their huge order and their sparseness, these systems can only be solved by iterative methods using powerful computers, e.g., vector supercomputers. Most of those methods are only attractive when used in combination with a so-called preconditioning matrix. Unfortunately, the more effective preconditioners, such as successive over-relaxation and incomplete decompositions, do not perform very well on most vector computers if used in a straightforward manner. In this paper it is shown how a rather high performance can be achieved for these preconditioners.

Data-dependent triangulations for scattered data interpolation and finite element approximation

Discontinuous Galerkin (DG) finite element methods were studied by many researchers for second-order elliptic partial differential equations, and a priori error estimates were established when the solution of the underlying problem is piecewise $H^{3/2+\epsilon}$ smooth with $\epsilon>0$. However, elliptic interface problems with intersecting interfaces do not possess such a smoothness. In this paper, we establish a quasi-optimal a priori error estimate for interface problems whose solutions are only $H^{1+\alpha}$ smooth with $\alpha\in(0,1)$ and, hence, fill a theoretical gap of the DG method for elliptic problems with low regularity. The second part of the paper deals with the design and analysis of robust residual- and recovery-based a posteriori error estimators. Theoretically, we show that the residual and recovery estimators studied in this paper are robust with respect to the DG norm, i.e., their reliability and efficiency bounds do not depend on the jump, provided that the distribution of coefficients is locally quasi-monotone.