The main goal of this paper is to build consistent and asymptotically normal estimators for the drift and volatility parameter of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space R and the bounded domain (0, π). First, we establish the exact regularity of the solution and its spatial derivative, which in turn, using power-variation arguments, allows building the desired estimators. Second, we propose two sets of estimators, based on sampling either the spatial derivative or the solution itself on a discrete space-time grid. Using the so-called Malliavin-Stein's method, we prove that these estimators are consistent and asymptotically normal as the spatial mesh-size vanishes. More importantly, we show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicitly. We conclude with some numerical experiments that illustrate the theoretical results.

An approach is based on the block discrete cosine transform (DCT). The novelty of this approach is that the transform coefficients of all image blocks are coded and transmitted in absolute magnitude order. The resulting ordered-by-magnitude transmission is accomplished without sacrificing coding efficiency by using partition priority coding. Coding and transmission are adaptive to the characteristics of each individual image. and therefore, very efficient. Another advantage of this approach is its high progression effectiveness. Since the largest transform coefficients that capture the most important characteristics of images are coded and transmitted first, this method is well suited for progressive image transmission. Further compression of the image-data is achieved by multiple distribution entropy coding, a technique based on arithmetic coding. Experiments show that the approach compares favorably with previously reported DCT and subband image codecs.

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In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first N Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as N → ∞.

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