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In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

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The failure of melanoma immunotherapy can be mediated by immunosuppression in the tumor microenvironment (TME), and insufficient activation of effector T cells against the tumor. Here, we show that inhibition of galectin-3 (gal-3) enhances the infiltration of T cells in TME and improves the sensitivity of anti-PD-L1 therapy. We identify that RNF8 downregulated the expression of gal-3 by K48-polyubiquitination and promoted gal-3 degradation via the ubiquitin–proteasome system. RNF8 deficiency in the host but sufficiency in implanted melanoma results in immune exclusion and tumor progression due to the upregulation of gal-3. Upregulation of gal-3 decreased the immune cell infiltration by restricting IL-12 and IFN-γ. Inhibition of gal-3 reverses immunosuppression and induces immune cell infiltration in the tumor microenvironment. Moreover, gal-3 inhibitor treatment can increase the sensitivity of PD-L1 inhibitors via increasing immune cell infiltration and enhancing immune response in tumors. This study reveals a previously unrecognized immunoregulation function of RNF8 and provides a promising strategy for the therapy of “cold” tumors. Tremendous effects of melanoma treatment can be achieved by facilitating immune cell infiltration combined with anti-PD-L1 treatment.