Questions tagged [regularization]
The regularization tag has no summary.
76 questions
0 votes
0 answers
66 views
Is the norm of the gradient of the Moreau-envelope non-decreasing in $\lambda$?
Let $f$ be an $\rho$-weakly convex function, $f:\mathbb{R}^d\to \mathbb{R}$. The Moreau envelope of $f$, $f_\lambda$ for a parameter $0<\lambda<1/\rho$ is defined as $$ f_{\lambda}(x) := \min_{y}...
- 1
1 vote
0 answers
111 views
Parallel between Pauli-Villars regularization and Wilson's lattice fermions – any deeper mathematical significance?
I recently noticed a curious parallelism between the Pauli-Villars regularization used in QFT and Wilson's formulation of lattice fermions. Basically, in Pauli-Villars regularization, one replaces (in ...
- 3,750
0 votes
0 answers
59 views
Effective 'generator' of semigroup restricted to non-invariant subspace
Let $H: \mathcal{D} \rightarrow \mathcal{H}$ be a densely defined, self-adjoint, non-negative operator. Let $P: \mathcal{H} \rightarrow \mathcal{H} $ be an orthogonal projection onto a subspace. We ...
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3 votes
0 answers
120 views
Effective action of unbounded operators on subspaces outside their domains of definition
Consider a densely defined, self-adjoint operator $$ H: \mathcal{D} \rightarrow \mathscr{H}. $$ Assume for simplicity that $H$ is nonnegative. We want to effectively restrict this operator $H$ to a ...
- 143
2 votes
1 answer
350 views
New (?) Regularization Method for Divergent Series [closed]
Playing with identities ($1$) and ($2$) from this blog post and infinite geometric series, I've noticed the following. For $x > 1$, the following series is convergent: $$\sum_{n=0}^{\infty} e^{(2n ...
1 vote
0 answers
181 views
Laplace transform
\begin{equation} \begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
- 21
1 vote
1 answer
264 views
Derivative of Moreau envelope in Hilbert space with respect to regularization parameter $\lambda$ using Hopf-Lax formula?
Let $\mathcal H$ be a Hilbert space, $f \colon \mathcal H \to (- \infty, \infty]$ a proper, convex, lower semi-continuous function and $\lambda > 0$. The $\lambda$-Moreau envelope of $f$ is $$ f_{\...
- 87
6 votes
0 answers
308 views
$\log\det$ asymptotics of a skew-circulant matrix with additive diagonal bimodal disorder
I'd like to share a problem that I have been dealing with for a longer time now. In the framework of quenched disorder in the square-lattice Ising model I want to calculate, for large even $M$, the ...
- 5,309
1 vote
0 answers
60 views
Construction of a regulariser for the boundary integral operator $\lambda\mathrm{Id} - K'$
$\newcommand\Id{\mathrm{Id}}$Assumptions and Notations : $\Omega$ is a bounded Lipschitz domain in $\mathbb R^2$, $\Gamma$ denotes its boundary and $n$ is the normal vector to the boundary $\Gamma$, ...
- 63
1 vote
0 answers
679 views
How to show that the trace of a regularized Laplacian defined on two sphere with radius $h\geq 1$ is diverging logarithmically?
Let $h,m\in[1,\infty)$. I would like to verify that the following sum diverges logarithmically \begin{equation} \sum_{d=0}^{\infty} \frac{2d+1}{2h^2(1+\frac{d(d+1)}{h^2})(1+\frac{d(d+1)}{h^2m ^2})^{2}}...
- 321
37 votes
2 answers
5k views
1+2+3+4+… and −⅛
Is there some deeper meaning to the following derivation (or rather one-parameter family of derivations) associating the divergent series $1+2+3+4+…$ with the value $-\frac 1 8$ (as opposed to the ...
- 20.1k
3 votes
0 answers
226 views
The divergent sum $\sum_{n=1}^\infty (-1)^n (n^2)! x^n$
Question I'm interested in assigning a value to the divergent series $F(x)=\sum_{n=1}^\infty (-1)^n (n^2)! x^n$. I'm hoping that (1) the definition for $F(x)$ has (one-sided) derivatives of $(-1)^n (n^...
- 1,800
2 votes
2 answers
338 views
Assigning values to divergent oscillating integrals
I have recently run into a number of divergent oscillating integrals in various contexts. Thus, I have been led to desire general methods for assigning values to divergent oscillating integrals. All ...
- 1,800
6 votes
0 answers
347 views
Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
- 10.4k
1 vote
2 answers
460 views
What's the true regularized value of product of all natural numbers?
Muñoz Garcia and Pérez-Marco - The product over all primes is $4\pi^2$ claims that the regularized value of product $\prod_{k=1}^\infty k$ is $\sqrt{2\pi}$ and of $\prod_{k=1}^\infty p_k$ over primes $...
- 10.4k
3 votes
0 answers
128 views
Regularised value of cardinality of non trivial Zeta zeros:
This is a straight forward question so apologies in advance Consider the following sums: $$\sum_k1_{\rho_k}$$ $$\sum_k{\rho_k}$$ (i.e. first sum counts non trivial zeros of Zeta function) I want to ...
- 790
4 votes
1 answer
462 views
Derivative of Cauchy PV is equivalent to Hadamard regularization?
Let $\mathcal C$ and $\mathcal H$ denote the Cauchy principal value and Hadamard finite part. According to the Wiki: $$ {\frac {\mathrm d}{\mathrm dx}}\left({\mathcal {C}}\int _{{a}}^{{b}}{\frac {...
0 votes
0 answers
51 views
Normalizing a parameter in a regression
I am thinking about the possibility of making a parameter in my regression, let's say the $\lambda$ in a ridge regression, somehow, inside a range : $\lambda \in [0,1]$. Do you have any ideas how I ...
- 1
3 votes
0 answers
389 views
Evaluating $\sum_{n=0}^\infty n^k n!$ in p-adics, and its connection to the summation of divergent series
Often, in the discussion of the regularization of the geometric series it is mentioned that $\sum_{n=0}^\infty p^n$ converges in the p-adics, and indeed, that it converges to $\frac{1}{1-p}$. I had ...
- 1,800
1 vote
0 answers
285 views
What's the regularized value of these divergent integrals: $\int_0^\infty \ln x \, dx$ and $\int_0^\infty \frac{\ln x}{x^2} \, dx$?
When playing with divergent integrals $\int_0^\infty f(x) \, dx$ and their transformations with operators $\int_0^\infty\mathcal{L}_t[t f(t)](x) \, dx$ and $\int_0^\infty\frac1x\mathcal{L}^{-1}_t[ f(t)...
- 10.4k
3 votes
2 answers
558 views
A proposition for summing divergent series, but how should partial summation be defined at non-natural values?
Introduction I have been in search of methods of assinging values to divergent series that have a nice intuitive or geometric interpretation. One fairly straightforward method I've considered for ...
- 1,800
12 votes
1 answer
1k views
Divergent series summation beyond natural boundaries
I'm hoping to investigate the effects of divergent summation methods on series which cannot be analytically continued due to a dense set of singularities. At least a priori, it doesn't seem that a ...
- 1,800
10 votes
2 answers
2k views
Value of divergent sum $\sum_{n=0}^\infty (-1)^n n^n$
I'm hoping to find a reasonable value to assign to the divergent series $\sum_{n=0}^\infty (-1)^n n^n$ and $\sum_{n=0}^\infty (-1)^n (xn)^n$. For the first one, I have obtained something around 0.71, ...
- 1,800
2 votes
0 answers
278 views
Hypermodulus and what mathematical objects have it
When researching divergent integrals, I decided to introduce a concept of "modulus" or "determinant" of divergent integral (and series). Basically, it is the exponent of the real ...
- 10.4k
5 votes
1 answer
308 views
Improving regularity of the boundary of a convex set in Riemannian manifolds
Let $X$ be a geodesically complete Riemannian manifold (we may assume that $X$ is simply connected and negatively curved, although I don't think it matters). Given a closed, convex subset $K \subset X$...
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3 votes
2 answers
343 views
Does this method analytically continue gap series series?
I was looking for ways to continue gap series, and it seemed to be that they could be continued outside of the boundary by simply turning $$f(x)= \sum_{n=0}^\infty x^{n^k}$$ into $$g(x) =- \sum_{n=1}^\...
- 1,800
1 vote
0 answers
321 views
How is this expression for the regularization of integrals of monomials, given in a paper, justified? How strong is argument in favor?
In this answer by Carlo Beenakker he cites the following regularization formula: $$\int_0^\infty x^p\,dx\mathrel{"="}\frac{(-1)^{p+1}}{(p+1)(p+2)},\;\;p=0,1,2,\dotsc,$$ citing Tafazoli - Calculation ...
- 10.4k
1 vote
0 answers
139 views
What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?
I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs. So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
- 10.4k
2 votes
0 answers
274 views
Did anyone ever propose the distinction between "divergent to infinity" as opposed to "divergent but with finite average"?
There are different regularization methods that allow us to ascribe finite values to divergent integrals, series or sequences. Still, in my view there is fundamental difference between divergent ...
- 10.4k
-1 votes
1 answer
330 views
A question on assigning finite values to divergent sums involving expression of primes
We know the following: $$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$ This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$. ...
- 149
2 votes
0 answers
215 views
Regularization of the area under hyperbola
So, I am trying to find the regularized value of the divergent integral $I=\int_1^\infty \sqrt{x^2-1}dx$. Since the area of $\int_0^1 \sqrt{1-x^2}dx=\frac\pi4$, I wonder whether the area under ...
- 10.4k
0 votes
1 answer
489 views
A set of divergent integrals that I think, equal to $-\gamma$
So, we take $\frac{\text{sgn}(x-1)}{x}$ and apply $\mathcal{L}_t[t f(t)](x)$ four times. The transform is known to keep area under the curve. These integrals, I think, are equal to minus Euler-...
- 10.4k
1 vote
2 answers
368 views
Can we meaningfully ascribe values to these divergent integrals?
My gut feeling is that $\int_0^\infty (1-\frac1{x^2})dx=0$ $\int_0^\infty (x-\frac2{x^3})dx=0$ $\int_0^\infty (x^2-\frac6{x^4})dx=0,$ etc, and in general, $\int_0^\infty (x^k-(k+1)!x^{-(k+2)})dx=0,$ ...
- 10.4k
6 votes
2 answers
1k views
On modified Euler product
Consider the modified Euler product as follows: $$F(s) = \prod_{p} \left( 1 - \frac{c}{p^s} \right)^{-\ln(p)}$$ Here $c$ is a constant My questions are Is there a compact representation for this ...
- 149
20 votes
4 answers
6k views
Is the pseudoinverse the same as least squares with regularization?
Given a linear system $Ax=b$, the pseudoinverse of $A$ is found as the matrix $A^+$ such that $x=A^+ b$ where $x$ solves the least squares problem $\min \| Ax - b \|^2 $ and $x \perp \mathcal{N}(A)$. ...
- 521
2 votes
1 answer
314 views
The zeta regularization of $\prod_{m=-\infty}^\infty (km+u)$
Background: I'm facing the computation of the zeta regularization of the infinite product given by $$\prod_{m=-\infty}^\infty (km+u)$$ for a real positive $k$ and $\Im(u)\neq 0$. From J. R. Quine, S. ...
- 582
2 votes
0 answers
124 views
Sparse signal recovery (nonlinear case)
Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
- 1,051
5 votes
0 answers
327 views
More or less universal formula for regularization of divergent integrals?
Is there a simple formula that would produce the regularized value for the most common divergent integrals? I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-...
- 10.4k
3 votes
2 answers
482 views
Theta-function in the lower half-plane
Standard theta function $$\vartheta(q)=\sum_{n=-\infty}^\infty q^{n^2} \qquad\qquad(1)$$ has a natural boundary of analyticity at $|q|=1$. This means that it can not be used to regularize expressions ...
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7 votes
3 answers
820 views
Is regularization of infinite sums by analytic continuation unique?
There are ill-posed summations that we can assign values to, take for concreteness, $$ S = \sum_{k=0}^\infty k $$ to which we can assign $-1/12$ by several methods. Is there a fundamental and rigorous ...
- 1,324
6 votes
2 answers
609 views
Less fundamental applications of Zeta regularization:
As we all know, zeta regularization is used in Quantum field theory and calculations regarding the Casimir effect. Are there less fundamental applications of zeta function regularization? By "less ...
- 375
-1 votes
1 answer
271 views
Are the shapes of the $\mathbb{R}^2$ plane and a disk of infinite radius different? Or otherwise, why their areas differ by $\frac\pi{12}$? [closed]
The calculation of the area of the $\mathbb{R}^2$ plane depends on filtering used. I think, the most natural filtering is along the radius in polar coordinates: $$S_{\mathbb{R}^2}=\int_0^\infty 2\pi ...
- 10.4k
0 votes
1 answer
243 views
Generalised limits via derivatives of integrals?
Assuming that $f$ is a continuous function, we have that $$f(x) = \frac{d}{dx}\int f(t)\,dt.$$ Assuming instead that $f$ has a removable singularity at $x=a$, and is otherwise continuous, we have ...
- 4,993
7 votes
2 answers
1k views
Regularizing the sum of all primes
In the spirit of a similar question for the harmonic series, is there a way to regularize the (divergent) sum of all primes? $$ \sum_{p \text{ prime}} p $$ Neither of these questions obtained a ...
- 2,440
4 votes
2 answers
1k views
Is there a sensible way to regularize $\int_0^\infty \tan x\, dx$?
I wonder if there is any sensible generalization of regularization which would be able to ascribe finite values to $\int_0^\infty \tan x \,dx$ and $\int_{-\infty}^0 \psi(x)dx$? Perticularly, since $\...
- 10.4k
4 votes
1 answer
291 views
Interesting questions for inverse parabolic problems
I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can ...
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4 votes
1 answer
181 views
Proximal Operator image of convex functionals
Let $\Gamma_0$ denote the set of lower-semi-continuous convex functionals on a Hilbert space $H$. What exactly is the image of $\Gamma_0$ under the proximal operator $$ \begin{aligned} &\Gamma_0\...
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0 votes
1 answer
421 views
How to derive the solution of Tikhonov Regularization via SVD [closed]
The solution to Tikhonov Regularization is $$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$ where $\sigma^2_{min}$ is the minimum of the singular values of $A$. Then we apply $SVD$ to $A$ such that, $$A=U\Sigma ...
- 5
3 votes
0 answers
378 views
New/useful method for summation of divergent series?
Questions $$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$ Also obeys (see background for argument): $$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
- 237
1 vote
1 answer
255 views
Why we cannot speak about the main or natural regularization?
Often when asking about a regularized value of an integral or series, I encounter a negative reaction of the sorts that "regularization is what you define it". But in practice if we consider some ...
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